Research ArticleAnalysis of Iterative Erasure Insertion and Decoding ofFHMFSK Systems without Channel State Information

Jinsoo Park 1 Gangsan Kim1 Hong-Yeop Song 1 Chanki Kim 2

Jong-Seon No2 and Suil Kim3

1Department of Electrical and Electronic Engineering Yonsei University Seoul 120-749 Republic of Korea2Department of Electrical and Computer Engineering Seoul National University Seoul 151-742 Republic of Korea3Agency for Defense Development Daejeon 305-600 Republic of Korea

Correspondence should be addressed to Hong-Yeop Song hysongyonseiackr

Received 4 September 2017 Accepted 27 December 2017 Published 24 January 2018

Academic Editor Kiseon Kim

Copyright copy 2018 Jinsoo Park et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We analyze the symbol measures for iterative erasure insertion and decoding of a Reed-Solomon coded SFHMFSK system overjamming channels In contrast to conventional erasure insertion schemes iterative schemes do not require any preoptimizedthreshold or channel state information at the receiver We confirm the performance improvement using a generalized minimumdistance (GMD) decoding method with three different symbol measures To analyze performance we propose a new analysisframework considering the ldquotrapped-errorrdquo probability From analysis and the simulation results we show that ratio-based GMDdecoding has the best performance among the one-dimensional iterative erasure insertion and decoding schemes

1 Introduction

To mitigate the threat of jamming attacks Reed-Solomon(RS) coded frequency-hopping spread-spectrum systemswith frequency shift keying (FSK) have beenwidely applied tomilitary communication systems [1] Although these systemscan reduce the effect of jamming attacks some signals arecorruptedwhen the jamming signal collides with the hoppingfrequency To recover information from the contaminatedsignals various channel codes have been applied with aninterleaver [1] In many previous studies [2ndash11] erasureinsertion schemes with an error and erasure decoder havebeen used to increase system reliability under this kind ofjamming The main aim of erasure insertion is to erase thecorrupted received symbols (or dwells) using the thresholdtest In most cases the receiver should select an optimalthreshold based on the system settings and the channelstate at the time If the receiver knows perfect channel stateinformation the performance can be improved using soft-decision-based soft-decoding algorithms [12ndash18] However itis very difficult to estimate the channel state information atthe receiver (CSIR) when the channel is jammed [7]

In this paper we consider an erasure insertion schemethat does not require any CSIR Using appropriate symbolmeasures we can improve the performance using severalwell-known iterative decoding algorithms [19ndash22] Thesealgorithms require the order of reliability of the receivedsymbols not exact soft measure of the symbols Specially weconsider ldquooutputrdquo ldquoratiordquo and ldquosumrdquo measures using gen-eralized minimum distance (GMD) decoding Some otherdecoding algorithms such as [23] can be applied after theiterative erasure insertion but it is out of our scope Thisiterative erasure insertion and decoding scheme does notguarantee independent processing of the received symbolstherefore we propose a new analysis framework in termsof trapped-error probability Finally we confirm that theratio measure works best and its concatenated scheme withanother measure improves the performance when the jam-ming signal is strong In [8] the authors proposed anotheriterative erasure insertion and decoding scheme but theyconsidered a partial-band jamming channel and a standardbounded-distance decoding In this paper we cover thepartial-band and partial-time jamming and we exploit GMDdecoding not a bounded-distance decoding

HindawiSecurity and Communication NetworksVolume 2018 Article ID 7318043 12 pageshttpsdoiorg10115520187318043

2 Security and Communication Networks

Frequencysynthesizer

Frequencysynthesizer

Frequency hopping sequence

Frequency hopping sequence

MFSKmodulatorInterleaver

Deinterleaver Non-cohMFSK demod

Encoder

Decoder Erasureinsertion

c c

wi xi(t)

ri(t)

nJ(t)

n(t)

w = (w0 w1 wNminus1)

Figure 1 System block diagram with erasure insertion component

This paper is organized as follows In Section 2 wedescribe the system model In Section 3 we review severalconventional erasure insertion schemes and describe the iter-ative erasure insertion and decoding schemes In Section 4the trapped-error probability of the measures is derivedIn Section 5 we compare the trapped-error probabilitiesand confirm the performances using computer simulationsFinally concluding remarks are given in Section 6

2 System Model

In this paper we consider the RS coded slow frequency-hopping 119872-ary FSK system over a partial-band (or partial-time) jamming channel At the transmitter 119870 informationsymbols are encoded by rate119870119873 RS encoder where119873 is thenumber of codeword symbols It is well known that (119873119870)RS codes can recover the received word if 119906 + 2V lt 119873minus119870+ 1and bounded-distance decoding is applied where 119906 and Vare the number of erasures and errors in the received wordrespectively The encoded codeword is represented by

w = (1199080 1199081 119908119873minus1) (1)

where 119908119894 119894 isin 0 119873 minus 1 is the 119894th codeword symbolWe assume that an119872-ary FSK modulated symbol representsthe single corresponding RS codeword symbol over 119866119865(119872)where 119872 = 2119887 and 119887 is a positive integer Furthermore weassume119873 = 119872 minus 1 because the conventional RS codes over119866119865(119872) have length119873 = 119872 minus 1

The system model is shown in Figure 1 The jammingsignal is inserted into the frequency domain with fraction(jamming duty ratio) 120588 which is the ratio of the jammingsignal bandwidth to the spread-spectrum bandwidth Weassume that the partial-band jamming signal has a Gaussiandistribution with zero-mean and1198731198692120588 = 1205902119869 power spectraldensity in the jamming bandwidth In the figure 120572 representsthe channel coefficients of the transmitter-receiver channelIf the transmitter-receiver channel is modeled as an additivewhite Gaussian noise (AWGN) channel then the channelcoefficient is set to 120572 = 1 The noise signal is modeled by aGaussian distribution with zero-mean and11987302 = 1205902119873 powerspectral density over the entire frequency domain

Throughout this paper we assume an ideal interleaverFrom this assumption every RS codeword symbol is trans-mitted over a different frequency hop and these symbolsare independently interfered with the jamming signal withprobability 120588 This assumption (and our result) covers everycase such that the received symbol is independently interferedwith probability 120588 such as partial-band and partial-timejamming Based on this assumption we will discuss thesymbols that are independently jammed with probability 120588

Let

119909119894 (119905) = radic2119864119904119879 cos (2120587 (119891ℎ + 119891119895) 119905) (2)

be a general form of the transmitted signal where 119864119904 119879and 119891ℎ are symbol energy symbol duration and a selectedhopping frequency respectively The orthogonal frequency119891119895 = 119895Δ119891 for 119895 isin 0 119872 minus 1 is the selected frequencyassociated with 119908119894 where Δ119891 = 1119879 If the transmittedsymbol is contaminated by a jamming signal the 119894th receivedsignal 119903119894(119905) is expressed as

119903119894 (119905) = 120572radic2119864119904119879 cos (2120587 (119891ℎ + 119891119895) 119905 + 120579) + 119899119869 (119905)+ 119899 (119905)

(3)

where 119899119869(119905) and 119899(119905) are the jamming signal and Gaussiannoise respectively A uniform random phase is denoted by 120579For the unjammed case 119899119869(119905) = 0 Without loss of generalitywe assume that 119864119904 = 1

The non-coherent MFSK demodulator calculates 119910119894119895 forthe 119894th symbol of the codeword and 119895thMFSK tone as follows[24]

119910119894119895 = (int1198790119903119894 (119905) cos (2120587 (119891ℎ + 119891119895) 119905) 119889119905)2

+ (int1198790119903119894 (119905) sin (2120587 (119891ℎ + 119891119895) 119905) 119889119905)2

(4)

The symbol index 119894 can be omitted if it is obvious based onthe context Thus 119910119895 denotes the value 119910119894119895 in this case Afterdeinterleaving all of the values of 119910119894119895 for 0 le 119894 le 119873 minus 1and 0 le 119895 le 119872 minus 1 are input to ldquoerasure insertionrdquo Thisblock erases several symbols with a selected rule which willbe described in Section 3

Security and Communication Networks 3

3 Erasure Insertion Schemes

31 Overview of Erasure Insertion Schemes

311 Conventional Schemes Next we briefly introduce thethree conventional erasure insertion schemes the ratiothreshold test (RTT) the output threshold test (OTT) andthe maximum output-ratio threshold test (MO-RTT) [4ndash7 25] Let maxsdot and max1015840sdot represent the first and secondmaximum values of a given set respectively For each 119894 from0 to 119873 minus 1 the 119894th received symbol is erased by RTT ifmax1015840119895119910119895max119895119910119895 gt 120582 where 120582 is a given threshold OTTerases the symbol if max119895119910119895 gt 120591 for a given threshold120591 Sometimes the OTT rule is applied as max119895119910119895 lt 120591[6] however this reversed inequality is not considered inthis paper We note that the conventional RTT and OTTare one-dimensional erasure insertion schemes because thedecision is made using a single measure On the other handMO-RTT is a two-dimensional erasure insertion schemecombining OTT and RTT MO-RTT generally exhibits betterperformance thanRTTorOTTwith their optimal thresholdsBefore we discuss the problems of conventional schemes weintroduce a new threshold test scheme that is sum-basedthreshold test (STT)This test erases the symbol ifsum119872minus1119895=0 119910119895 gt120583 where 120583 is a given threshold

Threshold test-based conventional schemes have used theone-time erasure insertion and the bounded-distance decod-ing with preoptimized thresholds The optimum thresholdsdepend on the channel state therefore the receiver shouldknow the channel state information at the time Because of thedifficulty of obtaining perfect CSIR over jamming channelswe consider the iterative erasure insertion scheme whichdoes not require any CSIR

312 Iterative Erasure Insertion and Decoding Schemes Toincrease the error correction capability of RS codes manyiterative decoding algorithms have been proposed [19ndash23]For an unjammed channel we can estimate the channelstate information using various techniques Thereafter wecan apply various decoding algorithms However if thejamming signal exists and CSIR is not available we mustaddress the jamming attack without CSIR To use the iter-ative decoding algorithms without CSIR we will considera simplified version of the GMD decoding algorithm [26]as an example The main idea of GMD decoding is iterativedecoding trials with erasures During each iteration GMDdecoding erases the least reliable symbol and attempts anerror and erasure decoding In this paper we use the symbolmeasures ratio (max1015840119895119910119895max119895119910119895) output (max119895119910119895) andsum (sum119872minus1119895=0 119910119895) for the GMD decoding algorithm Theseschemes will be called ratio-based GMD (R-GMD) output-based GMD (O-GMD) and sum-based GMD (S-GMD)decoding respectively Note that the three measures arecalculated without CSIR

In Algorithm 1 the one-dimensional GMD algorithm isdescribed using the three measures Here 119894 is the index of thecodeword symbol and 119895 is the index of 119872 detectors of theMFSK demodulator Initially the receiver decides that 119888119894 =

argmax119895119910119894119895 for each 119894 from 0 to119873 minus 1 In the beginning ofwhile loop the decoder attempts to recover the transmit wordw in (1) from c with error-only decoding This step preventsperformance degradation when there is no jamming signalAdditionally we will not consider the undetected errors ofthe decoded codeword because they seldom occur [27] Ifthe decoder declares a decoding failure it erases the mostsuspicious (likely to have been jammed) symbol 119888119894max

andreplaces it with an erasure 120598 where 119894max is determined by theselected decision rule For each measure 119894max is determinedas follows

Ratio measure

119894max = argmax119894notin119864

max1015840119895 119910119894119895max119895 119910119894119895 (5)

Output measure

119894max = argmax119894notin119864

119910119894119895 (6)

Sum measure

119894max = argmax119894notin119864

119872minus1sum119895=0

119910119894119895 (7)

After 119894max is determined it is added to 119864 which is thecurrent set of erased symbol indices Then error and erasuredecoding is applied As we noted in Section 2 (119873119870) RScodes can correct 119906 erasures and V errors if 119906 + 2V lt119873 minus 119870 + 1 is satisfied If the decoding is successful thenthe erasure insertion loop is terminated and the decodedcodeword becomes the output c Otherwise the algorithmcontinues to the second iteration of the while loop In thesecond iteration and thereafter all of the previously erasedsymbol indices (119864) are maintained and 119894max is determinedover all the unerased indices This iteration runs successivelyuntil the decoding succeeds or the number of erased symbolsis the same as the number of parity symbols that is |119864| =119873 minus 119870 The while loop in Algorithm 1 is described by thefeedback loop in Figure 1 We note that this iterative erasureinsertion and GMD decoding algorithm runs independentlyfor each codeword and does not make comparisons withany preoptimized threshold Furthermore if the conventionalerasure insertion scheme can recover the codeword then thecorresponding iterative scheme can recover the codeword

32 Two-Dimensional Schemes We introduced theMO-RTTat the beginning of Section 3 which has two-dimensionalmeasures the joint threshold tests of OTT and RTT Toexploit the diversity of multiple measures various combina-tions of iterative schemes with some different measures areconsidered In this paper however wewill focus only on serialconcatenation of R-GMD decoding and S-GMD decodingcalled ratio-to-sum-based GMD (RS-GMD) decoding FirstRS-GMD decoder runs R-GMD decoding If R-GMD decod-ing fails to decode with |119864| = 119873 minus 119870 + 1 then the receiver

4 Security and Communication Networks

InitializeDecide a vector c by 119888119894 = argmax119895119910119894119895 0 le 119894 le 119873 minus 1Set 119864 = 0

while |119864| le 119873 minus 119870 doTry to error-and-erasure decode cif decoding success then

c is the decoded codewordBreak while loop

else if decoding fail thenCase Measureratio 119894max = argmax119894notin119864max1015840119895119910119894119895max119895119910119894119895output 119894max = argmax119894notin119864119910119894119895sum 119894max = argmax119894notin119864sum119872minus1119895=0 119910119894119895

Erase 119894maxth codeword symbols ie set 119888119894max= 120598119864 larr 119864 cup 119894max

end ifend whileif |119864| le 119873 minus 119870 then

Decoding successOutput = c

else (|119864| = 119873 minus 119870 + 1)Decoding renounce

end if

Algorithm 1 Iterative erasure insertion and GMD decoding

Table 1 Thresholds 119911119901 of the three measures where119872 = 4 119901 = 01 120588 = 01 and 1198641198871198730 = 5 dB over AWGN and partial-band jammingchannels

119864119887119873119895 (dB) minus10 minus5 0 5 10 15 20119911119901 of R 0722 0722 0721 0719 0712 0695 0671119911119901 of O 4123 3625 3158 2462 2103 1993 1961119911119901 of S 4824 4377 4438 3400 2869 2623 2536

runs S-GMD decoding again from the beginning We willshow the performance improvement of RS-GMD decodingas compared to the performance of MO-RTT over variouschannel situations in Section 5 In Table 2 the conventionaliterative and two-dimensional schemes are classified into thefour types of measures

4 Trapped-Error Probability

For the conventional erasure insertion schemes that is RTTOTT and MO-RTT the error and erasure probabilities arederived in [4ndash6] And we have included the error and erasureprobabilities of STT in the Appendix Calculation of theerror and erasure probabilities of a given symbol cannotbe applied to the analysis of iterative schemes because theerasure insertion for one symbol is not independent of thatfor all other symbols For performance analysis of the iterativeschemes we propose trapped-error probability analysis basedon threemeasures ldquoratiordquo ldquooutputrdquo and ldquosumrdquoThe trapped-error probability is a new and useful analysis framework tocompare the quality of each measure

Consider a received word c of length 119873 Let 119885119894 be themeasure of one symbol of c for one of the threemeasures119885119894 =

max1015840119895119910119894119895max119895119910119894119895 119885119894 = max119895119910119894119895 and 119885119894 = sum119872minus1119895=0 119910119894119895for ratio output and sum measures respectively First wesort the received 119873 symbols in descending order accordingto 119885119894 values that are calculated by one selected way (amongthe three kinds of measures) Let 119901 denote the ratio of thewindow of interest to the length 119873 This ratio 119901 should bechosen in the range of 0 lt 119901 le (119873 minus 119870)119873 because the(119873119870) RS code can correct up to 119873 minus 119870 erasures Now wecount the number119873119901 of erroneous symbols in the uppermostportion of 119901119873 symbols Let 119873119905 denote the total number oferroneous symbols in the received word c of length119873 Thenthe trapped-error probability Γ is defined as the ratio between119873119901 and119873119905

Γ = 119873119901119873119905 (8)

For example let us assume that there are 119873 = 100 receivedsymbols with a total of 119873119905 = 15 erroneous symbols Let usassume that 119901 = 01 And we consider the first 10 (=pN)positions when all of the symbols are arranged in decreasingorder of their corresponding measures Let us assume that119873119901 = 8 1 and 5 erroneous symbols are placed among these10 positions for ratio output and summeasures respectively

Security and Communication Networks 5

Then the trapped-error probabilities are ΓR = 815 ΓO =115 and ΓS = 515 for these respective measures Inthis case because the iterative schemes successively erasethe symbols in decreasing order of each measure the ratiomeasure works best for trapping (or identifying) the erro-neous symbols and the output measure works worst For agiven 119901 a larger trapped-error probability indicates betterperformance

Without loss of generality we assume that all-zero code-word w = (0 0) is transmitted that is 1198910 is selected forevery symbol transmission If a transmitted symbol does notinterfere with the jamming signal and the AWGN channel isassumed then the probability density functions (PDFs) of 1199100and 119910119895 119895 isin 1 119872 minus 1 are derived as follows [24]

1198921198800 (119910) = 121205902119873 exp(minus119910 + 121205902119873 ) 1198680 (radic1199101205902119873 )

119892119880119895 (119910) = 121205902119873 exp(minus 11991021205902119873) (9)

For the jammed case

1198921198690 (119910) = 121205902 exp(minus119910 + 121205902 ) 1198680 (radic1199101205902 ) 119892119869119895 (119910) = 121205902 exp(minus 11991021205902 )

(10)

where 1205902 = 1205902119869 + 1205902119873 and 1198680(sdot) is the modified Bessel functionof order zero

If we assume Rayleigh fading channel withoutwithpartial-band jamming 1198921198800 and 1198921198690 are changed as follows[28 29]

1198921198800 (119910) = 121205902119873 + 1 exp(minus11991021205902119873 + 1)

1198921198690 (119910) = 121205902 + 1 exp (minus 11991021205902 + 1) (11)

Because 1198921198800(119910) 119892119880119872minus1(119910) and 1198921198690(119910) 119892119869119872minus1(119910) areindependent variables their joint PDFs can be represented asprod119872minus1119895=0 119892119880119895(119910119895) andprod119872minus1119895=0 119892119869119895(119910119895) respectively Their cumula-tive distribution functions (CDFs) are 119866119880119895(119910119895) and 119866119869119895(119910119895)for 119895 isin 0 119872 minus 1 respectively

Recall that 119885119894 is the measure of the symbol 119888119894 for one ofthe three measures For simplicity we will omit the subscript119894 119885 = max1015840119895119910119895max119895119910119895 119885 = max119895119910119895 or 119885 = sum119872minus1119895=0 119910119895for ratio output or sum measures respectively The CDF issimply denoted by 119865119885(119911) Define 1198670 as a representation ofthe event of a raw symbol error that is the event where thereexists at least one 119895 isin 1 119872minus1 that satisfies 1199100 lt 1199101198951198670rsquoscomplementary event is denoted by1198670 Now we consider the

conditional CDFs 119865119885|119867(119911 | 1198670) and 119865119885|119867(119911 | 1198670) where119867 isthe discrete event variable119867 isin 1198670 1198670 Then

119865119885 (119911) = Pr [1198670] 119865119885|119867 (119911 | 1198670)+ Pr [1198670] 119865119885|119867 (119911 | 1198670)

119891119885 (119911) = Pr [1198670] 119891119885|119867 (119911 | 1198670)+ Pr [1198670] 119891119885|119867 (119911 | 1198670)

(12)

where 119891s denote the corresponding PDFsDefine 119911119901 as the point that satisfies 119865119885(119911119901) = 1 minus 119901119911119901 is uniquely determined because 119865119885(119911) is monotonically

increasing from 0 to 1 Then the trapped-error probabilityΓ(119911119901) can be defined as follows

Γ (119911119901) = intinfin119911119901

119891119885|119867 (119911 | 1198670) 119889119911 = 1 minus 119865119885|119867 (119911119901 | 1198670) (13)

Now we will change (13) into a form involving1198670 insteadof 1198670 such that its computation can be easily performedFrom (12) we observe that

119865119885 (119911119901) = Pr [1198670] 119865119885|119867 (119911119901 | 1198670)+ Pr [1198670] 119865119885|119867 (119911119901 | 1198670) = 1 minus 119901 (14)

Using (14) the trapped-error probability in (13) can bewrittenas

Γ (119911119901) = 1 minus 1 minus 119901 minus Pr [1198670] 119865119885|119867 (119911119901 | 1198670)Pr [1198670] (15)

Using the relation Pr[1198670] + Pr[1198670] = 1 and after somemanipulation we have the following

Γ (119911119901) = 119901 minus Pr [1198670] + Pr [1198670] 119865119885|119867 (119911119901 | 1198670)1 minus Pr [1198670] (16)

Here Pr[1198670] can be calculated as follows

Pr [1198670] = Pr [1199100 gt 119910119895 forall119895 = 1 119872 minus 1]= 120588intinfin0int11991000sdot sdot sdot int11991000

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100+ 120588intinfin0int11991000sdot sdot sdot int11991000

119872minus1prod119895=0

119892119869119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100= 120588intinfin01198921198800 (1199100) (1198661198801 (1199100))119872minus1 1198891199100

+ 120588intinfin01198921198690 (1199100) (1198661198691 (1199100))119872minus1 1198891199100

(17)

where 120588 = 1 minus 120588

6 Security and Communication Networks

Let us assume AWGN and a partial-band jammingchannel Using a binomial expansion and some calculations[30] (17) becomes

120588119890(minus121205902119873) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902119873)119897 (119898 + 1)119897+1 119897 (

119872 minus 1119898 )

+ 120588119890(minus121205902) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902)119897 (119898 + 1)119897+1 119897 (

119872 minus 1119898 )

(18)

Similarly for a Rayleigh fading and partial-band jammingchannel (17) becomes

120588119872minus1sum119898=0

21205902119873(2 + 2119898) 1205902119873 + 119898 (minus1)119898 (119872 minus 1119898 )

+ 120588119872minus1sum119898=0

21205902(2 + 2119898) 1205902 + 119898 (minus1)119898 (119872 minus 1119898 )

(19)

Next we compute 119865119885|119867(119911119901 | 1198670) as follows

119865119885|119867 (119911119901 | 1198670)= Pr [119885 lt 119911119901 1199100 gt 1199101198951015840 forall1198951015840 = 1 119872 minus 1]

Pr [1198670] (20)

Now assume that we use the ratio measure Then thenumerator of (20) becomes

Pr[max1015840119895 119910119895max119895 119910119895 lt 119911119901 1199100 gt 1199101198951015840 forall1198951015840 = 1 119872 minus 1]

= Pr [max1015840119895 119910119895 lt 1199111199011199100]= Pr [119910119895 lt 1199111199011199100 forall119895 = 1 119872 minus 1]= 120588intinfin0int11991111990111991000

sdot sdot sdot int11991111990111991000

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100+ 120588intinfin0int11991111990111991000

sdot sdot sdot int11991111990111991000

119872minus1prod119895=0

119892119869119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100= 120588intinfin01198921198800 (1199100) (1198661198801 (1199111199011199100))(119872minus1) 1198891199100

+ 120588intinfin01198921198690 (1199100) (1198661198691 (1199111199011199100))(119872minus1) 1198891199100

(21)

For AWGN and Rayleigh fading channels with partial-bandjamming (21) becomes (22) and (23) respectively

120588119890(minus121205902119873) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902119873)119897 (119898119911119901 + 1)119897+1 119897 (

119872 minus 1119898 )

+ 120588119890(minus121205902) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902)119897 (119898119911119901 + 1)119897+1 119897 (

119872 minus 1119898 )

(22)

120588119872minus1sum119898=0

21205902119873(2 + 2119898119911119901) 1205902119873 + 119898119911119901 (minus1)119898 (119872 minus 1

119898 )

+ 120588119872minus1sum119898=0

21205902(2 + 2119898119911119901) 1205902 + 119898119911119901 (minus1)119898 (119872 minus 1

119898 ) (23)

For the output measure the numerator of (20) becomes

Pr [max119895 119910119895 lt 119911119901 1199100 gt 1199101198951015840 forall1198951015840 = 1 119872 minus 1]= Pr [1199100 lt 119911119901 1199100 gt 119910119895 forall119895 = 1 119872 minus 1]= 120588int1199111199010int11991000sdot sdot sdot int11991000

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100+ 120588int1199111199010int11991000sdot sdot sdot int11991000

119872minus1prod119895=0

119892119869119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100= 120588int11991111990101198921198800 (1199100) (1198661198801 (1199100))(119872minus1) 1198891199100

+ 120588int11991111990101198921198690 (1199100) (1198661198691 (1199100))(119872minus1) 1198891199100

(24)

With partial-band jamming (24) becomes (25) and (26)for AWGN and Rayleigh fading channels respectively

120588119890(minus121205902119873) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902119873)2119897+1 119897 (

119872 minus 1119898 )

sdot (( 21205902119873119898 + 1)119897+1

minus 119890(minus(119898+1)11991111990121205902119873) 119897sum119899=0

119911119899119901119899 ( 21205902119873119898 + 1)119897+1minus119899)

+ 120588119890(minus121205902) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902)2119897+1 119897 (

119872 minus 1119898 )

sdot (( 21205902119898 + 1)119897+1

minus 119890(minus(119898+1)11991111990121205902) 119897sum119899=0

119911119899119901119899 ( 21205902119898 + 1)119897+1minus119899)

(25)

Security and Communication Networks 7

120588119872minus1sum119898=0

21205902119873(2 + 2119898) 1205902119873 + 119898 (minus1)119898 (119872 minus 1119898 ) sdot (1

minus 119890(minus(((2+2119898)1205902119873+119898)(41205904119873+21205902119873))119911119901))+ 120588119872minus1sum119898=0

21205902(2 + 2119898) 1205902 + 119898 sdot (minus1)119898 (119872 minus 1119898 )(1

minus 119890(minus(((2+2119898)1205902+119898)(41205904+21205902))119911119901))

(26)

For the sum measure the numerator of (20) is given as (27)Actually (27) is the same as (A4) in the Appendix where 120583 =119911119901 Using the derived equations we can calculate the trapped-error probabilities of three measures

Pr[[119872minus1sum119895=0

119910119894119895 lt 119911119901 1199100 gt 119910119895 forall119895 = 1 119872 minus 1]] = Pr[[1199100 lt 119911119901

1199101 lt min (1199100 119911119901 minus 1199100) 119910119872minus1 lt min(1199100 119911119901 minus 119872minus2sum119895=0

119910119895)]]= 120588int1199111199010intmin(1199100 119911119901minus1199100)

0sdot sdot sdot intmin(1199100 119911119901minussum

119872minus2119895=0 119910119895)

0

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1sdot sdot sdot 1198891199100

(27)

+ 120588int1199111199010intmin(1199100 119911119901minus1199100)

0sdot sdot sdot intmin(1199100 119911119901minussum

119872minus2119895=0 119910119895)

0

119872minus1prod119895=0

119892119869119895 (119910119895) 119889119910119872minus1sdot sdot sdot 1198891199100

(28)

5 Simulation Results

51 Trapped-Error Probabilities In this subsection we con-firm the trapped-error probability analysis over AWGN andRayleigh fading channels with partial-band jamming Letus assume 4-FSK 120588 = 01 and 1198641198871198730 = 5 dB for theAWGN channel and 1198641198871198730 = 12 dB for the Rayleigh fadingchannel For the higher order of modulation such as 32-FSK which we will consider (27) requires 32-dimensionalintegration therefore we select 4-FSK as an example Tocalculate the probabilities Γ(119911119901) for each measure the input119911119901 should be obtained for the given set of system parametersFor this we further assume that 119901 = 01 and 02 for AWGNand Rayleigh fading channels respectively As we describedin Section 4 we found 119911119901 values that satisfy 119865119911(119911119901) =1 minus 119901 using simulations They are listed in Tables 1 and 3for various 119864119887119873119895 R O and S refer to ratio output andsum respectively In the investigation we transmitted 105symbols for various channel states and calculated their ratiooutput and sum measures We note that these trapped-errorprobabilities do not depend on a specific coding schemebecause these statistics are observed at the demodulatoroutput without coding

Figures 2 and 3 show the trapped-error probabilities ofthe three measures obtained using simulations and from (16)with the threshold values given in Tables 1 and 3 Solid anddashed lines indicate the probabilities obtained by simulation

Table 2 Class of erasure insertion schemes

Measure Threshold test IterativeRatio RTT R-GMDOutput OTT O-GMDSum STT S-GMDTwo-dimensional Ex MO-RTT Ex RS-GMD

Table 3Thresholds 119911119901 of the three measures where119872 = 4 119901 = 02120588 = 01 and 1198641198871198730 = 12 dB over Rayleigh fading and partial-bandjamming channels

119864119887119873119895 (dB) minus5 0 5 10 15 20119911119901 of R 0353 0345 0350 0329 0294 0263119911119901 of O 2271 2233 2053 1765 1681 1660119911119901 of S 2368 2342 2297 2069 1845 1776

and (16) respectively The cross circle and square marksrepresent Γ(119911119901) values of ratio output and sum measuresrespectively

These two figures confirm that the simulation is exactenough to closely follow the theoretical result in (16) Spe-cially ΓO in the analysis is almost the same as ΓO in thesimulation results This result implies the correctness of ourperformance simulation

In these figures the trapped-error probability of theratio measure is increased as SJR is increased therefore wepredict that the R-GMD decoding will perform better thanthe other one-dimensional schemes at high SJR Additionallywe predict that S-GMD decoding is better than O-GMDdecoding in the middle SJR region Recall that the higherthe trapped-error probability the better the performanceThis observation shows the influence on the two-dimensionalschemes because the trapped-error probability affects thenumber of decoding trials We will discuss it in the followingsimulation results

52 Performance over AWGN Channel Throughout perfor-mance simulations we consider 32-ary FSKmodulation with(31 20) RS code over a field of size 32 an ideal interleaverand a random hopping pattern We first consider partial-band jamming (or partial-time jamming with the same 120588)and AWGN channels In this case 120572 = 1 in Figure 1 ForFigures 4ndash7 the channel noise is fixed as we used 1198641198871198730 =5 dBVarious signals to jamming signal ratio119864119887119873119895 have beenconsidered for the simulations

In Figure 4 the performance of various erasure insertionschemes is plotted over the partial-band jamming channelwith 120588 = 01The dashed line shows the baseline performanceldquoWithout EIrdquo which does not use any erasure insertionschemeThe unmarked solid line represents the performanceof the MO-RTT which is the conventional erasure insertionschemeWenote that theMO-RTTuses preoptimized thresh-olds and CSIR To obtain the optimum thresholds for MO-RTT we optimized the thresholds for 119864119887119873119895 = 0sim30 dB with10 dB steps Solid lines with circle square cross plus anddiamond marks represent the performance of output-based

8 Security and Communication Networks

100 5 15 20minus5minus10

ΓR by simulationΓO by simulationΓS by simulation

ΓR by analysisΓO by analysisΓS by analysis

0

01

02

03

04

05Γ

06

07

08

09

1

EbNj (dB)

Figure 2 Trapped-error probabilities of ratio-based (ΓR) output-based (ΓO) and sum-based (ΓS) measures over partial-band jam-ming and AWGN channel

01

02

03

04

05

06

07

08

09

1

Γ

minus5 0 5 10 15 200

ΓR by simulationΓO by simulationΓS by simulation

ΓR by analysisΓO by analysisΓS by analysis

EbNj (dB)

Figure 3 Trapped-error probabilities of ratio-based (ΓR) output-based (ΓO) and sum-based (ΓS) measures over partial-band jam-ming and Rayleigh channels

sum-based ratio-based ratio-to-output-based and ratio-to-sum-based GMD decoding respectively In this figure wealso present the performance using log-likelihood ratio (LLR)based GMD decoding as the triangle-marked line This isthe performance of a system with perfect CSIR such asSNR SJR and the existence of a jamming signal This LLR-GMD decoding iteratively erases the least reliable symbolsbased on LLR as described in [19 22 26] Because it is wellknown that MO-RTT exhibits better performance than OTT

EbNj (dB)

10minus5

10minus4

10minus3

10minus2

10minus1

100

WER

5 10 15 20 250

Without EIMO-RTTO-GMDS-GMD

R-GMDRO-GMDRS-GMDLLR-GMD

Figure 4 Performance of various erasure insertion schemes overpartial-band jamming and AWGN channels with 120588 = 01 and1198641198871198730 = 5 dB

or RTT [5ndash7] the performances ofOTTandRTTare omittedBefore discussing the simulation result we note that RSO-GMD decoding which serially exploits the three measureshas almost the same performance as RS-GMD decodingtherefore we have also omitted it From Figure 4 we makethe following observations

(1) As we expect LLR-GMD decoding with CSIR showsthe best performance We will focus on the perfor-mance gap between LLR-GMD decoding with CSIRand the other erasure insertion schemes withoutCSIR To achieve WER = 10minus4 LLR-GMD decodingrequires 119864119887119873119895 = 19 dB The ratio-based GMDdecoding and its two-dimensional schemes achievethe target WER at 119864119887119873119895 = 21 dB which is muchcloser than that of MO-RTT Even if the jammingpower is decreased as 119864119887119873119895 = 25 dB the perfor-mance of MO-RTT is still far from WER = 10minus4 Inthe low SJR region only the two two-dimensionaliterative schemes can achieve WER = 10minus2 withoutCSIR We note that the one-dimensional iterativeschemes and MO-RTT require 119864119887119873119895 gt 9 dB toachieve WER = 10minus2

(2) The performance of RS-GMD decoding approachesthat of LLR-GMD decoding as SJR is increased At119864119887119873119895 = 20 dB LLR-GMD decoding with perfectCSIR has 35 less WER than RS-GMD decodingWe note that this gap cannot be closed even if thejamming power is decreased

(3) When the jamming power is decreased that isthe channel approaches the unjammed channel theperformance of R-GMD and RS-GMD decoding is

Security and Communication Networks 9

0

01

02

03

04

05

06

07

08

09

1

Γ

5 10 15 20 25 300

ΓRΓOΓS

EbNj (dB)

Figure 5 Trapped-error probability of various measures overpartial-band jamming and AWGN channels with 119901 = 03 120588 = 01and 1198641198871198730 = 5 dB

still better than the performance of ldquoWithout EIrdquoor MO-RTT This implies that the iterative erasureinsertion and decoding schemes do not decrease theperformance if there are no jamming signals Becauseof this we do not need to detect the jamming signal

In Figure 5 trapped-error probabilities of ratio outputand sum measures are represented which corresponds toFigure 4 To obtain the trapped-error probabilities SJR andSNR were scaled based on the code rate 2031 = 119873119870 whichis different from Section 51 We used 119901 = 03 and the otherparameters are the same as those used in Figure 4 In Figure 5ΓR is much larger than ΓO and ΓS The values of ΓR explain thegood performance of R-GMDdecodingWe find that ΓO goesto lt1 minus119870119873 asymp 03 at 119864119887119873119895 asymp 10 dB Any iterative schemethat has zero trapped-error probability at a given SJR has thesame performance as the performance of ldquoWithout EIrdquo Thisresult explains why O-GMD decoding has a performancedegradation region near 10 dB in Figure 4 Because theperformance of O-GMD decoding rapidly approaches theperformance of ldquoWithout EIrdquo its performance is degraded asSJR is increased We note that the performance of S-GMDdecoding slowly approaches the performance of ldquoWithoutEIrdquo therefore there is no performance degradation region

The performance gain of the iterative schemes in Fig-ure 4 cannot be obtained without drawbacks the iterativeschemes require more decoding trials To investigate howmany decoding trials are required we determine the averagenumber of decoding iterations of various iterative schemesvia simulations The results are shown in Figure 6 In factwe simulated iterative schemes that erase one symbol duringthe first loop and then erase two symbols for each remainingloop because this process does not decrease the performanceas described in [19] In general the average number of

1

11

12

13

14

15

16

17

18

Aver

age n

umbe

r of d

ecod

ing

tria

ls

2 4 6 8 10 12 14 16 18 200

O-GMDS-GMDR-GMD

RS-GMDLLR-GMD

EbNj (dB)

Figure 6 Average number of decoding iterations for various erasureinsertion schemes over partial-band jamming and AWGN channelswith 120588 = 01 and 1198641198871198730 = 5 dB

MO-RTTRS-SEI

10minus3

10minus2

10minus1

WER

02 03 04 05 06 07 08 0901

Figure 7 Performance comparison of MO-RTT and RS-GMDdecoding scheme over partial-band jamming and AWGN channelswith various values of 120588 1198641198871198730 = 5 dB and 119864119887119873119895 = 10 dB

decoding trials decreases rapidly as SJR increases At highSJR most of the received words are completely decoded by asingle trial of error-only decoding which holds true becausehigher SJR implies that the environment becomes less andless hostile For low SJR the average decoding trial of RS-GMD decoding is approximately 175 (at 119864119887119873119895 = 0 dB)which decreases as SJR increasesThis result indicates that theiterative erasure insertion and decoding scheme is practicallyimplementable We note that the trial number of O-GMDdecoding is increased at approximately 119864119887119873119895 asymp 10 dB

10 Security and Communication Networks

and that is consistent with the performance degradation ofO-GMD decoding in Figures 4 and 5 Because O-GMDdecoding could not accurately erase the erroneous symbolsthe decoder has to try more decoding Because of this resultwemust use S-GMDdecoding instead of O-GMDdecodingfor the two-dimensional scheme with R-GMD decodingNow recall the analysis results of Section 51 In Figure 2we observed that the trapped-error probability of the ratiomeasure is increased as SJR is increased and that of the summeasure is decreased For the case in which the trapped-errorprobabilities cross at the point of SJR if we want to havefewer decodings at the higher SJR then we must use R-GMDdecoding first in RS-GMDdecoding If we want to have fewerdecodings at the lower SJR then S-GMD decoding should beapplied first In Figures 4 and 6 because we are consideringthe SJR region in which ΓR gt ΓS we have first exploited R-GMD decoding for RS-GMD decoding

In Figure 7 the performances of MO-RTT and RS-GMDdecoding are shown over a partial-band jamming and anAWGN channel with various jamming duty ratios 120588 withfixed 1198641198871198730 = 5 dB and 119864119887119873119895 = 10 dB For all values of120588 RS-GMD decoding works much better than MO-RTT Wenote that two erasure insertion schemes are more efficient forsmaller 120588 channels53 Performance over Rayleigh Fading Channel Figure 8shows the performance of various erasure insertion schemesover partial-band jamming and Rayleigh fading channel with120588 = 01 and 1198641198871198730 = 12 dB For each symbol transmission120572 in Figure 1 is realized by an iid Rayleigh distribution Weassume that fading coefficients are not known to the receiverexcept for the system of MO-RTT

In Figure 8 RS-GMD decoding exhibits better per-formance than MO-RTT which is the same result as wefound over the AWGN channel For the same WER of 10minus3the jammer should spend 14 dB more 119873119895 The WER ofratio-based schemes (including two-dimensional schemes)approach 10minus4 Unlike the results for an AWGN channelR-GMD decoding is dominant and the contribution of S-GMD decoding is marginal Therefore we conclude that R-GMD decoding is sufficient for Rayleigh fading channelsAdditionally we present the performance of LLR-GMDdecoding which knows perfect CSIR As we discussed inSection 52 the performance of R-SEI approaches that of LLR-SEI and the gap is maintained for every SJR

In Figure 9 the trapped-error probabilities of the mea-sures are displayed over a Rayleigh fading channel with1198641198871198730 = 12 dB The other parameters are the same as thosein Figure 5 In this figure the trapped-error probability of theratio measure is 1 in all SJR regions In other words R-GMDdecoding is the best one-dimensional iterative scheme for thetarget system parameters As we found for the other trapped-error probability results the trapped-error probabilities ofsum and output measures decrease as SJR increases Wherethe trapped-error probabilities are lower than lt1 minus119870119873 asymp03 that is 119864119887119873119895 = 10 dB and 16 dB for output andsum respectively the performances of O-GMD and S-GMDdecoding follow the performance of ldquoWithout EIrdquo As we

10minus5

10minus4

10minus3

10minus2

10minus1

100

WER

5 10 15 20 25 300

Without EIMO-RTTO-GMDS-GMD

R-GMDRO-GMDRS-GMDLLR-GMD

EbNj (dB)

Figure 8 Performance of various erasure insertion schemes over aRayleigh fading channel where 120588 = 01 and 1198641198871198730 = 12 dB

0

01

02

03

04

05

06

07

08

09

1

5 10 15 20 25 300

Γ

ΓRΓOΓS

EbNj (dB)

Figure 9 Trapped-error probability of various iterative schemesover partial-band jamming and Rayleigh fading channels with 119901 =03 120588 = 01 and 1198641198871198730 = 12 dB

discussed above the fast decreasing trapped-error probabilityof the output measure causes the performance degradation ofO-GMD decoding at approximately 10 dB in Figure 9

6 Concluding Remarks

In this paper we considered iterative erasure insertion anddecoding schemes that do not require any preoptimizedthresholds or any CSIR Additionally we proposed a new

Security and Communication Networks 11

analysis framework for the ratio output and sum mea-sures From the simulation results we confirmed that theratio-based GMD decoding scheme and its two-dimensionalschemes have the best performance Using the trapped-error probability the performances of the iterative erasureinsertion and decoding schemes are explained

Appendix

Error and Erasure Probability of SumThreshold Test

Let us denote the symbol error and erasure probabilities ofSTT as 119875err and 119875ers respectively The symbol error eventoccurs when a received symbol is erroneous but is not erasedby STT The symbol erasure event occurs when a receivedsymbol is erased by STTThe two probabilities can be dividedinto two cases as follows

119875err = (1 minus 120588) 119875119880err + 120588119875119869err119875ers = (1 minus 120588) 119875119880ers + 120588119875119869ers (A1)

where 119869 and 119880 refer to the jammed and the unjammed casesrespectively We denote the symbol erase event of STT by119867119904that is 119911 = sum119872minus1119895=0 119910119895 gt 120583 Then 119875119880err can be represented asfollows

119875119880err = Pr [1198670 119867119904 | 119880]= Pr [119867119904 | 119880] minus Pr [1198670 119867119904 | 119880] (A2)

The first term Pr[119867119904 | 119880] and the second term Pr[1198670 119867119904 |119880] are given as (A3) and (A4) respectively Similarly119875119880ers = Pr[119867119904 | 119880] = 1 minus Pr[119867119904 | 119880] can be obtained using(A3) By replacing 119892119880119895 with 119892119869119895 119895 = 0 119872 minus 1 we cancalculate the error and erase probabilities of the unjammedcase For the case in which a jamming signal exists 119875119869err and119875119869ers can be calculated by substituting 119892119869119895(119910119895) for 119892119880119895(119910119895)119895 = 0 119872 minus 1 The derivation of average error probability119875err and erasure probability 119875ers is complete

Pr [119867119904 | 119880] = Pr [119911 lt 120583 | 119880] = Pr[[119872minus1sum119895=0

119910119895 lt 120583]] = int1205830int120583minus11991000

sdot sdot sdot int120583minussum119872minus2119895=0 1199101198950

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 11988911991011198891199100 (A3)

Pr [1198670 119867119904 | 119880] = Pr[[1199100 gt 119910119895 forall119895 = 1 119872 minus 1 119872minus1sum119895=0

119910119895 lt 120583]]= int1205830intmin(1199100 120583minus1199100)

0sdot sdot sdot intmin(1199100 120583minussum

119872minus2119895=0 119910119895)

0

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 11988911991011198891199100(A4)

Disclosure

The present address of Jinsoo Park is the Agency for DefenseDevelopment Daejeon Republic of Korea

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors gratefully acknowledge the support from Elec-tronic Warfare Research Center at the Gwangju Institute ofScience and Technology (GIST) originally funded by theDefense Acquisition Program Administration (DAPA) andthe Agency for Defense Development (ADD)

References

[1] L Milstein and M Simon ldquoSpread Spectrum Communi-cationsrdquo in The Mobile Communications Handbook SecondEdition vol 19990749 of Electrical Engineering Handbook CRCPress 1999

[2] C W Baum and M B Pursley ldquoBayesian Methods for ErasureInsertion in Frequency-Hop Communication Systems withPartial-Band Interferencerdquo IEEE Transactions on Communica-tions vol 40 no 7 pp 1231ndash1238 1992

[3] L-L Yang and L Hanzo ldquoPerformance analysis of codedM-aryorthogonal signaling using errors-and-erasures decoding overfrequency-selective fading channelsrdquo IEEE Journal on SelectedAreas in Communications vol 19 no 2 pp 211ndash221 2001

[4] A J Viterbi ldquoA Robust Ratio-Threshold Technique to MitigateTone and Partial Band Jamming in Coded MFSK Systemsrdquo inProceedings of the IEEE Military Communications ConferenceMILCOM 1982 pp 224-1ndash224-5 Boston MA October 1982

[5] L-L Yang and L Hanzo ldquoLow complexity erasure insertion inRS-coded SFH spread-spectrum communications with partial-band interference and nakagami-m fadingrdquo IEEE Transactionson Communications vol 50 no 6 pp 914ndash925 2002

[6] S Ahmed L-L Yang and L Hanzo ldquoErasure insertion in RS-coded SFH MFSK subjected to tone jamming and Rayleighfadingrdquo IEEE Transactions on Vehicular Technology vol 56 no6 I pp 3563ndash3571 2007

[7] H A Ngo S Ahmed L-L Yang and L Hanzo ldquoNon-coherent cooperative communications dispensing with channelestimation relying on erasure insertion aided Reed-Solomon

12 Security and Communication Networks

coded SFH M-ary FSK subjected to partial-band interferenceand Rayleigh fadingrdquo IEEE Transactions on Communicationsvol 60 no 8 pp 2177ndash2186 2012

[8] T G Macdonald and M B Pursley ldquoStaggered interleavingand iterative errors-and-erasures decoding for frequency-hoppacket radiordquo IEEE Transactions on Wireless Communicationsvol 2 no 1 pp 92ndash98 2003

[9] C-M Lee Y T Su and L-D Jeng ldquoPerformance analysis ofblock codes in hidden Markov channelsrdquo IEEE Transactions onCommunications vol 56 no 1 pp 1ndash4 2008

[10] A Aung K C Teh and K H Li ldquoDetection and counter-measure of interference in slow FHMFSK systems over fadingchannelsrdquo Physical Communication vol 10 pp 11ndash23 2014

[11] J G Kim ldquoError and erasure decoding scheme for RS codes inMFSK based FHSSMA communicationrdquo International Journalof Multimedia and Ubiquitous Engineering vol 10 no 10 pp11ndash20 2015

[12] R Koetter and A Vardy ldquoAlgebraic soft-decision decodingof Reed-Solomon codesrdquo Institute of Electrical and ElectronicsEngineers Transactions on InformationTheory vol 49 no 11 pp2809ndash2825 2003

[13] J Jiang and K R Narayanan ldquoIterative soft decoding of Reed-Solomon codesrdquo IEEE Communications Letters vol 8 no 4 pp244ndash246 2004

[14] P Sehier P Brelivet and I Aubry ldquoNew jammer estimationmethod adapted to FH-PSK waveformsrdquo in Proceedings of the1994 IEEE MILCOM Part 3 (of 3) pp 1027ndash1031 October 1994

[15] M Pursley and J Skinner ldquoDecoding strategies for turboproduct codes in frequency-hop wireless communicationsrdquo inProceedings of the IEEE International Conference on Communi-cations pp 2963ndash2968 Anchorage AK USA

[16] M B Pursley and J S Skinner ldquoAdaptive coding for frequency-hop transmission in mobile ad hoc networks with partial-bandinterferencerdquo IEEETransactions onCommunications vol 57 no3 pp 801ndash811 2009

[17] M R Masse M B Pursley and J S Skinner ldquoAdaptive codingfor frequency-hop transmission over fading channels withpartial-band interferencerdquo IEEE Transactions on Communica-tions vol 59 no 3 pp 854ndash862 2011

[18] H El Gamal and E Geraniotis ldquoIterative channel estimationand decoding for convolutionally coded anti-jam FH signalsrdquoIEEE Transactions on Communications vol 50 no 2 pp 321ndash331 2002

[19] G D Forney ldquoGeneralized Minimum Distance DecodingrdquoIEEE Transactions on Information Theory vol IT-12 no 2 pp125ndash131 1966

[20] D Chase ldquoA Class of Algorithms for Decoding Block CodeswithChannelMeasurement Informationrdquo IEEETransactions onInformation Theory vol 18 no 1 pp 170ndash182 1972

[21] H Tang Y LiuM Fossorier and S Lin ldquoOn combining Chase-2 and GMD decoding algorithms for nonbinary block codesrdquoIEEE Communications Letters vol 5 no 5 pp 209ndash211 2001

[22] S-W Lee and B V K V Kumar ldquoSoft-decision decoding ofreed-solomon codes using successive error-and-ersure decod-ingrdquo in Proceedings of the 2008 IEEEGlobal TelecommunicationsConference GLOBECOM2008 pp 3065ndash3069USADecember2008

[23] E R Berlekamp ldquoReadable Erasures Improve the Performanceof Reed-Solomon Codesrdquo IEEE Transactions on InformationTheory vol 24 no 5 pp 632-633 1978

[24] J G Proakis andM SalehiEssentials of Communication SystemsEngineering 2005

[25] W G Phoel ldquoIterative demodulation and decoding offrequency-hopped PSK in partial-band jammingrdquo IEEEJournal on Selected Areas in Communications vol 23 no 5 pp1026ndash1033 2005

[26] S Lin and D J Costello Error Control Coding Pearson 2ndedition 2004

[27] J K Wolf A M Michelson and A H Levesque ldquoOn theProbability of Undetected Error for Linear Block Codesrdquo IEEETransactions on Communications vol 30 no 2 pp 317ndash3241982

[28] A Ramesh A Chockalingam and L B Milstein ldquoPerformanceof noncoherent turbo detection on Rayleigh fading channelsrdquoin Proceedings of the IEEE Global Telecommunicatins ConferenceGLOBECOMrsquo01 pp 1193ndash1198 usa November 2001

[29] M C Valenti and S Cheng ldquoIterative demodulation anddecoding of turbo-codedM-ary noncoherent orthogonal mod-ulationrdquo IEEE Journal on Selected Areas in Communications vol23 no 9 pp 1739ndash1747 2005

[30] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press BostonMassUSA 5th edition 1994

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Security and Communication Networks 3

3 Erasure Insertion Schemes

31 Overview of Erasure Insertion Schemes

311 Conventional Schemes Next we briefly introduce thethree conventional erasure insertion schemes the ratiothreshold test (RTT) the output threshold test (OTT) andthe maximum output-ratio threshold test (MO-RTT) [4ndash7 25] Let maxsdot and max1015840sdot represent the first and secondmaximum values of a given set respectively For each 119894 from0 to 119873 minus 1 the 119894th received symbol is erased by RTT ifmax1015840119895119910119895max119895119910119895 gt 120582 where 120582 is a given threshold OTTerases the symbol if max119895119910119895 gt 120591 for a given threshold120591 Sometimes the OTT rule is applied as max119895119910119895 lt 120591[6] however this reversed inequality is not considered inthis paper We note that the conventional RTT and OTTare one-dimensional erasure insertion schemes because thedecision is made using a single measure On the other handMO-RTT is a two-dimensional erasure insertion schemecombining OTT and RTT MO-RTT generally exhibits betterperformance thanRTTorOTTwith their optimal thresholdsBefore we discuss the problems of conventional schemes weintroduce a new threshold test scheme that is sum-basedthreshold test (STT)This test erases the symbol ifsum119872minus1119895=0 119910119895 gt120583 where 120583 is a given threshold

Threshold test-based conventional schemes have used theone-time erasure insertion and the bounded-distance decod-ing with preoptimized thresholds The optimum thresholdsdepend on the channel state therefore the receiver shouldknow the channel state information at the time Because of thedifficulty of obtaining perfect CSIR over jamming channelswe consider the iterative erasure insertion scheme whichdoes not require any CSIR

312 Iterative Erasure Insertion and Decoding Schemes Toincrease the error correction capability of RS codes manyiterative decoding algorithms have been proposed [19ndash23]For an unjammed channel we can estimate the channelstate information using various techniques Thereafter wecan apply various decoding algorithms However if thejamming signal exists and CSIR is not available we mustaddress the jamming attack without CSIR To use the iter-ative decoding algorithms without CSIR we will considera simplified version of the GMD decoding algorithm [26]as an example The main idea of GMD decoding is iterativedecoding trials with erasures During each iteration GMDdecoding erases the least reliable symbol and attempts anerror and erasure decoding In this paper we use the symbolmeasures ratio (max1015840119895119910119895max119895119910119895) output (max119895119910119895) andsum (sum119872minus1119895=0 119910119895) for the GMD decoding algorithm Theseschemes will be called ratio-based GMD (R-GMD) output-based GMD (O-GMD) and sum-based GMD (S-GMD)decoding respectively Note that the three measures arecalculated without CSIR

In Algorithm 1 the one-dimensional GMD algorithm isdescribed using the three measures Here 119894 is the index of thecodeword symbol and 119895 is the index of 119872 detectors of theMFSK demodulator Initially the receiver decides that 119888119894 =

argmax119895119910119894119895 for each 119894 from 0 to119873 minus 1 In the beginning ofwhile loop the decoder attempts to recover the transmit wordw in (1) from c with error-only decoding This step preventsperformance degradation when there is no jamming signalAdditionally we will not consider the undetected errors ofthe decoded codeword because they seldom occur [27] Ifthe decoder declares a decoding failure it erases the mostsuspicious (likely to have been jammed) symbol 119888119894max

andreplaces it with an erasure 120598 where 119894max is determined by theselected decision rule For each measure 119894max is determinedas follows

Ratio measure

119894max = argmax119894notin119864

max1015840119895 119910119894119895max119895 119910119894119895 (5)

Output measure

119894max = argmax119894notin119864

119910119894119895 (6)

Sum measure

119894max = argmax119894notin119864

119872minus1sum119895=0

119910119894119895 (7)

After 119894max is determined it is added to 119864 which is thecurrent set of erased symbol indices Then error and erasuredecoding is applied As we noted in Section 2 (119873119870) RScodes can correct 119906 erasures and V errors if 119906 + 2V lt119873 minus 119870 + 1 is satisfied If the decoding is successful thenthe erasure insertion loop is terminated and the decodedcodeword becomes the output c Otherwise the algorithmcontinues to the second iteration of the while loop In thesecond iteration and thereafter all of the previously erasedsymbol indices (119864) are maintained and 119894max is determinedover all the unerased indices This iteration runs successivelyuntil the decoding succeeds or the number of erased symbolsis the same as the number of parity symbols that is |119864| =119873 minus 119870 The while loop in Algorithm 1 is described by thefeedback loop in Figure 1 We note that this iterative erasureinsertion and GMD decoding algorithm runs independentlyfor each codeword and does not make comparisons withany preoptimized threshold Furthermore if the conventionalerasure insertion scheme can recover the codeword then thecorresponding iterative scheme can recover the codeword

32 Two-Dimensional Schemes We introduced theMO-RTTat the beginning of Section 3 which has two-dimensionalmeasures the joint threshold tests of OTT and RTT Toexploit the diversity of multiple measures various combina-tions of iterative schemes with some different measures areconsidered In this paper however wewill focus only on serialconcatenation of R-GMD decoding and S-GMD decodingcalled ratio-to-sum-based GMD (RS-GMD) decoding FirstRS-GMD decoder runs R-GMD decoding If R-GMD decod-ing fails to decode with |119864| = 119873 minus 119870 + 1 then the receiver

4 Security and Communication Networks

InitializeDecide a vector c by 119888119894 = argmax119895119910119894119895 0 le 119894 le 119873 minus 1Set 119864 = 0

while |119864| le 119873 minus 119870 doTry to error-and-erasure decode cif decoding success then

c is the decoded codewordBreak while loop

else if decoding fail thenCase Measureratio 119894max = argmax119894notin119864max1015840119895119910119894119895max119895119910119894119895output 119894max = argmax119894notin119864119910119894119895sum 119894max = argmax119894notin119864sum119872minus1119895=0 119910119894119895

Erase 119894maxth codeword symbols ie set 119888119894max= 120598119864 larr 119864 cup 119894max

end ifend whileif |119864| le 119873 minus 119870 then

Decoding successOutput = c

else (|119864| = 119873 minus 119870 + 1)Decoding renounce

end if

Algorithm 1 Iterative erasure insertion and GMD decoding

Table 1 Thresholds 119911119901 of the three measures where119872 = 4 119901 = 01 120588 = 01 and 1198641198871198730 = 5 dB over AWGN and partial-band jammingchannels

119864119887119873119895 (dB) minus10 minus5 0 5 10 15 20119911119901 of R 0722 0722 0721 0719 0712 0695 0671119911119901 of O 4123 3625 3158 2462 2103 1993 1961119911119901 of S 4824 4377 4438 3400 2869 2623 2536

runs S-GMD decoding again from the beginning We willshow the performance improvement of RS-GMD decodingas compared to the performance of MO-RTT over variouschannel situations in Section 5 In Table 2 the conventionaliterative and two-dimensional schemes are classified into thefour types of measures

4 Trapped-Error Probability

For the conventional erasure insertion schemes that is RTTOTT and MO-RTT the error and erasure probabilities arederived in [4ndash6] And we have included the error and erasureprobabilities of STT in the Appendix Calculation of theerror and erasure probabilities of a given symbol cannotbe applied to the analysis of iterative schemes because theerasure insertion for one symbol is not independent of thatfor all other symbols For performance analysis of the iterativeschemes we propose trapped-error probability analysis basedon threemeasures ldquoratiordquo ldquooutputrdquo and ldquosumrdquoThe trapped-error probability is a new and useful analysis framework tocompare the quality of each measure

Consider a received word c of length 119873 Let 119885119894 be themeasure of one symbol of c for one of the threemeasures119885119894 =

max1015840119895119910119894119895max119895119910119894119895 119885119894 = max119895119910119894119895 and 119885119894 = sum119872minus1119895=0 119910119894119895for ratio output and sum measures respectively First wesort the received 119873 symbols in descending order accordingto 119885119894 values that are calculated by one selected way (amongthe three kinds of measures) Let 119901 denote the ratio of thewindow of interest to the length 119873 This ratio 119901 should bechosen in the range of 0 lt 119901 le (119873 minus 119870)119873 because the(119873119870) RS code can correct up to 119873 minus 119870 erasures Now wecount the number119873119901 of erroneous symbols in the uppermostportion of 119901119873 symbols Let 119873119905 denote the total number oferroneous symbols in the received word c of length119873 Thenthe trapped-error probability Γ is defined as the ratio between119873119901 and119873119905

Γ = 119873119901119873119905 (8)

For example let us assume that there are 119873 = 100 receivedsymbols with a total of 119873119905 = 15 erroneous symbols Let usassume that 119901 = 01 And we consider the first 10 (=pN)positions when all of the symbols are arranged in decreasingorder of their corresponding measures Let us assume that119873119901 = 8 1 and 5 erroneous symbols are placed among these10 positions for ratio output and summeasures respectively

Security and Communication Networks 5

Then the trapped-error probabilities are ΓR = 815 ΓO =115 and ΓS = 515 for these respective measures Inthis case because the iterative schemes successively erasethe symbols in decreasing order of each measure the ratiomeasure works best for trapping (or identifying) the erro-neous symbols and the output measure works worst For agiven 119901 a larger trapped-error probability indicates betterperformance

Without loss of generality we assume that all-zero code-word w = (0 0) is transmitted that is 1198910 is selected forevery symbol transmission If a transmitted symbol does notinterfere with the jamming signal and the AWGN channel isassumed then the probability density functions (PDFs) of 1199100and 119910119895 119895 isin 1 119872 minus 1 are derived as follows [24]

1198921198800 (119910) = 121205902119873 exp(minus119910 + 121205902119873 ) 1198680 (radic1199101205902119873 )

119892119880119895 (119910) = 121205902119873 exp(minus 11991021205902119873) (9)

For the jammed case

1198921198690 (119910) = 121205902 exp(minus119910 + 121205902 ) 1198680 (radic1199101205902 ) 119892119869119895 (119910) = 121205902 exp(minus 11991021205902 )

(10)

where 1205902 = 1205902119869 + 1205902119873 and 1198680(sdot) is the modified Bessel functionof order zero

If we assume Rayleigh fading channel withoutwithpartial-band jamming 1198921198800 and 1198921198690 are changed as follows[28 29]

1198921198800 (119910) = 121205902119873 + 1 exp(minus11991021205902119873 + 1)

1198921198690 (119910) = 121205902 + 1 exp (minus 11991021205902 + 1) (11)

Because 1198921198800(119910) 119892119880119872minus1(119910) and 1198921198690(119910) 119892119869119872minus1(119910) areindependent variables their joint PDFs can be represented asprod119872minus1119895=0 119892119880119895(119910119895) andprod119872minus1119895=0 119892119869119895(119910119895) respectively Their cumula-tive distribution functions (CDFs) are 119866119880119895(119910119895) and 119866119869119895(119910119895)for 119895 isin 0 119872 minus 1 respectively

Recall that 119885119894 is the measure of the symbol 119888119894 for one ofthe three measures For simplicity we will omit the subscript119894 119885 = max1015840119895119910119895max119895119910119895 119885 = max119895119910119895 or 119885 = sum119872minus1119895=0 119910119895for ratio output or sum measures respectively The CDF issimply denoted by 119865119885(119911) Define 1198670 as a representation ofthe event of a raw symbol error that is the event where thereexists at least one 119895 isin 1 119872minus1 that satisfies 1199100 lt 1199101198951198670rsquoscomplementary event is denoted by1198670 Now we consider the

conditional CDFs 119865119885|119867(119911 | 1198670) and 119865119885|119867(119911 | 1198670) where119867 isthe discrete event variable119867 isin 1198670 1198670 Then

119865119885 (119911) = Pr [1198670] 119865119885|119867 (119911 | 1198670)+ Pr [1198670] 119865119885|119867 (119911 | 1198670)

119891119885 (119911) = Pr [1198670] 119891119885|119867 (119911 | 1198670)+ Pr [1198670] 119891119885|119867 (119911 | 1198670)

(12)

where 119891s denote the corresponding PDFsDefine 119911119901 as the point that satisfies 119865119885(119911119901) = 1 minus 119901119911119901 is uniquely determined because 119865119885(119911) is monotonically

increasing from 0 to 1 Then the trapped-error probabilityΓ(119911119901) can be defined as follows

Γ (119911119901) = intinfin119911119901

119891119885|119867 (119911 | 1198670) 119889119911 = 1 minus 119865119885|119867 (119911119901 | 1198670) (13)

Now we will change (13) into a form involving1198670 insteadof 1198670 such that its computation can be easily performedFrom (12) we observe that

119865119885 (119911119901) = Pr [1198670] 119865119885|119867 (119911119901 | 1198670)+ Pr [1198670] 119865119885|119867 (119911119901 | 1198670) = 1 minus 119901 (14)

Using (14) the trapped-error probability in (13) can bewrittenas

Γ (119911119901) = 1 minus 1 minus 119901 minus Pr [1198670] 119865119885|119867 (119911119901 | 1198670)Pr [1198670] (15)

Using the relation Pr[1198670] + Pr[1198670] = 1 and after somemanipulation we have the following

Γ (119911119901) = 119901 minus Pr [1198670] + Pr [1198670] 119865119885|119867 (119911119901 | 1198670)1 minus Pr [1198670] (16)

Here Pr[1198670] can be calculated as follows

Pr [1198670] = Pr [1199100 gt 119910119895 forall119895 = 1 119872 minus 1]= 120588intinfin0int11991000sdot sdot sdot int11991000

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100+ 120588intinfin0int11991000sdot sdot sdot int11991000

119872minus1prod119895=0

119892119869119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100= 120588intinfin01198921198800 (1199100) (1198661198801 (1199100))119872minus1 1198891199100

+ 120588intinfin01198921198690 (1199100) (1198661198691 (1199100))119872minus1 1198891199100

(17)

where 120588 = 1 minus 120588

6 Security and Communication Networks

Let us assume AWGN and a partial-band jammingchannel Using a binomial expansion and some calculations[30] (17) becomes

120588119890(minus121205902119873) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902119873)119897 (119898 + 1)119897+1 119897 (

119872 minus 1119898 )

+ 120588119890(minus121205902) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902)119897 (119898 + 1)119897+1 119897 (

119872 minus 1119898 )

(18)

Similarly for a Rayleigh fading and partial-band jammingchannel (17) becomes

120588119872minus1sum119898=0

21205902119873(2 + 2119898) 1205902119873 + 119898 (minus1)119898 (119872 minus 1119898 )

+ 120588119872minus1sum119898=0

21205902(2 + 2119898) 1205902 + 119898 (minus1)119898 (119872 minus 1119898 )

(19)

Next we compute 119865119885|119867(119911119901 | 1198670) as follows

119865119885|119867 (119911119901 | 1198670)= Pr [119885 lt 119911119901 1199100 gt 1199101198951015840 forall1198951015840 = 1 119872 minus 1]

Pr [1198670] (20)

Now assume that we use the ratio measure Then thenumerator of (20) becomes

Pr[max1015840119895 119910119895max119895 119910119895 lt 119911119901 1199100 gt 1199101198951015840 forall1198951015840 = 1 119872 minus 1]

= Pr [max1015840119895 119910119895 lt 1199111199011199100]= Pr [119910119895 lt 1199111199011199100 forall119895 = 1 119872 minus 1]= 120588intinfin0int11991111990111991000

sdot sdot sdot int11991111990111991000

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100+ 120588intinfin0int11991111990111991000

sdot sdot sdot int11991111990111991000

119872minus1prod119895=0

119892119869119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100= 120588intinfin01198921198800 (1199100) (1198661198801 (1199111199011199100))(119872minus1) 1198891199100

+ 120588intinfin01198921198690 (1199100) (1198661198691 (1199111199011199100))(119872minus1) 1198891199100

(21)

For AWGN and Rayleigh fading channels with partial-bandjamming (21) becomes (22) and (23) respectively

120588119890(minus121205902119873) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902119873)119897 (119898119911119901 + 1)119897+1 119897 (

119872 minus 1119898 )

+ 120588119890(minus121205902) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902)119897 (119898119911119901 + 1)119897+1 119897 (

119872 minus 1119898 )

(22)

120588119872minus1sum119898=0

21205902119873(2 + 2119898119911119901) 1205902119873 + 119898119911119901 (minus1)119898 (119872 minus 1

119898 )

+ 120588119872minus1sum119898=0

21205902(2 + 2119898119911119901) 1205902 + 119898119911119901 (minus1)119898 (119872 minus 1

119898 ) (23)

For the output measure the numerator of (20) becomes

Pr [max119895 119910119895 lt 119911119901 1199100 gt 1199101198951015840 forall1198951015840 = 1 119872 minus 1]= Pr [1199100 lt 119911119901 1199100 gt 119910119895 forall119895 = 1 119872 minus 1]= 120588int1199111199010int11991000sdot sdot sdot int11991000

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100+ 120588int1199111199010int11991000sdot sdot sdot int11991000

119872minus1prod119895=0

119892119869119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100= 120588int11991111990101198921198800 (1199100) (1198661198801 (1199100))(119872minus1) 1198891199100

+ 120588int11991111990101198921198690 (1199100) (1198661198691 (1199100))(119872minus1) 1198891199100

(24)

With partial-band jamming (24) becomes (25) and (26)for AWGN and Rayleigh fading channels respectively

120588119890(minus121205902119873) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902119873)2119897+1 119897 (

119872 minus 1119898 )

sdot (( 21205902119873119898 + 1)119897+1

minus 119890(minus(119898+1)11991111990121205902119873) 119897sum119899=0

119911119899119901119899 ( 21205902119873119898 + 1)119897+1minus119899)

+ 120588119890(minus121205902) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902)2119897+1 119897 (

119872 minus 1119898 )

sdot (( 21205902119898 + 1)119897+1

minus 119890(minus(119898+1)11991111990121205902) 119897sum119899=0

119911119899119901119899 ( 21205902119898 + 1)119897+1minus119899)

(25)

Security and Communication Networks 7

120588119872minus1sum119898=0

21205902119873(2 + 2119898) 1205902119873 + 119898 (minus1)119898 (119872 minus 1119898 ) sdot (1

minus 119890(minus(((2+2119898)1205902119873+119898)(41205904119873+21205902119873))119911119901))+ 120588119872minus1sum119898=0

21205902(2 + 2119898) 1205902 + 119898 sdot (minus1)119898 (119872 minus 1119898 )(1

minus 119890(minus(((2+2119898)1205902+119898)(41205904+21205902))119911119901))

(26)

For the sum measure the numerator of (20) is given as (27)Actually (27) is the same as (A4) in the Appendix where 120583 =119911119901 Using the derived equations we can calculate the trapped-error probabilities of three measures

Pr[[119872minus1sum119895=0

119910119894119895 lt 119911119901 1199100 gt 119910119895 forall119895 = 1 119872 minus 1]] = Pr[[1199100 lt 119911119901

1199101 lt min (1199100 119911119901 minus 1199100) 119910119872minus1 lt min(1199100 119911119901 minus 119872minus2sum119895=0

119910119895)]]= 120588int1199111199010intmin(1199100 119911119901minus1199100)

0sdot sdot sdot intmin(1199100 119911119901minussum

119872minus2119895=0 119910119895)

0

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1sdot sdot sdot 1198891199100

(27)

+ 120588int1199111199010intmin(1199100 119911119901minus1199100)

0sdot sdot sdot intmin(1199100 119911119901minussum

119872minus2119895=0 119910119895)

0

119872minus1prod119895=0

119892119869119895 (119910119895) 119889119910119872minus1sdot sdot sdot 1198891199100

(28)

5 Simulation Results

51 Trapped-Error Probabilities In this subsection we con-firm the trapped-error probability analysis over AWGN andRayleigh fading channels with partial-band jamming Letus assume 4-FSK 120588 = 01 and 1198641198871198730 = 5 dB for theAWGN channel and 1198641198871198730 = 12 dB for the Rayleigh fadingchannel For the higher order of modulation such as 32-FSK which we will consider (27) requires 32-dimensionalintegration therefore we select 4-FSK as an example Tocalculate the probabilities Γ(119911119901) for each measure the input119911119901 should be obtained for the given set of system parametersFor this we further assume that 119901 = 01 and 02 for AWGNand Rayleigh fading channels respectively As we describedin Section 4 we found 119911119901 values that satisfy 119865119911(119911119901) =1 minus 119901 using simulations They are listed in Tables 1 and 3for various 119864119887119873119895 R O and S refer to ratio output andsum respectively In the investigation we transmitted 105symbols for various channel states and calculated their ratiooutput and sum measures We note that these trapped-errorprobabilities do not depend on a specific coding schemebecause these statistics are observed at the demodulatoroutput without coding

Figures 2 and 3 show the trapped-error probabilities ofthe three measures obtained using simulations and from (16)with the threshold values given in Tables 1 and 3 Solid anddashed lines indicate the probabilities obtained by simulation

Table 2 Class of erasure insertion schemes

Measure Threshold test IterativeRatio RTT R-GMDOutput OTT O-GMDSum STT S-GMDTwo-dimensional Ex MO-RTT Ex RS-GMD

Table 3Thresholds 119911119901 of the three measures where119872 = 4 119901 = 02120588 = 01 and 1198641198871198730 = 12 dB over Rayleigh fading and partial-bandjamming channels

119864119887119873119895 (dB) minus5 0 5 10 15 20119911119901 of R 0353 0345 0350 0329 0294 0263119911119901 of O 2271 2233 2053 1765 1681 1660119911119901 of S 2368 2342 2297 2069 1845 1776

and (16) respectively The cross circle and square marksrepresent Γ(119911119901) values of ratio output and sum measuresrespectively

These two figures confirm that the simulation is exactenough to closely follow the theoretical result in (16) Spe-cially ΓO in the analysis is almost the same as ΓO in thesimulation results This result implies the correctness of ourperformance simulation

In these figures the trapped-error probability of theratio measure is increased as SJR is increased therefore wepredict that the R-GMD decoding will perform better thanthe other one-dimensional schemes at high SJR Additionallywe predict that S-GMD decoding is better than O-GMDdecoding in the middle SJR region Recall that the higherthe trapped-error probability the better the performanceThis observation shows the influence on the two-dimensionalschemes because the trapped-error probability affects thenumber of decoding trials We will discuss it in the followingsimulation results

52 Performance over AWGN Channel Throughout perfor-mance simulations we consider 32-ary FSKmodulation with(31 20) RS code over a field of size 32 an ideal interleaverand a random hopping pattern We first consider partial-band jamming (or partial-time jamming with the same 120588)and AWGN channels In this case 120572 = 1 in Figure 1 ForFigures 4ndash7 the channel noise is fixed as we used 1198641198871198730 =5 dBVarious signals to jamming signal ratio119864119887119873119895 have beenconsidered for the simulations

In Figure 4 the performance of various erasure insertionschemes is plotted over the partial-band jamming channelwith 120588 = 01The dashed line shows the baseline performanceldquoWithout EIrdquo which does not use any erasure insertionschemeThe unmarked solid line represents the performanceof the MO-RTT which is the conventional erasure insertionschemeWenote that theMO-RTTuses preoptimized thresh-olds and CSIR To obtain the optimum thresholds for MO-RTT we optimized the thresholds for 119864119887119873119895 = 0sim30 dB with10 dB steps Solid lines with circle square cross plus anddiamond marks represent the performance of output-based

8 Security and Communication Networks

100 5 15 20minus5minus10

ΓR by simulationΓO by simulationΓS by simulation

ΓR by analysisΓO by analysisΓS by analysis

0

01

02

03

04

05Γ

06

07

08

09

1

EbNj (dB)

Figure 2 Trapped-error probabilities of ratio-based (ΓR) output-based (ΓO) and sum-based (ΓS) measures over partial-band jam-ming and AWGN channel

01

02

03

04

05

06

07

08

09

1

Γ

minus5 0 5 10 15 200

ΓR by simulationΓO by simulationΓS by simulation

ΓR by analysisΓO by analysisΓS by analysis

EbNj (dB)

Figure 3 Trapped-error probabilities of ratio-based (ΓR) output-based (ΓO) and sum-based (ΓS) measures over partial-band jam-ming and Rayleigh channels

sum-based ratio-based ratio-to-output-based and ratio-to-sum-based GMD decoding respectively In this figure wealso present the performance using log-likelihood ratio (LLR)based GMD decoding as the triangle-marked line This isthe performance of a system with perfect CSIR such asSNR SJR and the existence of a jamming signal This LLR-GMD decoding iteratively erases the least reliable symbolsbased on LLR as described in [19 22 26] Because it is wellknown that MO-RTT exhibits better performance than OTT

EbNj (dB)

10minus5

10minus4

10minus3

10minus2

10minus1

100

WER

5 10 15 20 250

Without EIMO-RTTO-GMDS-GMD

R-GMDRO-GMDRS-GMDLLR-GMD

Figure 4 Performance of various erasure insertion schemes overpartial-band jamming and AWGN channels with 120588 = 01 and1198641198871198730 = 5 dB

or RTT [5ndash7] the performances ofOTTandRTTare omittedBefore discussing the simulation result we note that RSO-GMD decoding which serially exploits the three measureshas almost the same performance as RS-GMD decodingtherefore we have also omitted it From Figure 4 we makethe following observations

(1) As we expect LLR-GMD decoding with CSIR showsthe best performance We will focus on the perfor-mance gap between LLR-GMD decoding with CSIRand the other erasure insertion schemes withoutCSIR To achieve WER = 10minus4 LLR-GMD decodingrequires 119864119887119873119895 = 19 dB The ratio-based GMDdecoding and its two-dimensional schemes achievethe target WER at 119864119887119873119895 = 21 dB which is muchcloser than that of MO-RTT Even if the jammingpower is decreased as 119864119887119873119895 = 25 dB the perfor-mance of MO-RTT is still far from WER = 10minus4 Inthe low SJR region only the two two-dimensionaliterative schemes can achieve WER = 10minus2 withoutCSIR We note that the one-dimensional iterativeschemes and MO-RTT require 119864119887119873119895 gt 9 dB toachieve WER = 10minus2

(2) The performance of RS-GMD decoding approachesthat of LLR-GMD decoding as SJR is increased At119864119887119873119895 = 20 dB LLR-GMD decoding with perfectCSIR has 35 less WER than RS-GMD decodingWe note that this gap cannot be closed even if thejamming power is decreased

(3) When the jamming power is decreased that isthe channel approaches the unjammed channel theperformance of R-GMD and RS-GMD decoding is

Security and Communication Networks 9

0

01

02

03

04

05

06

07

08

09

1

Γ

5 10 15 20 25 300

ΓRΓOΓS

EbNj (dB)

Figure 5 Trapped-error probability of various measures overpartial-band jamming and AWGN channels with 119901 = 03 120588 = 01and 1198641198871198730 = 5 dB

still better than the performance of ldquoWithout EIrdquoor MO-RTT This implies that the iterative erasureinsertion and decoding schemes do not decrease theperformance if there are no jamming signals Becauseof this we do not need to detect the jamming signal

In Figure 5 trapped-error probabilities of ratio outputand sum measures are represented which corresponds toFigure 4 To obtain the trapped-error probabilities SJR andSNR were scaled based on the code rate 2031 = 119873119870 whichis different from Section 51 We used 119901 = 03 and the otherparameters are the same as those used in Figure 4 In Figure 5ΓR is much larger than ΓO and ΓS The values of ΓR explain thegood performance of R-GMDdecodingWe find that ΓO goesto lt1 minus119870119873 asymp 03 at 119864119887119873119895 asymp 10 dB Any iterative schemethat has zero trapped-error probability at a given SJR has thesame performance as the performance of ldquoWithout EIrdquo Thisresult explains why O-GMD decoding has a performancedegradation region near 10 dB in Figure 4 Because theperformance of O-GMD decoding rapidly approaches theperformance of ldquoWithout EIrdquo its performance is degraded asSJR is increased We note that the performance of S-GMDdecoding slowly approaches the performance of ldquoWithoutEIrdquo therefore there is no performance degradation region

The performance gain of the iterative schemes in Fig-ure 4 cannot be obtained without drawbacks the iterativeschemes require more decoding trials To investigate howmany decoding trials are required we determine the averagenumber of decoding iterations of various iterative schemesvia simulations The results are shown in Figure 6 In factwe simulated iterative schemes that erase one symbol duringthe first loop and then erase two symbols for each remainingloop because this process does not decrease the performanceas described in [19] In general the average number of

1

11

12

13

14

15

16

17

18

Aver

age n

umbe

r of d

ecod

ing

tria

ls

2 4 6 8 10 12 14 16 18 200

O-GMDS-GMDR-GMD

RS-GMDLLR-GMD

EbNj (dB)

Figure 6 Average number of decoding iterations for various erasureinsertion schemes over partial-band jamming and AWGN channelswith 120588 = 01 and 1198641198871198730 = 5 dB

MO-RTTRS-SEI

10minus3

10minus2

10minus1

WER

02 03 04 05 06 07 08 0901

Figure 7 Performance comparison of MO-RTT and RS-GMDdecoding scheme over partial-band jamming and AWGN channelswith various values of 120588 1198641198871198730 = 5 dB and 119864119887119873119895 = 10 dB

decoding trials decreases rapidly as SJR increases At highSJR most of the received words are completely decoded by asingle trial of error-only decoding which holds true becausehigher SJR implies that the environment becomes less andless hostile For low SJR the average decoding trial of RS-GMD decoding is approximately 175 (at 119864119887119873119895 = 0 dB)which decreases as SJR increasesThis result indicates that theiterative erasure insertion and decoding scheme is practicallyimplementable We note that the trial number of O-GMDdecoding is increased at approximately 119864119887119873119895 asymp 10 dB

10 Security and Communication Networks

and that is consistent with the performance degradation ofO-GMD decoding in Figures 4 and 5 Because O-GMDdecoding could not accurately erase the erroneous symbolsthe decoder has to try more decoding Because of this resultwemust use S-GMDdecoding instead of O-GMDdecodingfor the two-dimensional scheme with R-GMD decodingNow recall the analysis results of Section 51 In Figure 2we observed that the trapped-error probability of the ratiomeasure is increased as SJR is increased and that of the summeasure is decreased For the case in which the trapped-errorprobabilities cross at the point of SJR if we want to havefewer decodings at the higher SJR then we must use R-GMDdecoding first in RS-GMDdecoding If we want to have fewerdecodings at the lower SJR then S-GMD decoding should beapplied first In Figures 4 and 6 because we are consideringthe SJR region in which ΓR gt ΓS we have first exploited R-GMD decoding for RS-GMD decoding

In Figure 7 the performances of MO-RTT and RS-GMDdecoding are shown over a partial-band jamming and anAWGN channel with various jamming duty ratios 120588 withfixed 1198641198871198730 = 5 dB and 119864119887119873119895 = 10 dB For all values of120588 RS-GMD decoding works much better than MO-RTT Wenote that two erasure insertion schemes are more efficient forsmaller 120588 channels53 Performance over Rayleigh Fading Channel Figure 8shows the performance of various erasure insertion schemesover partial-band jamming and Rayleigh fading channel with120588 = 01 and 1198641198871198730 = 12 dB For each symbol transmission120572 in Figure 1 is realized by an iid Rayleigh distribution Weassume that fading coefficients are not known to the receiverexcept for the system of MO-RTT

In Figure 8 RS-GMD decoding exhibits better per-formance than MO-RTT which is the same result as wefound over the AWGN channel For the same WER of 10minus3the jammer should spend 14 dB more 119873119895 The WER ofratio-based schemes (including two-dimensional schemes)approach 10minus4 Unlike the results for an AWGN channelR-GMD decoding is dominant and the contribution of S-GMD decoding is marginal Therefore we conclude that R-GMD decoding is sufficient for Rayleigh fading channelsAdditionally we present the performance of LLR-GMDdecoding which knows perfect CSIR As we discussed inSection 52 the performance of R-SEI approaches that of LLR-SEI and the gap is maintained for every SJR

In Figure 9 the trapped-error probabilities of the mea-sures are displayed over a Rayleigh fading channel with1198641198871198730 = 12 dB The other parameters are the same as thosein Figure 5 In this figure the trapped-error probability of theratio measure is 1 in all SJR regions In other words R-GMDdecoding is the best one-dimensional iterative scheme for thetarget system parameters As we found for the other trapped-error probability results the trapped-error probabilities ofsum and output measures decrease as SJR increases Wherethe trapped-error probabilities are lower than lt1 minus119870119873 asymp03 that is 119864119887119873119895 = 10 dB and 16 dB for output andsum respectively the performances of O-GMD and S-GMDdecoding follow the performance of ldquoWithout EIrdquo As we

10minus5

10minus4

10minus3

10minus2

10minus1

100

WER

5 10 15 20 25 300

Without EIMO-RTTO-GMDS-GMD

R-GMDRO-GMDRS-GMDLLR-GMD

EbNj (dB)

Figure 8 Performance of various erasure insertion schemes over aRayleigh fading channel where 120588 = 01 and 1198641198871198730 = 12 dB

0

01

02

03

04

05

06

07

08

09

1

5 10 15 20 25 300

Γ

ΓRΓOΓS

EbNj (dB)

Figure 9 Trapped-error probability of various iterative schemesover partial-band jamming and Rayleigh fading channels with 119901 =03 120588 = 01 and 1198641198871198730 = 12 dB

discussed above the fast decreasing trapped-error probabilityof the output measure causes the performance degradation ofO-GMD decoding at approximately 10 dB in Figure 9

6 Concluding Remarks

In this paper we considered iterative erasure insertion anddecoding schemes that do not require any preoptimizedthresholds or any CSIR Additionally we proposed a new

Security and Communication Networks 11

analysis framework for the ratio output and sum mea-sures From the simulation results we confirmed that theratio-based GMD decoding scheme and its two-dimensionalschemes have the best performance Using the trapped-error probability the performances of the iterative erasureinsertion and decoding schemes are explained

Appendix

Error and Erasure Probability of SumThreshold Test

Let us denote the symbol error and erasure probabilities ofSTT as 119875err and 119875ers respectively The symbol error eventoccurs when a received symbol is erroneous but is not erasedby STT The symbol erasure event occurs when a receivedsymbol is erased by STTThe two probabilities can be dividedinto two cases as follows

119875err = (1 minus 120588) 119875119880err + 120588119875119869err119875ers = (1 minus 120588) 119875119880ers + 120588119875119869ers (A1)

where 119869 and 119880 refer to the jammed and the unjammed casesrespectively We denote the symbol erase event of STT by119867119904that is 119911 = sum119872minus1119895=0 119910119895 gt 120583 Then 119875119880err can be represented asfollows

119875119880err = Pr [1198670 119867119904 | 119880]= Pr [119867119904 | 119880] minus Pr [1198670 119867119904 | 119880] (A2)

The first term Pr[119867119904 | 119880] and the second term Pr[1198670 119867119904 |119880] are given as (A3) and (A4) respectively Similarly119875119880ers = Pr[119867119904 | 119880] = 1 minus Pr[119867119904 | 119880] can be obtained using(A3) By replacing 119892119880119895 with 119892119869119895 119895 = 0 119872 minus 1 we cancalculate the error and erase probabilities of the unjammedcase For the case in which a jamming signal exists 119875119869err and119875119869ers can be calculated by substituting 119892119869119895(119910119895) for 119892119880119895(119910119895)119895 = 0 119872 minus 1 The derivation of average error probability119875err and erasure probability 119875ers is complete

Pr [119867119904 | 119880] = Pr [119911 lt 120583 | 119880] = Pr[[119872minus1sum119895=0

119910119895 lt 120583]] = int1205830int120583minus11991000

sdot sdot sdot int120583minussum119872minus2119895=0 1199101198950

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 11988911991011198891199100 (A3)

Pr [1198670 119867119904 | 119880] = Pr[[1199100 gt 119910119895 forall119895 = 1 119872 minus 1 119872minus1sum119895=0

119910119895 lt 120583]]= int1205830intmin(1199100 120583minus1199100)

0sdot sdot sdot intmin(1199100 120583minussum

119872minus2119895=0 119910119895)

0

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 11988911991011198891199100(A4)

Disclosure

The present address of Jinsoo Park is the Agency for DefenseDevelopment Daejeon Republic of Korea

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors gratefully acknowledge the support from Elec-tronic Warfare Research Center at the Gwangju Institute ofScience and Technology (GIST) originally funded by theDefense Acquisition Program Administration (DAPA) andthe Agency for Defense Development (ADD)

References

[1] L Milstein and M Simon ldquoSpread Spectrum Communi-cationsrdquo in The Mobile Communications Handbook SecondEdition vol 19990749 of Electrical Engineering Handbook CRCPress 1999

[2] C W Baum and M B Pursley ldquoBayesian Methods for ErasureInsertion in Frequency-Hop Communication Systems withPartial-Band Interferencerdquo IEEE Transactions on Communica-tions vol 40 no 7 pp 1231ndash1238 1992

[3] L-L Yang and L Hanzo ldquoPerformance analysis of codedM-aryorthogonal signaling using errors-and-erasures decoding overfrequency-selective fading channelsrdquo IEEE Journal on SelectedAreas in Communications vol 19 no 2 pp 211ndash221 2001

[4] A J Viterbi ldquoA Robust Ratio-Threshold Technique to MitigateTone and Partial Band Jamming in Coded MFSK Systemsrdquo inProceedings of the IEEE Military Communications ConferenceMILCOM 1982 pp 224-1ndash224-5 Boston MA October 1982

[5] L-L Yang and L Hanzo ldquoLow complexity erasure insertion inRS-coded SFH spread-spectrum communications with partial-band interference and nakagami-m fadingrdquo IEEE Transactionson Communications vol 50 no 6 pp 914ndash925 2002

[6] S Ahmed L-L Yang and L Hanzo ldquoErasure insertion in RS-coded SFH MFSK subjected to tone jamming and Rayleighfadingrdquo IEEE Transactions on Vehicular Technology vol 56 no6 I pp 3563ndash3571 2007

[7] H A Ngo S Ahmed L-L Yang and L Hanzo ldquoNon-coherent cooperative communications dispensing with channelestimation relying on erasure insertion aided Reed-Solomon

12 Security and Communication Networks

coded SFH M-ary FSK subjected to partial-band interferenceand Rayleigh fadingrdquo IEEE Transactions on Communicationsvol 60 no 8 pp 2177ndash2186 2012

[8] T G Macdonald and M B Pursley ldquoStaggered interleavingand iterative errors-and-erasures decoding for frequency-hoppacket radiordquo IEEE Transactions on Wireless Communicationsvol 2 no 1 pp 92ndash98 2003

[9] C-M Lee Y T Su and L-D Jeng ldquoPerformance analysis ofblock codes in hidden Markov channelsrdquo IEEE Transactions onCommunications vol 56 no 1 pp 1ndash4 2008

[10] A Aung K C Teh and K H Li ldquoDetection and counter-measure of interference in slow FHMFSK systems over fadingchannelsrdquo Physical Communication vol 10 pp 11ndash23 2014

[11] J G Kim ldquoError and erasure decoding scheme for RS codes inMFSK based FHSSMA communicationrdquo International Journalof Multimedia and Ubiquitous Engineering vol 10 no 10 pp11ndash20 2015

[12] R Koetter and A Vardy ldquoAlgebraic soft-decision decodingof Reed-Solomon codesrdquo Institute of Electrical and ElectronicsEngineers Transactions on InformationTheory vol 49 no 11 pp2809ndash2825 2003

[13] J Jiang and K R Narayanan ldquoIterative soft decoding of Reed-Solomon codesrdquo IEEE Communications Letters vol 8 no 4 pp244ndash246 2004

[14] P Sehier P Brelivet and I Aubry ldquoNew jammer estimationmethod adapted to FH-PSK waveformsrdquo in Proceedings of the1994 IEEE MILCOM Part 3 (of 3) pp 1027ndash1031 October 1994

[15] M Pursley and J Skinner ldquoDecoding strategies for turboproduct codes in frequency-hop wireless communicationsrdquo inProceedings of the IEEE International Conference on Communi-cations pp 2963ndash2968 Anchorage AK USA

[16] M B Pursley and J S Skinner ldquoAdaptive coding for frequency-hop transmission in mobile ad hoc networks with partial-bandinterferencerdquo IEEETransactions onCommunications vol 57 no3 pp 801ndash811 2009

[17] M R Masse M B Pursley and J S Skinner ldquoAdaptive codingfor frequency-hop transmission over fading channels withpartial-band interferencerdquo IEEE Transactions on Communica-tions vol 59 no 3 pp 854ndash862 2011

[18] H El Gamal and E Geraniotis ldquoIterative channel estimationand decoding for convolutionally coded anti-jam FH signalsrdquoIEEE Transactions on Communications vol 50 no 2 pp 321ndash331 2002

[19] G D Forney ldquoGeneralized Minimum Distance DecodingrdquoIEEE Transactions on Information Theory vol IT-12 no 2 pp125ndash131 1966

[20] D Chase ldquoA Class of Algorithms for Decoding Block CodeswithChannelMeasurement Informationrdquo IEEETransactions onInformation Theory vol 18 no 1 pp 170ndash182 1972

[21] H Tang Y LiuM Fossorier and S Lin ldquoOn combining Chase-2 and GMD decoding algorithms for nonbinary block codesrdquoIEEE Communications Letters vol 5 no 5 pp 209ndash211 2001

[22] S-W Lee and B V K V Kumar ldquoSoft-decision decoding ofreed-solomon codes using successive error-and-ersure decod-ingrdquo in Proceedings of the 2008 IEEEGlobal TelecommunicationsConference GLOBECOM2008 pp 3065ndash3069USADecember2008

[23] E R Berlekamp ldquoReadable Erasures Improve the Performanceof Reed-Solomon Codesrdquo IEEE Transactions on InformationTheory vol 24 no 5 pp 632-633 1978

[24] J G Proakis andM SalehiEssentials of Communication SystemsEngineering 2005

[25] W G Phoel ldquoIterative demodulation and decoding offrequency-hopped PSK in partial-band jammingrdquo IEEEJournal on Selected Areas in Communications vol 23 no 5 pp1026ndash1033 2005

[26] S Lin and D J Costello Error Control Coding Pearson 2ndedition 2004

[27] J K Wolf A M Michelson and A H Levesque ldquoOn theProbability of Undetected Error for Linear Block Codesrdquo IEEETransactions on Communications vol 30 no 2 pp 317ndash3241982

[28] A Ramesh A Chockalingam and L B Milstein ldquoPerformanceof noncoherent turbo detection on Rayleigh fading channelsrdquoin Proceedings of the IEEE Global Telecommunicatins ConferenceGLOBECOMrsquo01 pp 1193ndash1198 usa November 2001

[29] M C Valenti and S Cheng ldquoIterative demodulation anddecoding of turbo-codedM-ary noncoherent orthogonal mod-ulationrdquo IEEE Journal on Selected Areas in Communications vol23 no 9 pp 1739ndash1747 2005

[30] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press BostonMassUSA 5th edition 1994

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4 Security and Communication Networks

InitializeDecide a vector c by 119888119894 = argmax119895119910119894119895 0 le 119894 le 119873 minus 1Set 119864 = 0

while |119864| le 119873 minus 119870 doTry to error-and-erasure decode cif decoding success then

c is the decoded codewordBreak while loop

else if decoding fail thenCase Measureratio 119894max = argmax119894notin119864max1015840119895119910119894119895max119895119910119894119895output 119894max = argmax119894notin119864119910119894119895sum 119894max = argmax119894notin119864sum119872minus1119895=0 119910119894119895

Erase 119894maxth codeword symbols ie set 119888119894max= 120598119864 larr 119864 cup 119894max

end ifend whileif |119864| le 119873 minus 119870 then

Decoding successOutput = c

else (|119864| = 119873 minus 119870 + 1)Decoding renounce

end if

Algorithm 1 Iterative erasure insertion and GMD decoding

Table 1 Thresholds 119911119901 of the three measures where119872 = 4 119901 = 01 120588 = 01 and 1198641198871198730 = 5 dB over AWGN and partial-band jammingchannels

119864119887119873119895 (dB) minus10 minus5 0 5 10 15 20119911119901 of R 0722 0722 0721 0719 0712 0695 0671119911119901 of O 4123 3625 3158 2462 2103 1993 1961119911119901 of S 4824 4377 4438 3400 2869 2623 2536

runs S-GMD decoding again from the beginning We willshow the performance improvement of RS-GMD decodingas compared to the performance of MO-RTT over variouschannel situations in Section 5 In Table 2 the conventionaliterative and two-dimensional schemes are classified into thefour types of measures

4 Trapped-Error Probability

For the conventional erasure insertion schemes that is RTTOTT and MO-RTT the error and erasure probabilities arederived in [4ndash6] And we have included the error and erasureprobabilities of STT in the Appendix Calculation of theerror and erasure probabilities of a given symbol cannotbe applied to the analysis of iterative schemes because theerasure insertion for one symbol is not independent of thatfor all other symbols For performance analysis of the iterativeschemes we propose trapped-error probability analysis basedon threemeasures ldquoratiordquo ldquooutputrdquo and ldquosumrdquoThe trapped-error probability is a new and useful analysis framework tocompare the quality of each measure

Consider a received word c of length 119873 Let 119885119894 be themeasure of one symbol of c for one of the threemeasures119885119894 =

max1015840119895119910119894119895max119895119910119894119895 119885119894 = max119895119910119894119895 and 119885119894 = sum119872minus1119895=0 119910119894119895for ratio output and sum measures respectively First wesort the received 119873 symbols in descending order accordingto 119885119894 values that are calculated by one selected way (amongthe three kinds of measures) Let 119901 denote the ratio of thewindow of interest to the length 119873 This ratio 119901 should bechosen in the range of 0 lt 119901 le (119873 minus 119870)119873 because the(119873119870) RS code can correct up to 119873 minus 119870 erasures Now wecount the number119873119901 of erroneous symbols in the uppermostportion of 119901119873 symbols Let 119873119905 denote the total number oferroneous symbols in the received word c of length119873 Thenthe trapped-error probability Γ is defined as the ratio between119873119901 and119873119905

Γ = 119873119901119873119905 (8)

For example let us assume that there are 119873 = 100 receivedsymbols with a total of 119873119905 = 15 erroneous symbols Let usassume that 119901 = 01 And we consider the first 10 (=pN)positions when all of the symbols are arranged in decreasingorder of their corresponding measures Let us assume that119873119901 = 8 1 and 5 erroneous symbols are placed among these10 positions for ratio output and summeasures respectively

Security and Communication Networks 5

Then the trapped-error probabilities are ΓR = 815 ΓO =115 and ΓS = 515 for these respective measures Inthis case because the iterative schemes successively erasethe symbols in decreasing order of each measure the ratiomeasure works best for trapping (or identifying) the erro-neous symbols and the output measure works worst For agiven 119901 a larger trapped-error probability indicates betterperformance

Without loss of generality we assume that all-zero code-word w = (0 0) is transmitted that is 1198910 is selected forevery symbol transmission If a transmitted symbol does notinterfere with the jamming signal and the AWGN channel isassumed then the probability density functions (PDFs) of 1199100and 119910119895 119895 isin 1 119872 minus 1 are derived as follows [24]

1198921198800 (119910) = 121205902119873 exp(minus119910 + 121205902119873 ) 1198680 (radic1199101205902119873 )

119892119880119895 (119910) = 121205902119873 exp(minus 11991021205902119873) (9)

For the jammed case

1198921198690 (119910) = 121205902 exp(minus119910 + 121205902 ) 1198680 (radic1199101205902 ) 119892119869119895 (119910) = 121205902 exp(minus 11991021205902 )

(10)

where 1205902 = 1205902119869 + 1205902119873 and 1198680(sdot) is the modified Bessel functionof order zero

If we assume Rayleigh fading channel withoutwithpartial-band jamming 1198921198800 and 1198921198690 are changed as follows[28 29]

1198921198800 (119910) = 121205902119873 + 1 exp(minus11991021205902119873 + 1)

1198921198690 (119910) = 121205902 + 1 exp (minus 11991021205902 + 1) (11)

Because 1198921198800(119910) 119892119880119872minus1(119910) and 1198921198690(119910) 119892119869119872minus1(119910) areindependent variables their joint PDFs can be represented asprod119872minus1119895=0 119892119880119895(119910119895) andprod119872minus1119895=0 119892119869119895(119910119895) respectively Their cumula-tive distribution functions (CDFs) are 119866119880119895(119910119895) and 119866119869119895(119910119895)for 119895 isin 0 119872 minus 1 respectively

Recall that 119885119894 is the measure of the symbol 119888119894 for one ofthe three measures For simplicity we will omit the subscript119894 119885 = max1015840119895119910119895max119895119910119895 119885 = max119895119910119895 or 119885 = sum119872minus1119895=0 119910119895for ratio output or sum measures respectively The CDF issimply denoted by 119865119885(119911) Define 1198670 as a representation ofthe event of a raw symbol error that is the event where thereexists at least one 119895 isin 1 119872minus1 that satisfies 1199100 lt 1199101198951198670rsquoscomplementary event is denoted by1198670 Now we consider the

conditional CDFs 119865119885|119867(119911 | 1198670) and 119865119885|119867(119911 | 1198670) where119867 isthe discrete event variable119867 isin 1198670 1198670 Then

119865119885 (119911) = Pr [1198670] 119865119885|119867 (119911 | 1198670)+ Pr [1198670] 119865119885|119867 (119911 | 1198670)

119891119885 (119911) = Pr [1198670] 119891119885|119867 (119911 | 1198670)+ Pr [1198670] 119891119885|119867 (119911 | 1198670)

(12)

where 119891s denote the corresponding PDFsDefine 119911119901 as the point that satisfies 119865119885(119911119901) = 1 minus 119901119911119901 is uniquely determined because 119865119885(119911) is monotonically

increasing from 0 to 1 Then the trapped-error probabilityΓ(119911119901) can be defined as follows

Γ (119911119901) = intinfin119911119901

119891119885|119867 (119911 | 1198670) 119889119911 = 1 minus 119865119885|119867 (119911119901 | 1198670) (13)

Now we will change (13) into a form involving1198670 insteadof 1198670 such that its computation can be easily performedFrom (12) we observe that

119865119885 (119911119901) = Pr [1198670] 119865119885|119867 (119911119901 | 1198670)+ Pr [1198670] 119865119885|119867 (119911119901 | 1198670) = 1 minus 119901 (14)

Using (14) the trapped-error probability in (13) can bewrittenas

Γ (119911119901) = 1 minus 1 minus 119901 minus Pr [1198670] 119865119885|119867 (119911119901 | 1198670)Pr [1198670] (15)

Using the relation Pr[1198670] + Pr[1198670] = 1 and after somemanipulation we have the following

Γ (119911119901) = 119901 minus Pr [1198670] + Pr [1198670] 119865119885|119867 (119911119901 | 1198670)1 minus Pr [1198670] (16)

Here Pr[1198670] can be calculated as follows

Pr [1198670] = Pr [1199100 gt 119910119895 forall119895 = 1 119872 minus 1]= 120588intinfin0int11991000sdot sdot sdot int11991000

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100+ 120588intinfin0int11991000sdot sdot sdot int11991000

119872minus1prod119895=0

119892119869119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100= 120588intinfin01198921198800 (1199100) (1198661198801 (1199100))119872minus1 1198891199100

+ 120588intinfin01198921198690 (1199100) (1198661198691 (1199100))119872minus1 1198891199100

(17)

where 120588 = 1 minus 120588

6 Security and Communication Networks

Let us assume AWGN and a partial-band jammingchannel Using a binomial expansion and some calculations[30] (17) becomes

120588119890(minus121205902119873) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902119873)119897 (119898 + 1)119897+1 119897 (

119872 minus 1119898 )

+ 120588119890(minus121205902) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902)119897 (119898 + 1)119897+1 119897 (

119872 minus 1119898 )

(18)

Similarly for a Rayleigh fading and partial-band jammingchannel (17) becomes

120588119872minus1sum119898=0

21205902119873(2 + 2119898) 1205902119873 + 119898 (minus1)119898 (119872 minus 1119898 )

+ 120588119872minus1sum119898=0

21205902(2 + 2119898) 1205902 + 119898 (minus1)119898 (119872 minus 1119898 )

(19)

Next we compute 119865119885|119867(119911119901 | 1198670) as follows

119865119885|119867 (119911119901 | 1198670)= Pr [119885 lt 119911119901 1199100 gt 1199101198951015840 forall1198951015840 = 1 119872 minus 1]

Pr [1198670] (20)

Now assume that we use the ratio measure Then thenumerator of (20) becomes

Pr[max1015840119895 119910119895max119895 119910119895 lt 119911119901 1199100 gt 1199101198951015840 forall1198951015840 = 1 119872 minus 1]

= Pr [max1015840119895 119910119895 lt 1199111199011199100]= Pr [119910119895 lt 1199111199011199100 forall119895 = 1 119872 minus 1]= 120588intinfin0int11991111990111991000

sdot sdot sdot int11991111990111991000

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100+ 120588intinfin0int11991111990111991000

sdot sdot sdot int11991111990111991000

119872minus1prod119895=0

119892119869119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100= 120588intinfin01198921198800 (1199100) (1198661198801 (1199111199011199100))(119872minus1) 1198891199100

+ 120588intinfin01198921198690 (1199100) (1198661198691 (1199111199011199100))(119872minus1) 1198891199100

(21)

For AWGN and Rayleigh fading channels with partial-bandjamming (21) becomes (22) and (23) respectively

120588119890(minus121205902119873) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902119873)119897 (119898119911119901 + 1)119897+1 119897 (

119872 minus 1119898 )

+ 120588119890(minus121205902) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902)119897 (119898119911119901 + 1)119897+1 119897 (

119872 minus 1119898 )

(22)

120588119872minus1sum119898=0

21205902119873(2 + 2119898119911119901) 1205902119873 + 119898119911119901 (minus1)119898 (119872 minus 1

119898 )

+ 120588119872minus1sum119898=0

21205902(2 + 2119898119911119901) 1205902 + 119898119911119901 (minus1)119898 (119872 minus 1

119898 ) (23)

For the output measure the numerator of (20) becomes

Pr [max119895 119910119895 lt 119911119901 1199100 gt 1199101198951015840 forall1198951015840 = 1 119872 minus 1]= Pr [1199100 lt 119911119901 1199100 gt 119910119895 forall119895 = 1 119872 minus 1]= 120588int1199111199010int11991000sdot sdot sdot int11991000

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100+ 120588int1199111199010int11991000sdot sdot sdot int11991000

119872minus1prod119895=0

119892119869119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100= 120588int11991111990101198921198800 (1199100) (1198661198801 (1199100))(119872minus1) 1198891199100

+ 120588int11991111990101198921198690 (1199100) (1198661198691 (1199100))(119872minus1) 1198891199100

(24)

With partial-band jamming (24) becomes (25) and (26)for AWGN and Rayleigh fading channels respectively

120588119890(minus121205902119873) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902119873)2119897+1 119897 (

119872 minus 1119898 )

sdot (( 21205902119873119898 + 1)119897+1

minus 119890(minus(119898+1)11991111990121205902119873) 119897sum119899=0

119911119899119901119899 ( 21205902119873119898 + 1)119897+1minus119899)

+ 120588119890(minus121205902) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902)2119897+1 119897 (

119872 minus 1119898 )

sdot (( 21205902119898 + 1)119897+1

minus 119890(minus(119898+1)11991111990121205902) 119897sum119899=0

119911119899119901119899 ( 21205902119898 + 1)119897+1minus119899)

(25)

Security and Communication Networks 7

120588119872minus1sum119898=0

21205902119873(2 + 2119898) 1205902119873 + 119898 (minus1)119898 (119872 minus 1119898 ) sdot (1

minus 119890(minus(((2+2119898)1205902119873+119898)(41205904119873+21205902119873))119911119901))+ 120588119872minus1sum119898=0

21205902(2 + 2119898) 1205902 + 119898 sdot (minus1)119898 (119872 minus 1119898 )(1

minus 119890(minus(((2+2119898)1205902+119898)(41205904+21205902))119911119901))

(26)

For the sum measure the numerator of (20) is given as (27)Actually (27) is the same as (A4) in the Appendix where 120583 =119911119901 Using the derived equations we can calculate the trapped-error probabilities of three measures

Pr[[119872minus1sum119895=0

119910119894119895 lt 119911119901 1199100 gt 119910119895 forall119895 = 1 119872 minus 1]] = Pr[[1199100 lt 119911119901

1199101 lt min (1199100 119911119901 minus 1199100) 119910119872minus1 lt min(1199100 119911119901 minus 119872minus2sum119895=0

119910119895)]]= 120588int1199111199010intmin(1199100 119911119901minus1199100)

0sdot sdot sdot intmin(1199100 119911119901minussum

119872minus2119895=0 119910119895)

0

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1sdot sdot sdot 1198891199100

(27)

+ 120588int1199111199010intmin(1199100 119911119901minus1199100)

0sdot sdot sdot intmin(1199100 119911119901minussum

119872minus2119895=0 119910119895)

0

119872minus1prod119895=0

119892119869119895 (119910119895) 119889119910119872minus1sdot sdot sdot 1198891199100

(28)

5 Simulation Results

51 Trapped-Error Probabilities In this subsection we con-firm the trapped-error probability analysis over AWGN andRayleigh fading channels with partial-band jamming Letus assume 4-FSK 120588 = 01 and 1198641198871198730 = 5 dB for theAWGN channel and 1198641198871198730 = 12 dB for the Rayleigh fadingchannel For the higher order of modulation such as 32-FSK which we will consider (27) requires 32-dimensionalintegration therefore we select 4-FSK as an example Tocalculate the probabilities Γ(119911119901) for each measure the input119911119901 should be obtained for the given set of system parametersFor this we further assume that 119901 = 01 and 02 for AWGNand Rayleigh fading channels respectively As we describedin Section 4 we found 119911119901 values that satisfy 119865119911(119911119901) =1 minus 119901 using simulations They are listed in Tables 1 and 3for various 119864119887119873119895 R O and S refer to ratio output andsum respectively In the investigation we transmitted 105symbols for various channel states and calculated their ratiooutput and sum measures We note that these trapped-errorprobabilities do not depend on a specific coding schemebecause these statistics are observed at the demodulatoroutput without coding

Figures 2 and 3 show the trapped-error probabilities ofthe three measures obtained using simulations and from (16)with the threshold values given in Tables 1 and 3 Solid anddashed lines indicate the probabilities obtained by simulation

Table 2 Class of erasure insertion schemes

Measure Threshold test IterativeRatio RTT R-GMDOutput OTT O-GMDSum STT S-GMDTwo-dimensional Ex MO-RTT Ex RS-GMD

Table 3Thresholds 119911119901 of the three measures where119872 = 4 119901 = 02120588 = 01 and 1198641198871198730 = 12 dB over Rayleigh fading and partial-bandjamming channels

119864119887119873119895 (dB) minus5 0 5 10 15 20119911119901 of R 0353 0345 0350 0329 0294 0263119911119901 of O 2271 2233 2053 1765 1681 1660119911119901 of S 2368 2342 2297 2069 1845 1776

and (16) respectively The cross circle and square marksrepresent Γ(119911119901) values of ratio output and sum measuresrespectively

These two figures confirm that the simulation is exactenough to closely follow the theoretical result in (16) Spe-cially ΓO in the analysis is almost the same as ΓO in thesimulation results This result implies the correctness of ourperformance simulation

In these figures the trapped-error probability of theratio measure is increased as SJR is increased therefore wepredict that the R-GMD decoding will perform better thanthe other one-dimensional schemes at high SJR Additionallywe predict that S-GMD decoding is better than O-GMDdecoding in the middle SJR region Recall that the higherthe trapped-error probability the better the performanceThis observation shows the influence on the two-dimensionalschemes because the trapped-error probability affects thenumber of decoding trials We will discuss it in the followingsimulation results

52 Performance over AWGN Channel Throughout perfor-mance simulations we consider 32-ary FSKmodulation with(31 20) RS code over a field of size 32 an ideal interleaverand a random hopping pattern We first consider partial-band jamming (or partial-time jamming with the same 120588)and AWGN channels In this case 120572 = 1 in Figure 1 ForFigures 4ndash7 the channel noise is fixed as we used 1198641198871198730 =5 dBVarious signals to jamming signal ratio119864119887119873119895 have beenconsidered for the simulations

In Figure 4 the performance of various erasure insertionschemes is plotted over the partial-band jamming channelwith 120588 = 01The dashed line shows the baseline performanceldquoWithout EIrdquo which does not use any erasure insertionschemeThe unmarked solid line represents the performanceof the MO-RTT which is the conventional erasure insertionschemeWenote that theMO-RTTuses preoptimized thresh-olds and CSIR To obtain the optimum thresholds for MO-RTT we optimized the thresholds for 119864119887119873119895 = 0sim30 dB with10 dB steps Solid lines with circle square cross plus anddiamond marks represent the performance of output-based

8 Security and Communication Networks

100 5 15 20minus5minus10

ΓR by simulationΓO by simulationΓS by simulation

ΓR by analysisΓO by analysisΓS by analysis

0

01

02

03

04

05Γ

06

07

08

09

1

EbNj (dB)

Figure 2 Trapped-error probabilities of ratio-based (ΓR) output-based (ΓO) and sum-based (ΓS) measures over partial-band jam-ming and AWGN channel

01

02

03

04

05

06

07

08

09

1

Γ

minus5 0 5 10 15 200

ΓR by simulationΓO by simulationΓS by simulation

ΓR by analysisΓO by analysisΓS by analysis

EbNj (dB)

Figure 3 Trapped-error probabilities of ratio-based (ΓR) output-based (ΓO) and sum-based (ΓS) measures over partial-band jam-ming and Rayleigh channels

sum-based ratio-based ratio-to-output-based and ratio-to-sum-based GMD decoding respectively In this figure wealso present the performance using log-likelihood ratio (LLR)based GMD decoding as the triangle-marked line This isthe performance of a system with perfect CSIR such asSNR SJR and the existence of a jamming signal This LLR-GMD decoding iteratively erases the least reliable symbolsbased on LLR as described in [19 22 26] Because it is wellknown that MO-RTT exhibits better performance than OTT

EbNj (dB)

10minus5

10minus4

10minus3

10minus2

10minus1

100

WER

5 10 15 20 250

Without EIMO-RTTO-GMDS-GMD

R-GMDRO-GMDRS-GMDLLR-GMD

Figure 4 Performance of various erasure insertion schemes overpartial-band jamming and AWGN channels with 120588 = 01 and1198641198871198730 = 5 dB

or RTT [5ndash7] the performances ofOTTandRTTare omittedBefore discussing the simulation result we note that RSO-GMD decoding which serially exploits the three measureshas almost the same performance as RS-GMD decodingtherefore we have also omitted it From Figure 4 we makethe following observations

(1) As we expect LLR-GMD decoding with CSIR showsthe best performance We will focus on the perfor-mance gap between LLR-GMD decoding with CSIRand the other erasure insertion schemes withoutCSIR To achieve WER = 10minus4 LLR-GMD decodingrequires 119864119887119873119895 = 19 dB The ratio-based GMDdecoding and its two-dimensional schemes achievethe target WER at 119864119887119873119895 = 21 dB which is muchcloser than that of MO-RTT Even if the jammingpower is decreased as 119864119887119873119895 = 25 dB the perfor-mance of MO-RTT is still far from WER = 10minus4 Inthe low SJR region only the two two-dimensionaliterative schemes can achieve WER = 10minus2 withoutCSIR We note that the one-dimensional iterativeschemes and MO-RTT require 119864119887119873119895 gt 9 dB toachieve WER = 10minus2

(2) The performance of RS-GMD decoding approachesthat of LLR-GMD decoding as SJR is increased At119864119887119873119895 = 20 dB LLR-GMD decoding with perfectCSIR has 35 less WER than RS-GMD decodingWe note that this gap cannot be closed even if thejamming power is decreased

(3) When the jamming power is decreased that isthe channel approaches the unjammed channel theperformance of R-GMD and RS-GMD decoding is

Security and Communication Networks 9

0

01

02

03

04

05

06

07

08

09

1

Γ

5 10 15 20 25 300

ΓRΓOΓS

EbNj (dB)

Figure 5 Trapped-error probability of various measures overpartial-band jamming and AWGN channels with 119901 = 03 120588 = 01and 1198641198871198730 = 5 dB

still better than the performance of ldquoWithout EIrdquoor MO-RTT This implies that the iterative erasureinsertion and decoding schemes do not decrease theperformance if there are no jamming signals Becauseof this we do not need to detect the jamming signal

In Figure 5 trapped-error probabilities of ratio outputand sum measures are represented which corresponds toFigure 4 To obtain the trapped-error probabilities SJR andSNR were scaled based on the code rate 2031 = 119873119870 whichis different from Section 51 We used 119901 = 03 and the otherparameters are the same as those used in Figure 4 In Figure 5ΓR is much larger than ΓO and ΓS The values of ΓR explain thegood performance of R-GMDdecodingWe find that ΓO goesto lt1 minus119870119873 asymp 03 at 119864119887119873119895 asymp 10 dB Any iterative schemethat has zero trapped-error probability at a given SJR has thesame performance as the performance of ldquoWithout EIrdquo Thisresult explains why O-GMD decoding has a performancedegradation region near 10 dB in Figure 4 Because theperformance of O-GMD decoding rapidly approaches theperformance of ldquoWithout EIrdquo its performance is degraded asSJR is increased We note that the performance of S-GMDdecoding slowly approaches the performance of ldquoWithoutEIrdquo therefore there is no performance degradation region

The performance gain of the iterative schemes in Fig-ure 4 cannot be obtained without drawbacks the iterativeschemes require more decoding trials To investigate howmany decoding trials are required we determine the averagenumber of decoding iterations of various iterative schemesvia simulations The results are shown in Figure 6 In factwe simulated iterative schemes that erase one symbol duringthe first loop and then erase two symbols for each remainingloop because this process does not decrease the performanceas described in [19] In general the average number of

1

11

12

13

14

15

16

17

18

Aver

age n

umbe

r of d

ecod

ing

tria

ls

2 4 6 8 10 12 14 16 18 200

O-GMDS-GMDR-GMD

RS-GMDLLR-GMD

EbNj (dB)

Figure 6 Average number of decoding iterations for various erasureinsertion schemes over partial-band jamming and AWGN channelswith 120588 = 01 and 1198641198871198730 = 5 dB

MO-RTTRS-SEI

10minus3

10minus2

10minus1

WER

02 03 04 05 06 07 08 0901

Figure 7 Performance comparison of MO-RTT and RS-GMDdecoding scheme over partial-band jamming and AWGN channelswith various values of 120588 1198641198871198730 = 5 dB and 119864119887119873119895 = 10 dB

decoding trials decreases rapidly as SJR increases At highSJR most of the received words are completely decoded by asingle trial of error-only decoding which holds true becausehigher SJR implies that the environment becomes less andless hostile For low SJR the average decoding trial of RS-GMD decoding is approximately 175 (at 119864119887119873119895 = 0 dB)which decreases as SJR increasesThis result indicates that theiterative erasure insertion and decoding scheme is practicallyimplementable We note that the trial number of O-GMDdecoding is increased at approximately 119864119887119873119895 asymp 10 dB

10 Security and Communication Networks

and that is consistent with the performance degradation ofO-GMD decoding in Figures 4 and 5 Because O-GMDdecoding could not accurately erase the erroneous symbolsthe decoder has to try more decoding Because of this resultwemust use S-GMDdecoding instead of O-GMDdecodingfor the two-dimensional scheme with R-GMD decodingNow recall the analysis results of Section 51 In Figure 2we observed that the trapped-error probability of the ratiomeasure is increased as SJR is increased and that of the summeasure is decreased For the case in which the trapped-errorprobabilities cross at the point of SJR if we want to havefewer decodings at the higher SJR then we must use R-GMDdecoding first in RS-GMDdecoding If we want to have fewerdecodings at the lower SJR then S-GMD decoding should beapplied first In Figures 4 and 6 because we are consideringthe SJR region in which ΓR gt ΓS we have first exploited R-GMD decoding for RS-GMD decoding

In Figure 7 the performances of MO-RTT and RS-GMDdecoding are shown over a partial-band jamming and anAWGN channel with various jamming duty ratios 120588 withfixed 1198641198871198730 = 5 dB and 119864119887119873119895 = 10 dB For all values of120588 RS-GMD decoding works much better than MO-RTT Wenote that two erasure insertion schemes are more efficient forsmaller 120588 channels53 Performance over Rayleigh Fading Channel Figure 8shows the performance of various erasure insertion schemesover partial-band jamming and Rayleigh fading channel with120588 = 01 and 1198641198871198730 = 12 dB For each symbol transmission120572 in Figure 1 is realized by an iid Rayleigh distribution Weassume that fading coefficients are not known to the receiverexcept for the system of MO-RTT

In Figure 8 RS-GMD decoding exhibits better per-formance than MO-RTT which is the same result as wefound over the AWGN channel For the same WER of 10minus3the jammer should spend 14 dB more 119873119895 The WER ofratio-based schemes (including two-dimensional schemes)approach 10minus4 Unlike the results for an AWGN channelR-GMD decoding is dominant and the contribution of S-GMD decoding is marginal Therefore we conclude that R-GMD decoding is sufficient for Rayleigh fading channelsAdditionally we present the performance of LLR-GMDdecoding which knows perfect CSIR As we discussed inSection 52 the performance of R-SEI approaches that of LLR-SEI and the gap is maintained for every SJR

In Figure 9 the trapped-error probabilities of the mea-sures are displayed over a Rayleigh fading channel with1198641198871198730 = 12 dB The other parameters are the same as thosein Figure 5 In this figure the trapped-error probability of theratio measure is 1 in all SJR regions In other words R-GMDdecoding is the best one-dimensional iterative scheme for thetarget system parameters As we found for the other trapped-error probability results the trapped-error probabilities ofsum and output measures decrease as SJR increases Wherethe trapped-error probabilities are lower than lt1 minus119870119873 asymp03 that is 119864119887119873119895 = 10 dB and 16 dB for output andsum respectively the performances of O-GMD and S-GMDdecoding follow the performance of ldquoWithout EIrdquo As we

10minus5

10minus4

10minus3

10minus2

10minus1

100

WER

5 10 15 20 25 300

Without EIMO-RTTO-GMDS-GMD

R-GMDRO-GMDRS-GMDLLR-GMD

EbNj (dB)

Figure 8 Performance of various erasure insertion schemes over aRayleigh fading channel where 120588 = 01 and 1198641198871198730 = 12 dB

0

01

02

03

04

05

06

07

08

09

1

5 10 15 20 25 300

Γ

ΓRΓOΓS

EbNj (dB)

Figure 9 Trapped-error probability of various iterative schemesover partial-band jamming and Rayleigh fading channels with 119901 =03 120588 = 01 and 1198641198871198730 = 12 dB

discussed above the fast decreasing trapped-error probabilityof the output measure causes the performance degradation ofO-GMD decoding at approximately 10 dB in Figure 9

6 Concluding Remarks

In this paper we considered iterative erasure insertion anddecoding schemes that do not require any preoptimizedthresholds or any CSIR Additionally we proposed a new

Security and Communication Networks 11

analysis framework for the ratio output and sum mea-sures From the simulation results we confirmed that theratio-based GMD decoding scheme and its two-dimensionalschemes have the best performance Using the trapped-error probability the performances of the iterative erasureinsertion and decoding schemes are explained

Appendix

Error and Erasure Probability of SumThreshold Test

Let us denote the symbol error and erasure probabilities ofSTT as 119875err and 119875ers respectively The symbol error eventoccurs when a received symbol is erroneous but is not erasedby STT The symbol erasure event occurs when a receivedsymbol is erased by STTThe two probabilities can be dividedinto two cases as follows

119875err = (1 minus 120588) 119875119880err + 120588119875119869err119875ers = (1 minus 120588) 119875119880ers + 120588119875119869ers (A1)

where 119869 and 119880 refer to the jammed and the unjammed casesrespectively We denote the symbol erase event of STT by119867119904that is 119911 = sum119872minus1119895=0 119910119895 gt 120583 Then 119875119880err can be represented asfollows

119875119880err = Pr [1198670 119867119904 | 119880]= Pr [119867119904 | 119880] minus Pr [1198670 119867119904 | 119880] (A2)

The first term Pr[119867119904 | 119880] and the second term Pr[1198670 119867119904 |119880] are given as (A3) and (A4) respectively Similarly119875119880ers = Pr[119867119904 | 119880] = 1 minus Pr[119867119904 | 119880] can be obtained using(A3) By replacing 119892119880119895 with 119892119869119895 119895 = 0 119872 minus 1 we cancalculate the error and erase probabilities of the unjammedcase For the case in which a jamming signal exists 119875119869err and119875119869ers can be calculated by substituting 119892119869119895(119910119895) for 119892119880119895(119910119895)119895 = 0 119872 minus 1 The derivation of average error probability119875err and erasure probability 119875ers is complete

Pr [119867119904 | 119880] = Pr [119911 lt 120583 | 119880] = Pr[[119872minus1sum119895=0

119910119895 lt 120583]] = int1205830int120583minus11991000

sdot sdot sdot int120583minussum119872minus2119895=0 1199101198950

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 11988911991011198891199100 (A3)

Pr [1198670 119867119904 | 119880] = Pr[[1199100 gt 119910119895 forall119895 = 1 119872 minus 1 119872minus1sum119895=0

119910119895 lt 120583]]= int1205830intmin(1199100 120583minus1199100)

0sdot sdot sdot intmin(1199100 120583minussum

119872minus2119895=0 119910119895)

0

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 11988911991011198891199100(A4)

Disclosure

The present address of Jinsoo Park is the Agency for DefenseDevelopment Daejeon Republic of Korea

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors gratefully acknowledge the support from Elec-tronic Warfare Research Center at the Gwangju Institute ofScience and Technology (GIST) originally funded by theDefense Acquisition Program Administration (DAPA) andthe Agency for Defense Development (ADD)

References

[1] L Milstein and M Simon ldquoSpread Spectrum Communi-cationsrdquo in The Mobile Communications Handbook SecondEdition vol 19990749 of Electrical Engineering Handbook CRCPress 1999

[2] C W Baum and M B Pursley ldquoBayesian Methods for ErasureInsertion in Frequency-Hop Communication Systems withPartial-Band Interferencerdquo IEEE Transactions on Communica-tions vol 40 no 7 pp 1231ndash1238 1992

[3] L-L Yang and L Hanzo ldquoPerformance analysis of codedM-aryorthogonal signaling using errors-and-erasures decoding overfrequency-selective fading channelsrdquo IEEE Journal on SelectedAreas in Communications vol 19 no 2 pp 211ndash221 2001

[4] A J Viterbi ldquoA Robust Ratio-Threshold Technique to MitigateTone and Partial Band Jamming in Coded MFSK Systemsrdquo inProceedings of the IEEE Military Communications ConferenceMILCOM 1982 pp 224-1ndash224-5 Boston MA October 1982

[5] L-L Yang and L Hanzo ldquoLow complexity erasure insertion inRS-coded SFH spread-spectrum communications with partial-band interference and nakagami-m fadingrdquo IEEE Transactionson Communications vol 50 no 6 pp 914ndash925 2002

[6] S Ahmed L-L Yang and L Hanzo ldquoErasure insertion in RS-coded SFH MFSK subjected to tone jamming and Rayleighfadingrdquo IEEE Transactions on Vehicular Technology vol 56 no6 I pp 3563ndash3571 2007

[7] H A Ngo S Ahmed L-L Yang and L Hanzo ldquoNon-coherent cooperative communications dispensing with channelestimation relying on erasure insertion aided Reed-Solomon

12 Security and Communication Networks

coded SFH M-ary FSK subjected to partial-band interferenceand Rayleigh fadingrdquo IEEE Transactions on Communicationsvol 60 no 8 pp 2177ndash2186 2012

[8] T G Macdonald and M B Pursley ldquoStaggered interleavingand iterative errors-and-erasures decoding for frequency-hoppacket radiordquo IEEE Transactions on Wireless Communicationsvol 2 no 1 pp 92ndash98 2003

[9] C-M Lee Y T Su and L-D Jeng ldquoPerformance analysis ofblock codes in hidden Markov channelsrdquo IEEE Transactions onCommunications vol 56 no 1 pp 1ndash4 2008

[10] A Aung K C Teh and K H Li ldquoDetection and counter-measure of interference in slow FHMFSK systems over fadingchannelsrdquo Physical Communication vol 10 pp 11ndash23 2014

[11] J G Kim ldquoError and erasure decoding scheme for RS codes inMFSK based FHSSMA communicationrdquo International Journalof Multimedia and Ubiquitous Engineering vol 10 no 10 pp11ndash20 2015

[12] R Koetter and A Vardy ldquoAlgebraic soft-decision decodingof Reed-Solomon codesrdquo Institute of Electrical and ElectronicsEngineers Transactions on InformationTheory vol 49 no 11 pp2809ndash2825 2003

[13] J Jiang and K R Narayanan ldquoIterative soft decoding of Reed-Solomon codesrdquo IEEE Communications Letters vol 8 no 4 pp244ndash246 2004

[14] P Sehier P Brelivet and I Aubry ldquoNew jammer estimationmethod adapted to FH-PSK waveformsrdquo in Proceedings of the1994 IEEE MILCOM Part 3 (of 3) pp 1027ndash1031 October 1994

[15] M Pursley and J Skinner ldquoDecoding strategies for turboproduct codes in frequency-hop wireless communicationsrdquo inProceedings of the IEEE International Conference on Communi-cations pp 2963ndash2968 Anchorage AK USA

[16] M B Pursley and J S Skinner ldquoAdaptive coding for frequency-hop transmission in mobile ad hoc networks with partial-bandinterferencerdquo IEEETransactions onCommunications vol 57 no3 pp 801ndash811 2009

[17] M R Masse M B Pursley and J S Skinner ldquoAdaptive codingfor frequency-hop transmission over fading channels withpartial-band interferencerdquo IEEE Transactions on Communica-tions vol 59 no 3 pp 854ndash862 2011

[18] H El Gamal and E Geraniotis ldquoIterative channel estimationand decoding for convolutionally coded anti-jam FH signalsrdquoIEEE Transactions on Communications vol 50 no 2 pp 321ndash331 2002

[19] G D Forney ldquoGeneralized Minimum Distance DecodingrdquoIEEE Transactions on Information Theory vol IT-12 no 2 pp125ndash131 1966

[20] D Chase ldquoA Class of Algorithms for Decoding Block CodeswithChannelMeasurement Informationrdquo IEEETransactions onInformation Theory vol 18 no 1 pp 170ndash182 1972

[21] H Tang Y LiuM Fossorier and S Lin ldquoOn combining Chase-2 and GMD decoding algorithms for nonbinary block codesrdquoIEEE Communications Letters vol 5 no 5 pp 209ndash211 2001

[22] S-W Lee and B V K V Kumar ldquoSoft-decision decoding ofreed-solomon codes using successive error-and-ersure decod-ingrdquo in Proceedings of the 2008 IEEEGlobal TelecommunicationsConference GLOBECOM2008 pp 3065ndash3069USADecember2008

[23] E R Berlekamp ldquoReadable Erasures Improve the Performanceof Reed-Solomon Codesrdquo IEEE Transactions on InformationTheory vol 24 no 5 pp 632-633 1978

[24] J G Proakis andM SalehiEssentials of Communication SystemsEngineering 2005

[25] W G Phoel ldquoIterative demodulation and decoding offrequency-hopped PSK in partial-band jammingrdquo IEEEJournal on Selected Areas in Communications vol 23 no 5 pp1026ndash1033 2005

[26] S Lin and D J Costello Error Control Coding Pearson 2ndedition 2004

[27] J K Wolf A M Michelson and A H Levesque ldquoOn theProbability of Undetected Error for Linear Block Codesrdquo IEEETransactions on Communications vol 30 no 2 pp 317ndash3241982

[28] A Ramesh A Chockalingam and L B Milstein ldquoPerformanceof noncoherent turbo detection on Rayleigh fading channelsrdquoin Proceedings of the IEEE Global Telecommunicatins ConferenceGLOBECOMrsquo01 pp 1193ndash1198 usa November 2001

[29] M C Valenti and S Cheng ldquoIterative demodulation anddecoding of turbo-codedM-ary noncoherent orthogonal mod-ulationrdquo IEEE Journal on Selected Areas in Communications vol23 no 9 pp 1739ndash1747 2005

[30] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press BostonMassUSA 5th edition 1994

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Security and Communication Networks 5

Then the trapped-error probabilities are ΓR = 815 ΓO =115 and ΓS = 515 for these respective measures Inthis case because the iterative schemes successively erasethe symbols in decreasing order of each measure the ratiomeasure works best for trapping (or identifying) the erro-neous symbols and the output measure works worst For agiven 119901 a larger trapped-error probability indicates betterperformance

Without loss of generality we assume that all-zero code-word w = (0 0) is transmitted that is 1198910 is selected forevery symbol transmission If a transmitted symbol does notinterfere with the jamming signal and the AWGN channel isassumed then the probability density functions (PDFs) of 1199100and 119910119895 119895 isin 1 119872 minus 1 are derived as follows [24]

1198921198800 (119910) = 121205902119873 exp(minus119910 + 121205902119873 ) 1198680 (radic1199101205902119873 )

119892119880119895 (119910) = 121205902119873 exp(minus 11991021205902119873) (9)

For the jammed case

1198921198690 (119910) = 121205902 exp(minus119910 + 121205902 ) 1198680 (radic1199101205902 ) 119892119869119895 (119910) = 121205902 exp(minus 11991021205902 )

(10)

where 1205902 = 1205902119869 + 1205902119873 and 1198680(sdot) is the modified Bessel functionof order zero

If we assume Rayleigh fading channel withoutwithpartial-band jamming 1198921198800 and 1198921198690 are changed as follows[28 29]

1198921198800 (119910) = 121205902119873 + 1 exp(minus11991021205902119873 + 1)

1198921198690 (119910) = 121205902 + 1 exp (minus 11991021205902 + 1) (11)

Because 1198921198800(119910) 119892119880119872minus1(119910) and 1198921198690(119910) 119892119869119872minus1(119910) areindependent variables their joint PDFs can be represented asprod119872minus1119895=0 119892119880119895(119910119895) andprod119872minus1119895=0 119892119869119895(119910119895) respectively Their cumula-tive distribution functions (CDFs) are 119866119880119895(119910119895) and 119866119869119895(119910119895)for 119895 isin 0 119872 minus 1 respectively

Recall that 119885119894 is the measure of the symbol 119888119894 for one ofthe three measures For simplicity we will omit the subscript119894 119885 = max1015840119895119910119895max119895119910119895 119885 = max119895119910119895 or 119885 = sum119872minus1119895=0 119910119895for ratio output or sum measures respectively The CDF issimply denoted by 119865119885(119911) Define 1198670 as a representation ofthe event of a raw symbol error that is the event where thereexists at least one 119895 isin 1 119872minus1 that satisfies 1199100 lt 1199101198951198670rsquoscomplementary event is denoted by1198670 Now we consider the

conditional CDFs 119865119885|119867(119911 | 1198670) and 119865119885|119867(119911 | 1198670) where119867 isthe discrete event variable119867 isin 1198670 1198670 Then

119865119885 (119911) = Pr [1198670] 119865119885|119867 (119911 | 1198670)+ Pr [1198670] 119865119885|119867 (119911 | 1198670)

119891119885 (119911) = Pr [1198670] 119891119885|119867 (119911 | 1198670)+ Pr [1198670] 119891119885|119867 (119911 | 1198670)

(12)

where 119891s denote the corresponding PDFsDefine 119911119901 as the point that satisfies 119865119885(119911119901) = 1 minus 119901119911119901 is uniquely determined because 119865119885(119911) is monotonically

increasing from 0 to 1 Then the trapped-error probabilityΓ(119911119901) can be defined as follows

Γ (119911119901) = intinfin119911119901

119891119885|119867 (119911 | 1198670) 119889119911 = 1 minus 119865119885|119867 (119911119901 | 1198670) (13)

Now we will change (13) into a form involving1198670 insteadof 1198670 such that its computation can be easily performedFrom (12) we observe that

119865119885 (119911119901) = Pr [1198670] 119865119885|119867 (119911119901 | 1198670)+ Pr [1198670] 119865119885|119867 (119911119901 | 1198670) = 1 minus 119901 (14)

Using (14) the trapped-error probability in (13) can bewrittenas

Γ (119911119901) = 1 minus 1 minus 119901 minus Pr [1198670] 119865119885|119867 (119911119901 | 1198670)Pr [1198670] (15)

Using the relation Pr[1198670] + Pr[1198670] = 1 and after somemanipulation we have the following

Γ (119911119901) = 119901 minus Pr [1198670] + Pr [1198670] 119865119885|119867 (119911119901 | 1198670)1 minus Pr [1198670] (16)

Here Pr[1198670] can be calculated as follows

Pr [1198670] = Pr [1199100 gt 119910119895 forall119895 = 1 119872 minus 1]= 120588intinfin0int11991000sdot sdot sdot int11991000

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100+ 120588intinfin0int11991000sdot sdot sdot int11991000

119872minus1prod119895=0

119892119869119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100= 120588intinfin01198921198800 (1199100) (1198661198801 (1199100))119872minus1 1198891199100

+ 120588intinfin01198921198690 (1199100) (1198661198691 (1199100))119872minus1 1198891199100

(17)

where 120588 = 1 minus 120588

6 Security and Communication Networks

Let us assume AWGN and a partial-band jammingchannel Using a binomial expansion and some calculations[30] (17) becomes

120588119890(minus121205902119873) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902119873)119897 (119898 + 1)119897+1 119897 (

119872 minus 1119898 )

+ 120588119890(minus121205902) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902)119897 (119898 + 1)119897+1 119897 (

119872 minus 1119898 )

(18)

Similarly for a Rayleigh fading and partial-band jammingchannel (17) becomes

120588119872minus1sum119898=0

21205902119873(2 + 2119898) 1205902119873 + 119898 (minus1)119898 (119872 minus 1119898 )

+ 120588119872minus1sum119898=0

21205902(2 + 2119898) 1205902 + 119898 (minus1)119898 (119872 minus 1119898 )

(19)

Next we compute 119865119885|119867(119911119901 | 1198670) as follows

119865119885|119867 (119911119901 | 1198670)= Pr [119885 lt 119911119901 1199100 gt 1199101198951015840 forall1198951015840 = 1 119872 minus 1]

Pr [1198670] (20)

Now assume that we use the ratio measure Then thenumerator of (20) becomes

Pr[max1015840119895 119910119895max119895 119910119895 lt 119911119901 1199100 gt 1199101198951015840 forall1198951015840 = 1 119872 minus 1]

= Pr [max1015840119895 119910119895 lt 1199111199011199100]= Pr [119910119895 lt 1199111199011199100 forall119895 = 1 119872 minus 1]= 120588intinfin0int11991111990111991000

sdot sdot sdot int11991111990111991000

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100+ 120588intinfin0int11991111990111991000

sdot sdot sdot int11991111990111991000

119872minus1prod119895=0

119892119869119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100= 120588intinfin01198921198800 (1199100) (1198661198801 (1199111199011199100))(119872minus1) 1198891199100

+ 120588intinfin01198921198690 (1199100) (1198661198691 (1199111199011199100))(119872minus1) 1198891199100

(21)

For AWGN and Rayleigh fading channels with partial-bandjamming (21) becomes (22) and (23) respectively

120588119890(minus121205902119873) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902119873)119897 (119898119911119901 + 1)119897+1 119897 (

119872 minus 1119898 )

+ 120588119890(minus121205902) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902)119897 (119898119911119901 + 1)119897+1 119897 (

119872 minus 1119898 )

(22)

120588119872minus1sum119898=0

21205902119873(2 + 2119898119911119901) 1205902119873 + 119898119911119901 (minus1)119898 (119872 minus 1

119898 )

+ 120588119872minus1sum119898=0

21205902(2 + 2119898119911119901) 1205902 + 119898119911119901 (minus1)119898 (119872 minus 1

119898 ) (23)

For the output measure the numerator of (20) becomes

Pr [max119895 119910119895 lt 119911119901 1199100 gt 1199101198951015840 forall1198951015840 = 1 119872 minus 1]= Pr [1199100 lt 119911119901 1199100 gt 119910119895 forall119895 = 1 119872 minus 1]= 120588int1199111199010int11991000sdot sdot sdot int11991000

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100+ 120588int1199111199010int11991000sdot sdot sdot int11991000

119872minus1prod119895=0

119892119869119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100= 120588int11991111990101198921198800 (1199100) (1198661198801 (1199100))(119872minus1) 1198891199100

+ 120588int11991111990101198921198690 (1199100) (1198661198691 (1199100))(119872minus1) 1198891199100

(24)

With partial-band jamming (24) becomes (25) and (26)for AWGN and Rayleigh fading channels respectively

120588119890(minus121205902119873) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902119873)2119897+1 119897 (

119872 minus 1119898 )

sdot (( 21205902119873119898 + 1)119897+1

minus 119890(minus(119898+1)11991111990121205902119873) 119897sum119899=0

119911119899119901119899 ( 21205902119873119898 + 1)119897+1minus119899)

+ 120588119890(minus121205902) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902)2119897+1 119897 (

119872 minus 1119898 )

sdot (( 21205902119898 + 1)119897+1

minus 119890(minus(119898+1)11991111990121205902) 119897sum119899=0

119911119899119901119899 ( 21205902119898 + 1)119897+1minus119899)

(25)

Security and Communication Networks 7

120588119872minus1sum119898=0

21205902119873(2 + 2119898) 1205902119873 + 119898 (minus1)119898 (119872 minus 1119898 ) sdot (1

minus 119890(minus(((2+2119898)1205902119873+119898)(41205904119873+21205902119873))119911119901))+ 120588119872minus1sum119898=0

21205902(2 + 2119898) 1205902 + 119898 sdot (minus1)119898 (119872 minus 1119898 )(1

minus 119890(minus(((2+2119898)1205902+119898)(41205904+21205902))119911119901))

(26)

For the sum measure the numerator of (20) is given as (27)Actually (27) is the same as (A4) in the Appendix where 120583 =119911119901 Using the derived equations we can calculate the trapped-error probabilities of three measures

Pr[[119872minus1sum119895=0

119910119894119895 lt 119911119901 1199100 gt 119910119895 forall119895 = 1 119872 minus 1]] = Pr[[1199100 lt 119911119901

1199101 lt min (1199100 119911119901 minus 1199100) 119910119872minus1 lt min(1199100 119911119901 minus 119872minus2sum119895=0

119910119895)]]= 120588int1199111199010intmin(1199100 119911119901minus1199100)

0sdot sdot sdot intmin(1199100 119911119901minussum

119872minus2119895=0 119910119895)

0

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1sdot sdot sdot 1198891199100

(27)

+ 120588int1199111199010intmin(1199100 119911119901minus1199100)

0sdot sdot sdot intmin(1199100 119911119901minussum

119872minus2119895=0 119910119895)

0

119872minus1prod119895=0

119892119869119895 (119910119895) 119889119910119872minus1sdot sdot sdot 1198891199100

(28)

5 Simulation Results

51 Trapped-Error Probabilities In this subsection we con-firm the trapped-error probability analysis over AWGN andRayleigh fading channels with partial-band jamming Letus assume 4-FSK 120588 = 01 and 1198641198871198730 = 5 dB for theAWGN channel and 1198641198871198730 = 12 dB for the Rayleigh fadingchannel For the higher order of modulation such as 32-FSK which we will consider (27) requires 32-dimensionalintegration therefore we select 4-FSK as an example Tocalculate the probabilities Γ(119911119901) for each measure the input119911119901 should be obtained for the given set of system parametersFor this we further assume that 119901 = 01 and 02 for AWGNand Rayleigh fading channels respectively As we describedin Section 4 we found 119911119901 values that satisfy 119865119911(119911119901) =1 minus 119901 using simulations They are listed in Tables 1 and 3for various 119864119887119873119895 R O and S refer to ratio output andsum respectively In the investigation we transmitted 105symbols for various channel states and calculated their ratiooutput and sum measures We note that these trapped-errorprobabilities do not depend on a specific coding schemebecause these statistics are observed at the demodulatoroutput without coding

Figures 2 and 3 show the trapped-error probabilities ofthe three measures obtained using simulations and from (16)with the threshold values given in Tables 1 and 3 Solid anddashed lines indicate the probabilities obtained by simulation

Table 2 Class of erasure insertion schemes

Measure Threshold test IterativeRatio RTT R-GMDOutput OTT O-GMDSum STT S-GMDTwo-dimensional Ex MO-RTT Ex RS-GMD

Table 3Thresholds 119911119901 of the three measures where119872 = 4 119901 = 02120588 = 01 and 1198641198871198730 = 12 dB over Rayleigh fading and partial-bandjamming channels

119864119887119873119895 (dB) minus5 0 5 10 15 20119911119901 of R 0353 0345 0350 0329 0294 0263119911119901 of O 2271 2233 2053 1765 1681 1660119911119901 of S 2368 2342 2297 2069 1845 1776

and (16) respectively The cross circle and square marksrepresent Γ(119911119901) values of ratio output and sum measuresrespectively

These two figures confirm that the simulation is exactenough to closely follow the theoretical result in (16) Spe-cially ΓO in the analysis is almost the same as ΓO in thesimulation results This result implies the correctness of ourperformance simulation

In these figures the trapped-error probability of theratio measure is increased as SJR is increased therefore wepredict that the R-GMD decoding will perform better thanthe other one-dimensional schemes at high SJR Additionallywe predict that S-GMD decoding is better than O-GMDdecoding in the middle SJR region Recall that the higherthe trapped-error probability the better the performanceThis observation shows the influence on the two-dimensionalschemes because the trapped-error probability affects thenumber of decoding trials We will discuss it in the followingsimulation results

52 Performance over AWGN Channel Throughout perfor-mance simulations we consider 32-ary FSKmodulation with(31 20) RS code over a field of size 32 an ideal interleaverand a random hopping pattern We first consider partial-band jamming (or partial-time jamming with the same 120588)and AWGN channels In this case 120572 = 1 in Figure 1 ForFigures 4ndash7 the channel noise is fixed as we used 1198641198871198730 =5 dBVarious signals to jamming signal ratio119864119887119873119895 have beenconsidered for the simulations

In Figure 4 the performance of various erasure insertionschemes is plotted over the partial-band jamming channelwith 120588 = 01The dashed line shows the baseline performanceldquoWithout EIrdquo which does not use any erasure insertionschemeThe unmarked solid line represents the performanceof the MO-RTT which is the conventional erasure insertionschemeWenote that theMO-RTTuses preoptimized thresh-olds and CSIR To obtain the optimum thresholds for MO-RTT we optimized the thresholds for 119864119887119873119895 = 0sim30 dB with10 dB steps Solid lines with circle square cross plus anddiamond marks represent the performance of output-based

8 Security and Communication Networks

100 5 15 20minus5minus10

ΓR by simulationΓO by simulationΓS by simulation

ΓR by analysisΓO by analysisΓS by analysis

0

01

02

03

04

05Γ

06

07

08

09

1

EbNj (dB)

Figure 2 Trapped-error probabilities of ratio-based (ΓR) output-based (ΓO) and sum-based (ΓS) measures over partial-band jam-ming and AWGN channel

01

02

03

04

05

06

07

08

09

1

Γ

minus5 0 5 10 15 200

ΓR by simulationΓO by simulationΓS by simulation

ΓR by analysisΓO by analysisΓS by analysis

EbNj (dB)

Figure 3 Trapped-error probabilities of ratio-based (ΓR) output-based (ΓO) and sum-based (ΓS) measures over partial-band jam-ming and Rayleigh channels

sum-based ratio-based ratio-to-output-based and ratio-to-sum-based GMD decoding respectively In this figure wealso present the performance using log-likelihood ratio (LLR)based GMD decoding as the triangle-marked line This isthe performance of a system with perfect CSIR such asSNR SJR and the existence of a jamming signal This LLR-GMD decoding iteratively erases the least reliable symbolsbased on LLR as described in [19 22 26] Because it is wellknown that MO-RTT exhibits better performance than OTT

EbNj (dB)

10minus5

10minus4

10minus3

10minus2

10minus1

100

WER

5 10 15 20 250

Without EIMO-RTTO-GMDS-GMD

R-GMDRO-GMDRS-GMDLLR-GMD

Figure 4 Performance of various erasure insertion schemes overpartial-band jamming and AWGN channels with 120588 = 01 and1198641198871198730 = 5 dB

or RTT [5ndash7] the performances ofOTTandRTTare omittedBefore discussing the simulation result we note that RSO-GMD decoding which serially exploits the three measureshas almost the same performance as RS-GMD decodingtherefore we have also omitted it From Figure 4 we makethe following observations

(1) As we expect LLR-GMD decoding with CSIR showsthe best performance We will focus on the perfor-mance gap between LLR-GMD decoding with CSIRand the other erasure insertion schemes withoutCSIR To achieve WER = 10minus4 LLR-GMD decodingrequires 119864119887119873119895 = 19 dB The ratio-based GMDdecoding and its two-dimensional schemes achievethe target WER at 119864119887119873119895 = 21 dB which is muchcloser than that of MO-RTT Even if the jammingpower is decreased as 119864119887119873119895 = 25 dB the perfor-mance of MO-RTT is still far from WER = 10minus4 Inthe low SJR region only the two two-dimensionaliterative schemes can achieve WER = 10minus2 withoutCSIR We note that the one-dimensional iterativeschemes and MO-RTT require 119864119887119873119895 gt 9 dB toachieve WER = 10minus2

(2) The performance of RS-GMD decoding approachesthat of LLR-GMD decoding as SJR is increased At119864119887119873119895 = 20 dB LLR-GMD decoding with perfectCSIR has 35 less WER than RS-GMD decodingWe note that this gap cannot be closed even if thejamming power is decreased

(3) When the jamming power is decreased that isthe channel approaches the unjammed channel theperformance of R-GMD and RS-GMD decoding is

Security and Communication Networks 9

0

01

02

03

04

05

06

07

08

09

1

Γ

5 10 15 20 25 300

ΓRΓOΓS

EbNj (dB)

Figure 5 Trapped-error probability of various measures overpartial-band jamming and AWGN channels with 119901 = 03 120588 = 01and 1198641198871198730 = 5 dB

still better than the performance of ldquoWithout EIrdquoor MO-RTT This implies that the iterative erasureinsertion and decoding schemes do not decrease theperformance if there are no jamming signals Becauseof this we do not need to detect the jamming signal

In Figure 5 trapped-error probabilities of ratio outputand sum measures are represented which corresponds toFigure 4 To obtain the trapped-error probabilities SJR andSNR were scaled based on the code rate 2031 = 119873119870 whichis different from Section 51 We used 119901 = 03 and the otherparameters are the same as those used in Figure 4 In Figure 5ΓR is much larger than ΓO and ΓS The values of ΓR explain thegood performance of R-GMDdecodingWe find that ΓO goesto lt1 minus119870119873 asymp 03 at 119864119887119873119895 asymp 10 dB Any iterative schemethat has zero trapped-error probability at a given SJR has thesame performance as the performance of ldquoWithout EIrdquo Thisresult explains why O-GMD decoding has a performancedegradation region near 10 dB in Figure 4 Because theperformance of O-GMD decoding rapidly approaches theperformance of ldquoWithout EIrdquo its performance is degraded asSJR is increased We note that the performance of S-GMDdecoding slowly approaches the performance of ldquoWithoutEIrdquo therefore there is no performance degradation region

The performance gain of the iterative schemes in Fig-ure 4 cannot be obtained without drawbacks the iterativeschemes require more decoding trials To investigate howmany decoding trials are required we determine the averagenumber of decoding iterations of various iterative schemesvia simulations The results are shown in Figure 6 In factwe simulated iterative schemes that erase one symbol duringthe first loop and then erase two symbols for each remainingloop because this process does not decrease the performanceas described in [19] In general the average number of

1

11

12

13

14

15

16

17

18

Aver

age n

umbe

r of d

ecod

ing

tria

ls

2 4 6 8 10 12 14 16 18 200

O-GMDS-GMDR-GMD

RS-GMDLLR-GMD

EbNj (dB)

Figure 6 Average number of decoding iterations for various erasureinsertion schemes over partial-band jamming and AWGN channelswith 120588 = 01 and 1198641198871198730 = 5 dB

MO-RTTRS-SEI

10minus3

10minus2

10minus1

WER

02 03 04 05 06 07 08 0901

Figure 7 Performance comparison of MO-RTT and RS-GMDdecoding scheme over partial-band jamming and AWGN channelswith various values of 120588 1198641198871198730 = 5 dB and 119864119887119873119895 = 10 dB

decoding trials decreases rapidly as SJR increases At highSJR most of the received words are completely decoded by asingle trial of error-only decoding which holds true becausehigher SJR implies that the environment becomes less andless hostile For low SJR the average decoding trial of RS-GMD decoding is approximately 175 (at 119864119887119873119895 = 0 dB)which decreases as SJR increasesThis result indicates that theiterative erasure insertion and decoding scheme is practicallyimplementable We note that the trial number of O-GMDdecoding is increased at approximately 119864119887119873119895 asymp 10 dB

10 Security and Communication Networks

and that is consistent with the performance degradation ofO-GMD decoding in Figures 4 and 5 Because O-GMDdecoding could not accurately erase the erroneous symbolsthe decoder has to try more decoding Because of this resultwemust use S-GMDdecoding instead of O-GMDdecodingfor the two-dimensional scheme with R-GMD decodingNow recall the analysis results of Section 51 In Figure 2we observed that the trapped-error probability of the ratiomeasure is increased as SJR is increased and that of the summeasure is decreased For the case in which the trapped-errorprobabilities cross at the point of SJR if we want to havefewer decodings at the higher SJR then we must use R-GMDdecoding first in RS-GMDdecoding If we want to have fewerdecodings at the lower SJR then S-GMD decoding should beapplied first In Figures 4 and 6 because we are consideringthe SJR region in which ΓR gt ΓS we have first exploited R-GMD decoding for RS-GMD decoding

In Figure 7 the performances of MO-RTT and RS-GMDdecoding are shown over a partial-band jamming and anAWGN channel with various jamming duty ratios 120588 withfixed 1198641198871198730 = 5 dB and 119864119887119873119895 = 10 dB For all values of120588 RS-GMD decoding works much better than MO-RTT Wenote that two erasure insertion schemes are more efficient forsmaller 120588 channels53 Performance over Rayleigh Fading Channel Figure 8shows the performance of various erasure insertion schemesover partial-band jamming and Rayleigh fading channel with120588 = 01 and 1198641198871198730 = 12 dB For each symbol transmission120572 in Figure 1 is realized by an iid Rayleigh distribution Weassume that fading coefficients are not known to the receiverexcept for the system of MO-RTT

In Figure 8 RS-GMD decoding exhibits better per-formance than MO-RTT which is the same result as wefound over the AWGN channel For the same WER of 10minus3the jammer should spend 14 dB more 119873119895 The WER ofratio-based schemes (including two-dimensional schemes)approach 10minus4 Unlike the results for an AWGN channelR-GMD decoding is dominant and the contribution of S-GMD decoding is marginal Therefore we conclude that R-GMD decoding is sufficient for Rayleigh fading channelsAdditionally we present the performance of LLR-GMDdecoding which knows perfect CSIR As we discussed inSection 52 the performance of R-SEI approaches that of LLR-SEI and the gap is maintained for every SJR

In Figure 9 the trapped-error probabilities of the mea-sures are displayed over a Rayleigh fading channel with1198641198871198730 = 12 dB The other parameters are the same as thosein Figure 5 In this figure the trapped-error probability of theratio measure is 1 in all SJR regions In other words R-GMDdecoding is the best one-dimensional iterative scheme for thetarget system parameters As we found for the other trapped-error probability results the trapped-error probabilities ofsum and output measures decrease as SJR increases Wherethe trapped-error probabilities are lower than lt1 minus119870119873 asymp03 that is 119864119887119873119895 = 10 dB and 16 dB for output andsum respectively the performances of O-GMD and S-GMDdecoding follow the performance of ldquoWithout EIrdquo As we

10minus5

10minus4

10minus3

10minus2

10minus1

100

WER

5 10 15 20 25 300

Without EIMO-RTTO-GMDS-GMD

R-GMDRO-GMDRS-GMDLLR-GMD

EbNj (dB)

Figure 8 Performance of various erasure insertion schemes over aRayleigh fading channel where 120588 = 01 and 1198641198871198730 = 12 dB

0

01

02

03

04

05

06

07

08

09

1

5 10 15 20 25 300

Γ

ΓRΓOΓS

EbNj (dB)

Figure 9 Trapped-error probability of various iterative schemesover partial-band jamming and Rayleigh fading channels with 119901 =03 120588 = 01 and 1198641198871198730 = 12 dB

discussed above the fast decreasing trapped-error probabilityof the output measure causes the performance degradation ofO-GMD decoding at approximately 10 dB in Figure 9

6 Concluding Remarks

In this paper we considered iterative erasure insertion anddecoding schemes that do not require any preoptimizedthresholds or any CSIR Additionally we proposed a new

Security and Communication Networks 11

analysis framework for the ratio output and sum mea-sures From the simulation results we confirmed that theratio-based GMD decoding scheme and its two-dimensionalschemes have the best performance Using the trapped-error probability the performances of the iterative erasureinsertion and decoding schemes are explained

Appendix

Error and Erasure Probability of SumThreshold Test

Let us denote the symbol error and erasure probabilities ofSTT as 119875err and 119875ers respectively The symbol error eventoccurs when a received symbol is erroneous but is not erasedby STT The symbol erasure event occurs when a receivedsymbol is erased by STTThe two probabilities can be dividedinto two cases as follows

119875err = (1 minus 120588) 119875119880err + 120588119875119869err119875ers = (1 minus 120588) 119875119880ers + 120588119875119869ers (A1)

where 119869 and 119880 refer to the jammed and the unjammed casesrespectively We denote the symbol erase event of STT by119867119904that is 119911 = sum119872minus1119895=0 119910119895 gt 120583 Then 119875119880err can be represented asfollows

119875119880err = Pr [1198670 119867119904 | 119880]= Pr [119867119904 | 119880] minus Pr [1198670 119867119904 | 119880] (A2)

The first term Pr[119867119904 | 119880] and the second term Pr[1198670 119867119904 |119880] are given as (A3) and (A4) respectively Similarly119875119880ers = Pr[119867119904 | 119880] = 1 minus Pr[119867119904 | 119880] can be obtained using(A3) By replacing 119892119880119895 with 119892119869119895 119895 = 0 119872 minus 1 we cancalculate the error and erase probabilities of the unjammedcase For the case in which a jamming signal exists 119875119869err and119875119869ers can be calculated by substituting 119892119869119895(119910119895) for 119892119880119895(119910119895)119895 = 0 119872 minus 1 The derivation of average error probability119875err and erasure probability 119875ers is complete

Pr [119867119904 | 119880] = Pr [119911 lt 120583 | 119880] = Pr[[119872minus1sum119895=0

119910119895 lt 120583]] = int1205830int120583minus11991000

sdot sdot sdot int120583minussum119872minus2119895=0 1199101198950

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 11988911991011198891199100 (A3)

Pr [1198670 119867119904 | 119880] = Pr[[1199100 gt 119910119895 forall119895 = 1 119872 minus 1 119872minus1sum119895=0

119910119895 lt 120583]]= int1205830intmin(1199100 120583minus1199100)

0sdot sdot sdot intmin(1199100 120583minussum

119872minus2119895=0 119910119895)

0

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 11988911991011198891199100(A4)

Disclosure

The present address of Jinsoo Park is the Agency for DefenseDevelopment Daejeon Republic of Korea

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors gratefully acknowledge the support from Elec-tronic Warfare Research Center at the Gwangju Institute ofScience and Technology (GIST) originally funded by theDefense Acquisition Program Administration (DAPA) andthe Agency for Defense Development (ADD)

References

[1] L Milstein and M Simon ldquoSpread Spectrum Communi-cationsrdquo in The Mobile Communications Handbook SecondEdition vol 19990749 of Electrical Engineering Handbook CRCPress 1999

[2] C W Baum and M B Pursley ldquoBayesian Methods for ErasureInsertion in Frequency-Hop Communication Systems withPartial-Band Interferencerdquo IEEE Transactions on Communica-tions vol 40 no 7 pp 1231ndash1238 1992

[3] L-L Yang and L Hanzo ldquoPerformance analysis of codedM-aryorthogonal signaling using errors-and-erasures decoding overfrequency-selective fading channelsrdquo IEEE Journal on SelectedAreas in Communications vol 19 no 2 pp 211ndash221 2001

[4] A J Viterbi ldquoA Robust Ratio-Threshold Technique to MitigateTone and Partial Band Jamming in Coded MFSK Systemsrdquo inProceedings of the IEEE Military Communications ConferenceMILCOM 1982 pp 224-1ndash224-5 Boston MA October 1982

[5] L-L Yang and L Hanzo ldquoLow complexity erasure insertion inRS-coded SFH spread-spectrum communications with partial-band interference and nakagami-m fadingrdquo IEEE Transactionson Communications vol 50 no 6 pp 914ndash925 2002

[6] S Ahmed L-L Yang and L Hanzo ldquoErasure insertion in RS-coded SFH MFSK subjected to tone jamming and Rayleighfadingrdquo IEEE Transactions on Vehicular Technology vol 56 no6 I pp 3563ndash3571 2007

[7] H A Ngo S Ahmed L-L Yang and L Hanzo ldquoNon-coherent cooperative communications dispensing with channelestimation relying on erasure insertion aided Reed-Solomon

12 Security and Communication Networks

coded SFH M-ary FSK subjected to partial-band interferenceand Rayleigh fadingrdquo IEEE Transactions on Communicationsvol 60 no 8 pp 2177ndash2186 2012

[8] T G Macdonald and M B Pursley ldquoStaggered interleavingand iterative errors-and-erasures decoding for frequency-hoppacket radiordquo IEEE Transactions on Wireless Communicationsvol 2 no 1 pp 92ndash98 2003

[9] C-M Lee Y T Su and L-D Jeng ldquoPerformance analysis ofblock codes in hidden Markov channelsrdquo IEEE Transactions onCommunications vol 56 no 1 pp 1ndash4 2008

[10] A Aung K C Teh and K H Li ldquoDetection and counter-measure of interference in slow FHMFSK systems over fadingchannelsrdquo Physical Communication vol 10 pp 11ndash23 2014

[11] J G Kim ldquoError and erasure decoding scheme for RS codes inMFSK based FHSSMA communicationrdquo International Journalof Multimedia and Ubiquitous Engineering vol 10 no 10 pp11ndash20 2015

[12] R Koetter and A Vardy ldquoAlgebraic soft-decision decodingof Reed-Solomon codesrdquo Institute of Electrical and ElectronicsEngineers Transactions on InformationTheory vol 49 no 11 pp2809ndash2825 2003

[13] J Jiang and K R Narayanan ldquoIterative soft decoding of Reed-Solomon codesrdquo IEEE Communications Letters vol 8 no 4 pp244ndash246 2004

[14] P Sehier P Brelivet and I Aubry ldquoNew jammer estimationmethod adapted to FH-PSK waveformsrdquo in Proceedings of the1994 IEEE MILCOM Part 3 (of 3) pp 1027ndash1031 October 1994

[15] M Pursley and J Skinner ldquoDecoding strategies for turboproduct codes in frequency-hop wireless communicationsrdquo inProceedings of the IEEE International Conference on Communi-cations pp 2963ndash2968 Anchorage AK USA

[16] M B Pursley and J S Skinner ldquoAdaptive coding for frequency-hop transmission in mobile ad hoc networks with partial-bandinterferencerdquo IEEETransactions onCommunications vol 57 no3 pp 801ndash811 2009

[17] M R Masse M B Pursley and J S Skinner ldquoAdaptive codingfor frequency-hop transmission over fading channels withpartial-band interferencerdquo IEEE Transactions on Communica-tions vol 59 no 3 pp 854ndash862 2011

[18] H El Gamal and E Geraniotis ldquoIterative channel estimationand decoding for convolutionally coded anti-jam FH signalsrdquoIEEE Transactions on Communications vol 50 no 2 pp 321ndash331 2002

[19] G D Forney ldquoGeneralized Minimum Distance DecodingrdquoIEEE Transactions on Information Theory vol IT-12 no 2 pp125ndash131 1966

[20] D Chase ldquoA Class of Algorithms for Decoding Block CodeswithChannelMeasurement Informationrdquo IEEETransactions onInformation Theory vol 18 no 1 pp 170ndash182 1972

[21] H Tang Y LiuM Fossorier and S Lin ldquoOn combining Chase-2 and GMD decoding algorithms for nonbinary block codesrdquoIEEE Communications Letters vol 5 no 5 pp 209ndash211 2001

[22] S-W Lee and B V K V Kumar ldquoSoft-decision decoding ofreed-solomon codes using successive error-and-ersure decod-ingrdquo in Proceedings of the 2008 IEEEGlobal TelecommunicationsConference GLOBECOM2008 pp 3065ndash3069USADecember2008

[23] E R Berlekamp ldquoReadable Erasures Improve the Performanceof Reed-Solomon Codesrdquo IEEE Transactions on InformationTheory vol 24 no 5 pp 632-633 1978

[24] J G Proakis andM SalehiEssentials of Communication SystemsEngineering 2005

[25] W G Phoel ldquoIterative demodulation and decoding offrequency-hopped PSK in partial-band jammingrdquo IEEEJournal on Selected Areas in Communications vol 23 no 5 pp1026ndash1033 2005

[26] S Lin and D J Costello Error Control Coding Pearson 2ndedition 2004

[27] J K Wolf A M Michelson and A H Levesque ldquoOn theProbability of Undetected Error for Linear Block Codesrdquo IEEETransactions on Communications vol 30 no 2 pp 317ndash3241982

[28] A Ramesh A Chockalingam and L B Milstein ldquoPerformanceof noncoherent turbo detection on Rayleigh fading channelsrdquoin Proceedings of the IEEE Global Telecommunicatins ConferenceGLOBECOMrsquo01 pp 1193ndash1198 usa November 2001

[29] M C Valenti and S Cheng ldquoIterative demodulation anddecoding of turbo-codedM-ary noncoherent orthogonal mod-ulationrdquo IEEE Journal on Selected Areas in Communications vol23 no 9 pp 1739ndash1747 2005

[30] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press BostonMassUSA 5th edition 1994

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6 Security and Communication Networks

Let us assume AWGN and a partial-band jammingchannel Using a binomial expansion and some calculations[30] (17) becomes

120588119890(minus121205902119873) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902119873)119897 (119898 + 1)119897+1 119897 (

119872 minus 1119898 )

+ 120588119890(minus121205902) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902)119897 (119898 + 1)119897+1 119897 (

119872 minus 1119898 )

(18)

Similarly for a Rayleigh fading and partial-band jammingchannel (17) becomes

120588119872minus1sum119898=0

21205902119873(2 + 2119898) 1205902119873 + 119898 (minus1)119898 (119872 minus 1119898 )

+ 120588119872minus1sum119898=0

21205902(2 + 2119898) 1205902 + 119898 (minus1)119898 (119872 minus 1119898 )

(19)

Next we compute 119865119885|119867(119911119901 | 1198670) as follows

119865119885|119867 (119911119901 | 1198670)= Pr [119885 lt 119911119901 1199100 gt 1199101198951015840 forall1198951015840 = 1 119872 minus 1]

Pr [1198670] (20)

Now assume that we use the ratio measure Then thenumerator of (20) becomes

Pr[max1015840119895 119910119895max119895 119910119895 lt 119911119901 1199100 gt 1199101198951015840 forall1198951015840 = 1 119872 minus 1]

= Pr [max1015840119895 119910119895 lt 1199111199011199100]= Pr [119910119895 lt 1199111199011199100 forall119895 = 1 119872 minus 1]= 120588intinfin0int11991111990111991000

sdot sdot sdot int11991111990111991000

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100+ 120588intinfin0int11991111990111991000

sdot sdot sdot int11991111990111991000

119872minus1prod119895=0

119892119869119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100= 120588intinfin01198921198800 (1199100) (1198661198801 (1199111199011199100))(119872minus1) 1198891199100

+ 120588intinfin01198921198690 (1199100) (1198661198691 (1199111199011199100))(119872minus1) 1198891199100

(21)

For AWGN and Rayleigh fading channels with partial-bandjamming (21) becomes (22) and (23) respectively

120588119890(minus121205902119873) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902119873)119897 (119898119911119901 + 1)119897+1 119897 (

119872 minus 1119898 )

+ 120588119890(minus121205902) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902)119897 (119898119911119901 + 1)119897+1 119897 (

119872 minus 1119898 )

(22)

120588119872minus1sum119898=0

21205902119873(2 + 2119898119911119901) 1205902119873 + 119898119911119901 (minus1)119898 (119872 minus 1

119898 )

+ 120588119872minus1sum119898=0

21205902(2 + 2119898119911119901) 1205902 + 119898119911119901 (minus1)119898 (119872 minus 1

119898 ) (23)

For the output measure the numerator of (20) becomes

Pr [max119895 119910119895 lt 119911119901 1199100 gt 1199101198951015840 forall1198951015840 = 1 119872 minus 1]= Pr [1199100 lt 119911119901 1199100 gt 119910119895 forall119895 = 1 119872 minus 1]= 120588int1199111199010int11991000sdot sdot sdot int11991000

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100+ 120588int1199111199010int11991000sdot sdot sdot int11991000

119872minus1prod119895=0

119892119869119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 1198891199100= 120588int11991111990101198921198800 (1199100) (1198661198801 (1199100))(119872minus1) 1198891199100

+ 120588int11991111990101198921198690 (1199100) (1198661198691 (1199100))(119872minus1) 1198891199100

(24)

With partial-band jamming (24) becomes (25) and (26)for AWGN and Rayleigh fading channels respectively

120588119890(minus121205902119873) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902119873)2119897+1 119897 (

119872 minus 1119898 )

sdot (( 21205902119873119898 + 1)119897+1

minus 119890(minus(119898+1)11991111990121205902119873) 119897sum119899=0

119911119899119901119899 ( 21205902119873119898 + 1)119897+1minus119899)

+ 120588119890(minus121205902) infinsum119897=0

119872minus1sum119898=0

(minus1)119898(21205902)2119897+1 119897 (

119872 minus 1119898 )

sdot (( 21205902119898 + 1)119897+1

minus 119890(minus(119898+1)11991111990121205902) 119897sum119899=0

119911119899119901119899 ( 21205902119898 + 1)119897+1minus119899)

(25)

Security and Communication Networks 7

120588119872minus1sum119898=0

21205902119873(2 + 2119898) 1205902119873 + 119898 (minus1)119898 (119872 minus 1119898 ) sdot (1

minus 119890(minus(((2+2119898)1205902119873+119898)(41205904119873+21205902119873))119911119901))+ 120588119872minus1sum119898=0

21205902(2 + 2119898) 1205902 + 119898 sdot (minus1)119898 (119872 minus 1119898 )(1

minus 119890(minus(((2+2119898)1205902+119898)(41205904+21205902))119911119901))

(26)

For the sum measure the numerator of (20) is given as (27)Actually (27) is the same as (A4) in the Appendix where 120583 =119911119901 Using the derived equations we can calculate the trapped-error probabilities of three measures

Pr[[119872minus1sum119895=0

119910119894119895 lt 119911119901 1199100 gt 119910119895 forall119895 = 1 119872 minus 1]] = Pr[[1199100 lt 119911119901

1199101 lt min (1199100 119911119901 minus 1199100) 119910119872minus1 lt min(1199100 119911119901 minus 119872minus2sum119895=0

119910119895)]]= 120588int1199111199010intmin(1199100 119911119901minus1199100)

0sdot sdot sdot intmin(1199100 119911119901minussum

119872minus2119895=0 119910119895)

0

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1sdot sdot sdot 1198891199100

(27)

+ 120588int1199111199010intmin(1199100 119911119901minus1199100)

0sdot sdot sdot intmin(1199100 119911119901minussum

119872minus2119895=0 119910119895)

0

119872minus1prod119895=0

119892119869119895 (119910119895) 119889119910119872minus1sdot sdot sdot 1198891199100

(28)

5 Simulation Results

51 Trapped-Error Probabilities In this subsection we con-firm the trapped-error probability analysis over AWGN andRayleigh fading channels with partial-band jamming Letus assume 4-FSK 120588 = 01 and 1198641198871198730 = 5 dB for theAWGN channel and 1198641198871198730 = 12 dB for the Rayleigh fadingchannel For the higher order of modulation such as 32-FSK which we will consider (27) requires 32-dimensionalintegration therefore we select 4-FSK as an example Tocalculate the probabilities Γ(119911119901) for each measure the input119911119901 should be obtained for the given set of system parametersFor this we further assume that 119901 = 01 and 02 for AWGNand Rayleigh fading channels respectively As we describedin Section 4 we found 119911119901 values that satisfy 119865119911(119911119901) =1 minus 119901 using simulations They are listed in Tables 1 and 3for various 119864119887119873119895 R O and S refer to ratio output andsum respectively In the investigation we transmitted 105symbols for various channel states and calculated their ratiooutput and sum measures We note that these trapped-errorprobabilities do not depend on a specific coding schemebecause these statistics are observed at the demodulatoroutput without coding

Figures 2 and 3 show the trapped-error probabilities ofthe three measures obtained using simulations and from (16)with the threshold values given in Tables 1 and 3 Solid anddashed lines indicate the probabilities obtained by simulation

Table 2 Class of erasure insertion schemes

Measure Threshold test IterativeRatio RTT R-GMDOutput OTT O-GMDSum STT S-GMDTwo-dimensional Ex MO-RTT Ex RS-GMD

Table 3Thresholds 119911119901 of the three measures where119872 = 4 119901 = 02120588 = 01 and 1198641198871198730 = 12 dB over Rayleigh fading and partial-bandjamming channels

119864119887119873119895 (dB) minus5 0 5 10 15 20119911119901 of R 0353 0345 0350 0329 0294 0263119911119901 of O 2271 2233 2053 1765 1681 1660119911119901 of S 2368 2342 2297 2069 1845 1776

and (16) respectively The cross circle and square marksrepresent Γ(119911119901) values of ratio output and sum measuresrespectively

These two figures confirm that the simulation is exactenough to closely follow the theoretical result in (16) Spe-cially ΓO in the analysis is almost the same as ΓO in thesimulation results This result implies the correctness of ourperformance simulation

In these figures the trapped-error probability of theratio measure is increased as SJR is increased therefore wepredict that the R-GMD decoding will perform better thanthe other one-dimensional schemes at high SJR Additionallywe predict that S-GMD decoding is better than O-GMDdecoding in the middle SJR region Recall that the higherthe trapped-error probability the better the performanceThis observation shows the influence on the two-dimensionalschemes because the trapped-error probability affects thenumber of decoding trials We will discuss it in the followingsimulation results

52 Performance over AWGN Channel Throughout perfor-mance simulations we consider 32-ary FSKmodulation with(31 20) RS code over a field of size 32 an ideal interleaverand a random hopping pattern We first consider partial-band jamming (or partial-time jamming with the same 120588)and AWGN channels In this case 120572 = 1 in Figure 1 ForFigures 4ndash7 the channel noise is fixed as we used 1198641198871198730 =5 dBVarious signals to jamming signal ratio119864119887119873119895 have beenconsidered for the simulations

In Figure 4 the performance of various erasure insertionschemes is plotted over the partial-band jamming channelwith 120588 = 01The dashed line shows the baseline performanceldquoWithout EIrdquo which does not use any erasure insertionschemeThe unmarked solid line represents the performanceof the MO-RTT which is the conventional erasure insertionschemeWenote that theMO-RTTuses preoptimized thresh-olds and CSIR To obtain the optimum thresholds for MO-RTT we optimized the thresholds for 119864119887119873119895 = 0sim30 dB with10 dB steps Solid lines with circle square cross plus anddiamond marks represent the performance of output-based

8 Security and Communication Networks

100 5 15 20minus5minus10

ΓR by simulationΓO by simulationΓS by simulation

ΓR by analysisΓO by analysisΓS by analysis

0

01

02

03

04

05Γ

06

07

08

09

1

EbNj (dB)

Figure 2 Trapped-error probabilities of ratio-based (ΓR) output-based (ΓO) and sum-based (ΓS) measures over partial-band jam-ming and AWGN channel

01

02

03

04

05

06

07

08

09

1

Γ

minus5 0 5 10 15 200

ΓR by simulationΓO by simulationΓS by simulation

ΓR by analysisΓO by analysisΓS by analysis

EbNj (dB)

Figure 3 Trapped-error probabilities of ratio-based (ΓR) output-based (ΓO) and sum-based (ΓS) measures over partial-band jam-ming and Rayleigh channels

sum-based ratio-based ratio-to-output-based and ratio-to-sum-based GMD decoding respectively In this figure wealso present the performance using log-likelihood ratio (LLR)based GMD decoding as the triangle-marked line This isthe performance of a system with perfect CSIR such asSNR SJR and the existence of a jamming signal This LLR-GMD decoding iteratively erases the least reliable symbolsbased on LLR as described in [19 22 26] Because it is wellknown that MO-RTT exhibits better performance than OTT

EbNj (dB)

10minus5

10minus4

10minus3

10minus2

10minus1

100

WER

5 10 15 20 250

Without EIMO-RTTO-GMDS-GMD

R-GMDRO-GMDRS-GMDLLR-GMD

Figure 4 Performance of various erasure insertion schemes overpartial-band jamming and AWGN channels with 120588 = 01 and1198641198871198730 = 5 dB

or RTT [5ndash7] the performances ofOTTandRTTare omittedBefore discussing the simulation result we note that RSO-GMD decoding which serially exploits the three measureshas almost the same performance as RS-GMD decodingtherefore we have also omitted it From Figure 4 we makethe following observations

(1) As we expect LLR-GMD decoding with CSIR showsthe best performance We will focus on the perfor-mance gap between LLR-GMD decoding with CSIRand the other erasure insertion schemes withoutCSIR To achieve WER = 10minus4 LLR-GMD decodingrequires 119864119887119873119895 = 19 dB The ratio-based GMDdecoding and its two-dimensional schemes achievethe target WER at 119864119887119873119895 = 21 dB which is muchcloser than that of MO-RTT Even if the jammingpower is decreased as 119864119887119873119895 = 25 dB the perfor-mance of MO-RTT is still far from WER = 10minus4 Inthe low SJR region only the two two-dimensionaliterative schemes can achieve WER = 10minus2 withoutCSIR We note that the one-dimensional iterativeschemes and MO-RTT require 119864119887119873119895 gt 9 dB toachieve WER = 10minus2

(2) The performance of RS-GMD decoding approachesthat of LLR-GMD decoding as SJR is increased At119864119887119873119895 = 20 dB LLR-GMD decoding with perfectCSIR has 35 less WER than RS-GMD decodingWe note that this gap cannot be closed even if thejamming power is decreased

(3) When the jamming power is decreased that isthe channel approaches the unjammed channel theperformance of R-GMD and RS-GMD decoding is

Security and Communication Networks 9

0

01

02

03

04

05

06

07

08

09

1

Γ

5 10 15 20 25 300

ΓRΓOΓS

EbNj (dB)

Figure 5 Trapped-error probability of various measures overpartial-band jamming and AWGN channels with 119901 = 03 120588 = 01and 1198641198871198730 = 5 dB

still better than the performance of ldquoWithout EIrdquoor MO-RTT This implies that the iterative erasureinsertion and decoding schemes do not decrease theperformance if there are no jamming signals Becauseof this we do not need to detect the jamming signal

In Figure 5 trapped-error probabilities of ratio outputand sum measures are represented which corresponds toFigure 4 To obtain the trapped-error probabilities SJR andSNR were scaled based on the code rate 2031 = 119873119870 whichis different from Section 51 We used 119901 = 03 and the otherparameters are the same as those used in Figure 4 In Figure 5ΓR is much larger than ΓO and ΓS The values of ΓR explain thegood performance of R-GMDdecodingWe find that ΓO goesto lt1 minus119870119873 asymp 03 at 119864119887119873119895 asymp 10 dB Any iterative schemethat has zero trapped-error probability at a given SJR has thesame performance as the performance of ldquoWithout EIrdquo Thisresult explains why O-GMD decoding has a performancedegradation region near 10 dB in Figure 4 Because theperformance of O-GMD decoding rapidly approaches theperformance of ldquoWithout EIrdquo its performance is degraded asSJR is increased We note that the performance of S-GMDdecoding slowly approaches the performance of ldquoWithoutEIrdquo therefore there is no performance degradation region

The performance gain of the iterative schemes in Fig-ure 4 cannot be obtained without drawbacks the iterativeschemes require more decoding trials To investigate howmany decoding trials are required we determine the averagenumber of decoding iterations of various iterative schemesvia simulations The results are shown in Figure 6 In factwe simulated iterative schemes that erase one symbol duringthe first loop and then erase two symbols for each remainingloop because this process does not decrease the performanceas described in [19] In general the average number of

1

11

12

13

14

15

16

17

18

Aver

age n

umbe

r of d

ecod

ing

tria

ls

2 4 6 8 10 12 14 16 18 200

O-GMDS-GMDR-GMD

RS-GMDLLR-GMD

EbNj (dB)

Figure 6 Average number of decoding iterations for various erasureinsertion schemes over partial-band jamming and AWGN channelswith 120588 = 01 and 1198641198871198730 = 5 dB

MO-RTTRS-SEI

10minus3

10minus2

10minus1

WER

02 03 04 05 06 07 08 0901

Figure 7 Performance comparison of MO-RTT and RS-GMDdecoding scheme over partial-band jamming and AWGN channelswith various values of 120588 1198641198871198730 = 5 dB and 119864119887119873119895 = 10 dB

decoding trials decreases rapidly as SJR increases At highSJR most of the received words are completely decoded by asingle trial of error-only decoding which holds true becausehigher SJR implies that the environment becomes less andless hostile For low SJR the average decoding trial of RS-GMD decoding is approximately 175 (at 119864119887119873119895 = 0 dB)which decreases as SJR increasesThis result indicates that theiterative erasure insertion and decoding scheme is practicallyimplementable We note that the trial number of O-GMDdecoding is increased at approximately 119864119887119873119895 asymp 10 dB

10 Security and Communication Networks

and that is consistent with the performance degradation ofO-GMD decoding in Figures 4 and 5 Because O-GMDdecoding could not accurately erase the erroneous symbolsthe decoder has to try more decoding Because of this resultwemust use S-GMDdecoding instead of O-GMDdecodingfor the two-dimensional scheme with R-GMD decodingNow recall the analysis results of Section 51 In Figure 2we observed that the trapped-error probability of the ratiomeasure is increased as SJR is increased and that of the summeasure is decreased For the case in which the trapped-errorprobabilities cross at the point of SJR if we want to havefewer decodings at the higher SJR then we must use R-GMDdecoding first in RS-GMDdecoding If we want to have fewerdecodings at the lower SJR then S-GMD decoding should beapplied first In Figures 4 and 6 because we are consideringthe SJR region in which ΓR gt ΓS we have first exploited R-GMD decoding for RS-GMD decoding

In Figure 7 the performances of MO-RTT and RS-GMDdecoding are shown over a partial-band jamming and anAWGN channel with various jamming duty ratios 120588 withfixed 1198641198871198730 = 5 dB and 119864119887119873119895 = 10 dB For all values of120588 RS-GMD decoding works much better than MO-RTT Wenote that two erasure insertion schemes are more efficient forsmaller 120588 channels53 Performance over Rayleigh Fading Channel Figure 8shows the performance of various erasure insertion schemesover partial-band jamming and Rayleigh fading channel with120588 = 01 and 1198641198871198730 = 12 dB For each symbol transmission120572 in Figure 1 is realized by an iid Rayleigh distribution Weassume that fading coefficients are not known to the receiverexcept for the system of MO-RTT

In Figure 8 RS-GMD decoding exhibits better per-formance than MO-RTT which is the same result as wefound over the AWGN channel For the same WER of 10minus3the jammer should spend 14 dB more 119873119895 The WER ofratio-based schemes (including two-dimensional schemes)approach 10minus4 Unlike the results for an AWGN channelR-GMD decoding is dominant and the contribution of S-GMD decoding is marginal Therefore we conclude that R-GMD decoding is sufficient for Rayleigh fading channelsAdditionally we present the performance of LLR-GMDdecoding which knows perfect CSIR As we discussed inSection 52 the performance of R-SEI approaches that of LLR-SEI and the gap is maintained for every SJR

In Figure 9 the trapped-error probabilities of the mea-sures are displayed over a Rayleigh fading channel with1198641198871198730 = 12 dB The other parameters are the same as thosein Figure 5 In this figure the trapped-error probability of theratio measure is 1 in all SJR regions In other words R-GMDdecoding is the best one-dimensional iterative scheme for thetarget system parameters As we found for the other trapped-error probability results the trapped-error probabilities ofsum and output measures decrease as SJR increases Wherethe trapped-error probabilities are lower than lt1 minus119870119873 asymp03 that is 119864119887119873119895 = 10 dB and 16 dB for output andsum respectively the performances of O-GMD and S-GMDdecoding follow the performance of ldquoWithout EIrdquo As we

10minus5

10minus4

10minus3

10minus2

10minus1

100

WER

5 10 15 20 25 300

Without EIMO-RTTO-GMDS-GMD

R-GMDRO-GMDRS-GMDLLR-GMD

EbNj (dB)

Figure 8 Performance of various erasure insertion schemes over aRayleigh fading channel where 120588 = 01 and 1198641198871198730 = 12 dB

0

01

02

03

04

05

06

07

08

09

1

5 10 15 20 25 300

Γ

ΓRΓOΓS

EbNj (dB)

Figure 9 Trapped-error probability of various iterative schemesover partial-band jamming and Rayleigh fading channels with 119901 =03 120588 = 01 and 1198641198871198730 = 12 dB

discussed above the fast decreasing trapped-error probabilityof the output measure causes the performance degradation ofO-GMD decoding at approximately 10 dB in Figure 9

6 Concluding Remarks

In this paper we considered iterative erasure insertion anddecoding schemes that do not require any preoptimizedthresholds or any CSIR Additionally we proposed a new

Security and Communication Networks 11

analysis framework for the ratio output and sum mea-sures From the simulation results we confirmed that theratio-based GMD decoding scheme and its two-dimensionalschemes have the best performance Using the trapped-error probability the performances of the iterative erasureinsertion and decoding schemes are explained

Appendix

Error and Erasure Probability of SumThreshold Test

Let us denote the symbol error and erasure probabilities ofSTT as 119875err and 119875ers respectively The symbol error eventoccurs when a received symbol is erroneous but is not erasedby STT The symbol erasure event occurs when a receivedsymbol is erased by STTThe two probabilities can be dividedinto two cases as follows

119875err = (1 minus 120588) 119875119880err + 120588119875119869err119875ers = (1 minus 120588) 119875119880ers + 120588119875119869ers (A1)

where 119869 and 119880 refer to the jammed and the unjammed casesrespectively We denote the symbol erase event of STT by119867119904that is 119911 = sum119872minus1119895=0 119910119895 gt 120583 Then 119875119880err can be represented asfollows

119875119880err = Pr [1198670 119867119904 | 119880]= Pr [119867119904 | 119880] minus Pr [1198670 119867119904 | 119880] (A2)

The first term Pr[119867119904 | 119880] and the second term Pr[1198670 119867119904 |119880] are given as (A3) and (A4) respectively Similarly119875119880ers = Pr[119867119904 | 119880] = 1 minus Pr[119867119904 | 119880] can be obtained using(A3) By replacing 119892119880119895 with 119892119869119895 119895 = 0 119872 minus 1 we cancalculate the error and erase probabilities of the unjammedcase For the case in which a jamming signal exists 119875119869err and119875119869ers can be calculated by substituting 119892119869119895(119910119895) for 119892119880119895(119910119895)119895 = 0 119872 minus 1 The derivation of average error probability119875err and erasure probability 119875ers is complete

Pr [119867119904 | 119880] = Pr [119911 lt 120583 | 119880] = Pr[[119872minus1sum119895=0

119910119895 lt 120583]] = int1205830int120583minus11991000

sdot sdot sdot int120583minussum119872minus2119895=0 1199101198950

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 11988911991011198891199100 (A3)

Pr [1198670 119867119904 | 119880] = Pr[[1199100 gt 119910119895 forall119895 = 1 119872 minus 1 119872minus1sum119895=0

119910119895 lt 120583]]= int1205830intmin(1199100 120583minus1199100)

0sdot sdot sdot intmin(1199100 120583minussum

119872minus2119895=0 119910119895)

0

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 11988911991011198891199100(A4)

Disclosure

The present address of Jinsoo Park is the Agency for DefenseDevelopment Daejeon Republic of Korea

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors gratefully acknowledge the support from Elec-tronic Warfare Research Center at the Gwangju Institute ofScience and Technology (GIST) originally funded by theDefense Acquisition Program Administration (DAPA) andthe Agency for Defense Development (ADD)

References

[1] L Milstein and M Simon ldquoSpread Spectrum Communi-cationsrdquo in The Mobile Communications Handbook SecondEdition vol 19990749 of Electrical Engineering Handbook CRCPress 1999

[2] C W Baum and M B Pursley ldquoBayesian Methods for ErasureInsertion in Frequency-Hop Communication Systems withPartial-Band Interferencerdquo IEEE Transactions on Communica-tions vol 40 no 7 pp 1231ndash1238 1992

[3] L-L Yang and L Hanzo ldquoPerformance analysis of codedM-aryorthogonal signaling using errors-and-erasures decoding overfrequency-selective fading channelsrdquo IEEE Journal on SelectedAreas in Communications vol 19 no 2 pp 211ndash221 2001

[4] A J Viterbi ldquoA Robust Ratio-Threshold Technique to MitigateTone and Partial Band Jamming in Coded MFSK Systemsrdquo inProceedings of the IEEE Military Communications ConferenceMILCOM 1982 pp 224-1ndash224-5 Boston MA October 1982

[5] L-L Yang and L Hanzo ldquoLow complexity erasure insertion inRS-coded SFH spread-spectrum communications with partial-band interference and nakagami-m fadingrdquo IEEE Transactionson Communications vol 50 no 6 pp 914ndash925 2002

[6] S Ahmed L-L Yang and L Hanzo ldquoErasure insertion in RS-coded SFH MFSK subjected to tone jamming and Rayleighfadingrdquo IEEE Transactions on Vehicular Technology vol 56 no6 I pp 3563ndash3571 2007

[7] H A Ngo S Ahmed L-L Yang and L Hanzo ldquoNon-coherent cooperative communications dispensing with channelestimation relying on erasure insertion aided Reed-Solomon

12 Security and Communication Networks

coded SFH M-ary FSK subjected to partial-band interferenceand Rayleigh fadingrdquo IEEE Transactions on Communicationsvol 60 no 8 pp 2177ndash2186 2012

[8] T G Macdonald and M B Pursley ldquoStaggered interleavingand iterative errors-and-erasures decoding for frequency-hoppacket radiordquo IEEE Transactions on Wireless Communicationsvol 2 no 1 pp 92ndash98 2003

[9] C-M Lee Y T Su and L-D Jeng ldquoPerformance analysis ofblock codes in hidden Markov channelsrdquo IEEE Transactions onCommunications vol 56 no 1 pp 1ndash4 2008

[10] A Aung K C Teh and K H Li ldquoDetection and counter-measure of interference in slow FHMFSK systems over fadingchannelsrdquo Physical Communication vol 10 pp 11ndash23 2014

[11] J G Kim ldquoError and erasure decoding scheme for RS codes inMFSK based FHSSMA communicationrdquo International Journalof Multimedia and Ubiquitous Engineering vol 10 no 10 pp11ndash20 2015

[12] R Koetter and A Vardy ldquoAlgebraic soft-decision decodingof Reed-Solomon codesrdquo Institute of Electrical and ElectronicsEngineers Transactions on InformationTheory vol 49 no 11 pp2809ndash2825 2003

[13] J Jiang and K R Narayanan ldquoIterative soft decoding of Reed-Solomon codesrdquo IEEE Communications Letters vol 8 no 4 pp244ndash246 2004

[14] P Sehier P Brelivet and I Aubry ldquoNew jammer estimationmethod adapted to FH-PSK waveformsrdquo in Proceedings of the1994 IEEE MILCOM Part 3 (of 3) pp 1027ndash1031 October 1994

[15] M Pursley and J Skinner ldquoDecoding strategies for turboproduct codes in frequency-hop wireless communicationsrdquo inProceedings of the IEEE International Conference on Communi-cations pp 2963ndash2968 Anchorage AK USA

[16] M B Pursley and J S Skinner ldquoAdaptive coding for frequency-hop transmission in mobile ad hoc networks with partial-bandinterferencerdquo IEEETransactions onCommunications vol 57 no3 pp 801ndash811 2009

[17] M R Masse M B Pursley and J S Skinner ldquoAdaptive codingfor frequency-hop transmission over fading channels withpartial-band interferencerdquo IEEE Transactions on Communica-tions vol 59 no 3 pp 854ndash862 2011

[18] H El Gamal and E Geraniotis ldquoIterative channel estimationand decoding for convolutionally coded anti-jam FH signalsrdquoIEEE Transactions on Communications vol 50 no 2 pp 321ndash331 2002

[19] G D Forney ldquoGeneralized Minimum Distance DecodingrdquoIEEE Transactions on Information Theory vol IT-12 no 2 pp125ndash131 1966

[20] D Chase ldquoA Class of Algorithms for Decoding Block CodeswithChannelMeasurement Informationrdquo IEEETransactions onInformation Theory vol 18 no 1 pp 170ndash182 1972

[21] H Tang Y LiuM Fossorier and S Lin ldquoOn combining Chase-2 and GMD decoding algorithms for nonbinary block codesrdquoIEEE Communications Letters vol 5 no 5 pp 209ndash211 2001

[22] S-W Lee and B V K V Kumar ldquoSoft-decision decoding ofreed-solomon codes using successive error-and-ersure decod-ingrdquo in Proceedings of the 2008 IEEEGlobal TelecommunicationsConference GLOBECOM2008 pp 3065ndash3069USADecember2008

[23] E R Berlekamp ldquoReadable Erasures Improve the Performanceof Reed-Solomon Codesrdquo IEEE Transactions on InformationTheory vol 24 no 5 pp 632-633 1978

[24] J G Proakis andM SalehiEssentials of Communication SystemsEngineering 2005

[25] W G Phoel ldquoIterative demodulation and decoding offrequency-hopped PSK in partial-band jammingrdquo IEEEJournal on Selected Areas in Communications vol 23 no 5 pp1026ndash1033 2005

[26] S Lin and D J Costello Error Control Coding Pearson 2ndedition 2004

[27] J K Wolf A M Michelson and A H Levesque ldquoOn theProbability of Undetected Error for Linear Block Codesrdquo IEEETransactions on Communications vol 30 no 2 pp 317ndash3241982

[28] A Ramesh A Chockalingam and L B Milstein ldquoPerformanceof noncoherent turbo detection on Rayleigh fading channelsrdquoin Proceedings of the IEEE Global Telecommunicatins ConferenceGLOBECOMrsquo01 pp 1193ndash1198 usa November 2001

[29] M C Valenti and S Cheng ldquoIterative demodulation anddecoding of turbo-codedM-ary noncoherent orthogonal mod-ulationrdquo IEEE Journal on Selected Areas in Communications vol23 no 9 pp 1739ndash1747 2005

[30] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press BostonMassUSA 5th edition 1994

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Security and Communication Networks 7

120588119872minus1sum119898=0

21205902119873(2 + 2119898) 1205902119873 + 119898 (minus1)119898 (119872 minus 1119898 ) sdot (1

minus 119890(minus(((2+2119898)1205902119873+119898)(41205904119873+21205902119873))119911119901))+ 120588119872minus1sum119898=0

21205902(2 + 2119898) 1205902 + 119898 sdot (minus1)119898 (119872 minus 1119898 )(1

minus 119890(minus(((2+2119898)1205902+119898)(41205904+21205902))119911119901))

(26)

For the sum measure the numerator of (20) is given as (27)Actually (27) is the same as (A4) in the Appendix where 120583 =119911119901 Using the derived equations we can calculate the trapped-error probabilities of three measures

Pr[[119872minus1sum119895=0

119910119894119895 lt 119911119901 1199100 gt 119910119895 forall119895 = 1 119872 minus 1]] = Pr[[1199100 lt 119911119901

1199101 lt min (1199100 119911119901 minus 1199100) 119910119872minus1 lt min(1199100 119911119901 minus 119872minus2sum119895=0

119910119895)]]= 120588int1199111199010intmin(1199100 119911119901minus1199100)

0sdot sdot sdot intmin(1199100 119911119901minussum

119872minus2119895=0 119910119895)

0

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1sdot sdot sdot 1198891199100

(27)

+ 120588int1199111199010intmin(1199100 119911119901minus1199100)

0sdot sdot sdot intmin(1199100 119911119901minussum

119872minus2119895=0 119910119895)

0

119872minus1prod119895=0

119892119869119895 (119910119895) 119889119910119872minus1sdot sdot sdot 1198891199100

(28)

5 Simulation Results

51 Trapped-Error Probabilities In this subsection we con-firm the trapped-error probability analysis over AWGN andRayleigh fading channels with partial-band jamming Letus assume 4-FSK 120588 = 01 and 1198641198871198730 = 5 dB for theAWGN channel and 1198641198871198730 = 12 dB for the Rayleigh fadingchannel For the higher order of modulation such as 32-FSK which we will consider (27) requires 32-dimensionalintegration therefore we select 4-FSK as an example Tocalculate the probabilities Γ(119911119901) for each measure the input119911119901 should be obtained for the given set of system parametersFor this we further assume that 119901 = 01 and 02 for AWGNand Rayleigh fading channels respectively As we describedin Section 4 we found 119911119901 values that satisfy 119865119911(119911119901) =1 minus 119901 using simulations They are listed in Tables 1 and 3for various 119864119887119873119895 R O and S refer to ratio output andsum respectively In the investigation we transmitted 105symbols for various channel states and calculated their ratiooutput and sum measures We note that these trapped-errorprobabilities do not depend on a specific coding schemebecause these statistics are observed at the demodulatoroutput without coding

Figures 2 and 3 show the trapped-error probabilities ofthe three measures obtained using simulations and from (16)with the threshold values given in Tables 1 and 3 Solid anddashed lines indicate the probabilities obtained by simulation

Table 2 Class of erasure insertion schemes

Measure Threshold test IterativeRatio RTT R-GMDOutput OTT O-GMDSum STT S-GMDTwo-dimensional Ex MO-RTT Ex RS-GMD

Table 3Thresholds 119911119901 of the three measures where119872 = 4 119901 = 02120588 = 01 and 1198641198871198730 = 12 dB over Rayleigh fading and partial-bandjamming channels

119864119887119873119895 (dB) minus5 0 5 10 15 20119911119901 of R 0353 0345 0350 0329 0294 0263119911119901 of O 2271 2233 2053 1765 1681 1660119911119901 of S 2368 2342 2297 2069 1845 1776

and (16) respectively The cross circle and square marksrepresent Γ(119911119901) values of ratio output and sum measuresrespectively

These two figures confirm that the simulation is exactenough to closely follow the theoretical result in (16) Spe-cially ΓO in the analysis is almost the same as ΓO in thesimulation results This result implies the correctness of ourperformance simulation

In these figures the trapped-error probability of theratio measure is increased as SJR is increased therefore wepredict that the R-GMD decoding will perform better thanthe other one-dimensional schemes at high SJR Additionallywe predict that S-GMD decoding is better than O-GMDdecoding in the middle SJR region Recall that the higherthe trapped-error probability the better the performanceThis observation shows the influence on the two-dimensionalschemes because the trapped-error probability affects thenumber of decoding trials We will discuss it in the followingsimulation results

52 Performance over AWGN Channel Throughout perfor-mance simulations we consider 32-ary FSKmodulation with(31 20) RS code over a field of size 32 an ideal interleaverand a random hopping pattern We first consider partial-band jamming (or partial-time jamming with the same 120588)and AWGN channels In this case 120572 = 1 in Figure 1 ForFigures 4ndash7 the channel noise is fixed as we used 1198641198871198730 =5 dBVarious signals to jamming signal ratio119864119887119873119895 have beenconsidered for the simulations

In Figure 4 the performance of various erasure insertionschemes is plotted over the partial-band jamming channelwith 120588 = 01The dashed line shows the baseline performanceldquoWithout EIrdquo which does not use any erasure insertionschemeThe unmarked solid line represents the performanceof the MO-RTT which is the conventional erasure insertionschemeWenote that theMO-RTTuses preoptimized thresh-olds and CSIR To obtain the optimum thresholds for MO-RTT we optimized the thresholds for 119864119887119873119895 = 0sim30 dB with10 dB steps Solid lines with circle square cross plus anddiamond marks represent the performance of output-based

8 Security and Communication Networks

100 5 15 20minus5minus10

ΓR by simulationΓO by simulationΓS by simulation

ΓR by analysisΓO by analysisΓS by analysis

0

01

02

03

04

05Γ

06

07

08

09

1

EbNj (dB)

Figure 2 Trapped-error probabilities of ratio-based (ΓR) output-based (ΓO) and sum-based (ΓS) measures over partial-band jam-ming and AWGN channel

01

02

03

04

05

06

07

08

09

1

Γ

minus5 0 5 10 15 200

ΓR by simulationΓO by simulationΓS by simulation

ΓR by analysisΓO by analysisΓS by analysis

EbNj (dB)

Figure 3 Trapped-error probabilities of ratio-based (ΓR) output-based (ΓO) and sum-based (ΓS) measures over partial-band jam-ming and Rayleigh channels

sum-based ratio-based ratio-to-output-based and ratio-to-sum-based GMD decoding respectively In this figure wealso present the performance using log-likelihood ratio (LLR)based GMD decoding as the triangle-marked line This isthe performance of a system with perfect CSIR such asSNR SJR and the existence of a jamming signal This LLR-GMD decoding iteratively erases the least reliable symbolsbased on LLR as described in [19 22 26] Because it is wellknown that MO-RTT exhibits better performance than OTT

EbNj (dB)

10minus5

10minus4

10minus3

10minus2

10minus1

100

WER

5 10 15 20 250

Without EIMO-RTTO-GMDS-GMD

R-GMDRO-GMDRS-GMDLLR-GMD

Figure 4 Performance of various erasure insertion schemes overpartial-band jamming and AWGN channels with 120588 = 01 and1198641198871198730 = 5 dB

or RTT [5ndash7] the performances ofOTTandRTTare omittedBefore discussing the simulation result we note that RSO-GMD decoding which serially exploits the three measureshas almost the same performance as RS-GMD decodingtherefore we have also omitted it From Figure 4 we makethe following observations

(1) As we expect LLR-GMD decoding with CSIR showsthe best performance We will focus on the perfor-mance gap between LLR-GMD decoding with CSIRand the other erasure insertion schemes withoutCSIR To achieve WER = 10minus4 LLR-GMD decodingrequires 119864119887119873119895 = 19 dB The ratio-based GMDdecoding and its two-dimensional schemes achievethe target WER at 119864119887119873119895 = 21 dB which is muchcloser than that of MO-RTT Even if the jammingpower is decreased as 119864119887119873119895 = 25 dB the perfor-mance of MO-RTT is still far from WER = 10minus4 Inthe low SJR region only the two two-dimensionaliterative schemes can achieve WER = 10minus2 withoutCSIR We note that the one-dimensional iterativeschemes and MO-RTT require 119864119887119873119895 gt 9 dB toachieve WER = 10minus2

(2) The performance of RS-GMD decoding approachesthat of LLR-GMD decoding as SJR is increased At119864119887119873119895 = 20 dB LLR-GMD decoding with perfectCSIR has 35 less WER than RS-GMD decodingWe note that this gap cannot be closed even if thejamming power is decreased

(3) When the jamming power is decreased that isthe channel approaches the unjammed channel theperformance of R-GMD and RS-GMD decoding is

Security and Communication Networks 9

0

01

02

03

04

05

06

07

08

09

1

Γ

5 10 15 20 25 300

ΓRΓOΓS

EbNj (dB)

Figure 5 Trapped-error probability of various measures overpartial-band jamming and AWGN channels with 119901 = 03 120588 = 01and 1198641198871198730 = 5 dB

still better than the performance of ldquoWithout EIrdquoor MO-RTT This implies that the iterative erasureinsertion and decoding schemes do not decrease theperformance if there are no jamming signals Becauseof this we do not need to detect the jamming signal

In Figure 5 trapped-error probabilities of ratio outputand sum measures are represented which corresponds toFigure 4 To obtain the trapped-error probabilities SJR andSNR were scaled based on the code rate 2031 = 119873119870 whichis different from Section 51 We used 119901 = 03 and the otherparameters are the same as those used in Figure 4 In Figure 5ΓR is much larger than ΓO and ΓS The values of ΓR explain thegood performance of R-GMDdecodingWe find that ΓO goesto lt1 minus119870119873 asymp 03 at 119864119887119873119895 asymp 10 dB Any iterative schemethat has zero trapped-error probability at a given SJR has thesame performance as the performance of ldquoWithout EIrdquo Thisresult explains why O-GMD decoding has a performancedegradation region near 10 dB in Figure 4 Because theperformance of O-GMD decoding rapidly approaches theperformance of ldquoWithout EIrdquo its performance is degraded asSJR is increased We note that the performance of S-GMDdecoding slowly approaches the performance of ldquoWithoutEIrdquo therefore there is no performance degradation region

The performance gain of the iterative schemes in Fig-ure 4 cannot be obtained without drawbacks the iterativeschemes require more decoding trials To investigate howmany decoding trials are required we determine the averagenumber of decoding iterations of various iterative schemesvia simulations The results are shown in Figure 6 In factwe simulated iterative schemes that erase one symbol duringthe first loop and then erase two symbols for each remainingloop because this process does not decrease the performanceas described in [19] In general the average number of

1

11

12

13

14

15

16

17

18

Aver

age n

umbe

r of d

ecod

ing

tria

ls

2 4 6 8 10 12 14 16 18 200

O-GMDS-GMDR-GMD

RS-GMDLLR-GMD

EbNj (dB)

Figure 6 Average number of decoding iterations for various erasureinsertion schemes over partial-band jamming and AWGN channelswith 120588 = 01 and 1198641198871198730 = 5 dB

MO-RTTRS-SEI

10minus3

10minus2

10minus1

WER

02 03 04 05 06 07 08 0901

Figure 7 Performance comparison of MO-RTT and RS-GMDdecoding scheme over partial-band jamming and AWGN channelswith various values of 120588 1198641198871198730 = 5 dB and 119864119887119873119895 = 10 dB

decoding trials decreases rapidly as SJR increases At highSJR most of the received words are completely decoded by asingle trial of error-only decoding which holds true becausehigher SJR implies that the environment becomes less andless hostile For low SJR the average decoding trial of RS-GMD decoding is approximately 175 (at 119864119887119873119895 = 0 dB)which decreases as SJR increasesThis result indicates that theiterative erasure insertion and decoding scheme is practicallyimplementable We note that the trial number of O-GMDdecoding is increased at approximately 119864119887119873119895 asymp 10 dB

10 Security and Communication Networks

and that is consistent with the performance degradation ofO-GMD decoding in Figures 4 and 5 Because O-GMDdecoding could not accurately erase the erroneous symbolsthe decoder has to try more decoding Because of this resultwemust use S-GMDdecoding instead of O-GMDdecodingfor the two-dimensional scheme with R-GMD decodingNow recall the analysis results of Section 51 In Figure 2we observed that the trapped-error probability of the ratiomeasure is increased as SJR is increased and that of the summeasure is decreased For the case in which the trapped-errorprobabilities cross at the point of SJR if we want to havefewer decodings at the higher SJR then we must use R-GMDdecoding first in RS-GMDdecoding If we want to have fewerdecodings at the lower SJR then S-GMD decoding should beapplied first In Figures 4 and 6 because we are consideringthe SJR region in which ΓR gt ΓS we have first exploited R-GMD decoding for RS-GMD decoding

In Figure 7 the performances of MO-RTT and RS-GMDdecoding are shown over a partial-band jamming and anAWGN channel with various jamming duty ratios 120588 withfixed 1198641198871198730 = 5 dB and 119864119887119873119895 = 10 dB For all values of120588 RS-GMD decoding works much better than MO-RTT Wenote that two erasure insertion schemes are more efficient forsmaller 120588 channels53 Performance over Rayleigh Fading Channel Figure 8shows the performance of various erasure insertion schemesover partial-band jamming and Rayleigh fading channel with120588 = 01 and 1198641198871198730 = 12 dB For each symbol transmission120572 in Figure 1 is realized by an iid Rayleigh distribution Weassume that fading coefficients are not known to the receiverexcept for the system of MO-RTT

In Figure 8 RS-GMD decoding exhibits better per-formance than MO-RTT which is the same result as wefound over the AWGN channel For the same WER of 10minus3the jammer should spend 14 dB more 119873119895 The WER ofratio-based schemes (including two-dimensional schemes)approach 10minus4 Unlike the results for an AWGN channelR-GMD decoding is dominant and the contribution of S-GMD decoding is marginal Therefore we conclude that R-GMD decoding is sufficient for Rayleigh fading channelsAdditionally we present the performance of LLR-GMDdecoding which knows perfect CSIR As we discussed inSection 52 the performance of R-SEI approaches that of LLR-SEI and the gap is maintained for every SJR

In Figure 9 the trapped-error probabilities of the mea-sures are displayed over a Rayleigh fading channel with1198641198871198730 = 12 dB The other parameters are the same as thosein Figure 5 In this figure the trapped-error probability of theratio measure is 1 in all SJR regions In other words R-GMDdecoding is the best one-dimensional iterative scheme for thetarget system parameters As we found for the other trapped-error probability results the trapped-error probabilities ofsum and output measures decrease as SJR increases Wherethe trapped-error probabilities are lower than lt1 minus119870119873 asymp03 that is 119864119887119873119895 = 10 dB and 16 dB for output andsum respectively the performances of O-GMD and S-GMDdecoding follow the performance of ldquoWithout EIrdquo As we

10minus5

10minus4

10minus3

10minus2

10minus1

100

WER

5 10 15 20 25 300

Without EIMO-RTTO-GMDS-GMD

R-GMDRO-GMDRS-GMDLLR-GMD

EbNj (dB)

Figure 8 Performance of various erasure insertion schemes over aRayleigh fading channel where 120588 = 01 and 1198641198871198730 = 12 dB

0

01

02

03

04

05

06

07

08

09

1

5 10 15 20 25 300

Γ

ΓRΓOΓS

EbNj (dB)

Figure 9 Trapped-error probability of various iterative schemesover partial-band jamming and Rayleigh fading channels with 119901 =03 120588 = 01 and 1198641198871198730 = 12 dB

discussed above the fast decreasing trapped-error probabilityof the output measure causes the performance degradation ofO-GMD decoding at approximately 10 dB in Figure 9

6 Concluding Remarks

In this paper we considered iterative erasure insertion anddecoding schemes that do not require any preoptimizedthresholds or any CSIR Additionally we proposed a new

Security and Communication Networks 11

analysis framework for the ratio output and sum mea-sures From the simulation results we confirmed that theratio-based GMD decoding scheme and its two-dimensionalschemes have the best performance Using the trapped-error probability the performances of the iterative erasureinsertion and decoding schemes are explained

Appendix

Error and Erasure Probability of SumThreshold Test

Let us denote the symbol error and erasure probabilities ofSTT as 119875err and 119875ers respectively The symbol error eventoccurs when a received symbol is erroneous but is not erasedby STT The symbol erasure event occurs when a receivedsymbol is erased by STTThe two probabilities can be dividedinto two cases as follows

119875err = (1 minus 120588) 119875119880err + 120588119875119869err119875ers = (1 minus 120588) 119875119880ers + 120588119875119869ers (A1)

where 119869 and 119880 refer to the jammed and the unjammed casesrespectively We denote the symbol erase event of STT by119867119904that is 119911 = sum119872minus1119895=0 119910119895 gt 120583 Then 119875119880err can be represented asfollows

119875119880err = Pr [1198670 119867119904 | 119880]= Pr [119867119904 | 119880] minus Pr [1198670 119867119904 | 119880] (A2)

The first term Pr[119867119904 | 119880] and the second term Pr[1198670 119867119904 |119880] are given as (A3) and (A4) respectively Similarly119875119880ers = Pr[119867119904 | 119880] = 1 minus Pr[119867119904 | 119880] can be obtained using(A3) By replacing 119892119880119895 with 119892119869119895 119895 = 0 119872 minus 1 we cancalculate the error and erase probabilities of the unjammedcase For the case in which a jamming signal exists 119875119869err and119875119869ers can be calculated by substituting 119892119869119895(119910119895) for 119892119880119895(119910119895)119895 = 0 119872 minus 1 The derivation of average error probability119875err and erasure probability 119875ers is complete

Pr [119867119904 | 119880] = Pr [119911 lt 120583 | 119880] = Pr[[119872minus1sum119895=0

119910119895 lt 120583]] = int1205830int120583minus11991000

sdot sdot sdot int120583minussum119872minus2119895=0 1199101198950

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 11988911991011198891199100 (A3)

Pr [1198670 119867119904 | 119880] = Pr[[1199100 gt 119910119895 forall119895 = 1 119872 minus 1 119872minus1sum119895=0

119910119895 lt 120583]]= int1205830intmin(1199100 120583minus1199100)

0sdot sdot sdot intmin(1199100 120583minussum

119872minus2119895=0 119910119895)

0

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 11988911991011198891199100(A4)

Disclosure

The present address of Jinsoo Park is the Agency for DefenseDevelopment Daejeon Republic of Korea

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors gratefully acknowledge the support from Elec-tronic Warfare Research Center at the Gwangju Institute ofScience and Technology (GIST) originally funded by theDefense Acquisition Program Administration (DAPA) andthe Agency for Defense Development (ADD)

References

[1] L Milstein and M Simon ldquoSpread Spectrum Communi-cationsrdquo in The Mobile Communications Handbook SecondEdition vol 19990749 of Electrical Engineering Handbook CRCPress 1999

[2] C W Baum and M B Pursley ldquoBayesian Methods for ErasureInsertion in Frequency-Hop Communication Systems withPartial-Band Interferencerdquo IEEE Transactions on Communica-tions vol 40 no 7 pp 1231ndash1238 1992

[3] L-L Yang and L Hanzo ldquoPerformance analysis of codedM-aryorthogonal signaling using errors-and-erasures decoding overfrequency-selective fading channelsrdquo IEEE Journal on SelectedAreas in Communications vol 19 no 2 pp 211ndash221 2001

[4] A J Viterbi ldquoA Robust Ratio-Threshold Technique to MitigateTone and Partial Band Jamming in Coded MFSK Systemsrdquo inProceedings of the IEEE Military Communications ConferenceMILCOM 1982 pp 224-1ndash224-5 Boston MA October 1982

[5] L-L Yang and L Hanzo ldquoLow complexity erasure insertion inRS-coded SFH spread-spectrum communications with partial-band interference and nakagami-m fadingrdquo IEEE Transactionson Communications vol 50 no 6 pp 914ndash925 2002

[6] S Ahmed L-L Yang and L Hanzo ldquoErasure insertion in RS-coded SFH MFSK subjected to tone jamming and Rayleighfadingrdquo IEEE Transactions on Vehicular Technology vol 56 no6 I pp 3563ndash3571 2007

[7] H A Ngo S Ahmed L-L Yang and L Hanzo ldquoNon-coherent cooperative communications dispensing with channelestimation relying on erasure insertion aided Reed-Solomon

12 Security and Communication Networks

coded SFH M-ary FSK subjected to partial-band interferenceand Rayleigh fadingrdquo IEEE Transactions on Communicationsvol 60 no 8 pp 2177ndash2186 2012

[8] T G Macdonald and M B Pursley ldquoStaggered interleavingand iterative errors-and-erasures decoding for frequency-hoppacket radiordquo IEEE Transactions on Wireless Communicationsvol 2 no 1 pp 92ndash98 2003

[9] C-M Lee Y T Su and L-D Jeng ldquoPerformance analysis ofblock codes in hidden Markov channelsrdquo IEEE Transactions onCommunications vol 56 no 1 pp 1ndash4 2008

[10] A Aung K C Teh and K H Li ldquoDetection and counter-measure of interference in slow FHMFSK systems over fadingchannelsrdquo Physical Communication vol 10 pp 11ndash23 2014

[11] J G Kim ldquoError and erasure decoding scheme for RS codes inMFSK based FHSSMA communicationrdquo International Journalof Multimedia and Ubiquitous Engineering vol 10 no 10 pp11ndash20 2015

[12] R Koetter and A Vardy ldquoAlgebraic soft-decision decodingof Reed-Solomon codesrdquo Institute of Electrical and ElectronicsEngineers Transactions on InformationTheory vol 49 no 11 pp2809ndash2825 2003

[13] J Jiang and K R Narayanan ldquoIterative soft decoding of Reed-Solomon codesrdquo IEEE Communications Letters vol 8 no 4 pp244ndash246 2004

[14] P Sehier P Brelivet and I Aubry ldquoNew jammer estimationmethod adapted to FH-PSK waveformsrdquo in Proceedings of the1994 IEEE MILCOM Part 3 (of 3) pp 1027ndash1031 October 1994

[15] M Pursley and J Skinner ldquoDecoding strategies for turboproduct codes in frequency-hop wireless communicationsrdquo inProceedings of the IEEE International Conference on Communi-cations pp 2963ndash2968 Anchorage AK USA

[16] M B Pursley and J S Skinner ldquoAdaptive coding for frequency-hop transmission in mobile ad hoc networks with partial-bandinterferencerdquo IEEETransactions onCommunications vol 57 no3 pp 801ndash811 2009

[17] M R Masse M B Pursley and J S Skinner ldquoAdaptive codingfor frequency-hop transmission over fading channels withpartial-band interferencerdquo IEEE Transactions on Communica-tions vol 59 no 3 pp 854ndash862 2011

[18] H El Gamal and E Geraniotis ldquoIterative channel estimationand decoding for convolutionally coded anti-jam FH signalsrdquoIEEE Transactions on Communications vol 50 no 2 pp 321ndash331 2002

[19] G D Forney ldquoGeneralized Minimum Distance DecodingrdquoIEEE Transactions on Information Theory vol IT-12 no 2 pp125ndash131 1966

[20] D Chase ldquoA Class of Algorithms for Decoding Block CodeswithChannelMeasurement Informationrdquo IEEETransactions onInformation Theory vol 18 no 1 pp 170ndash182 1972

[21] H Tang Y LiuM Fossorier and S Lin ldquoOn combining Chase-2 and GMD decoding algorithms for nonbinary block codesrdquoIEEE Communications Letters vol 5 no 5 pp 209ndash211 2001

[22] S-W Lee and B V K V Kumar ldquoSoft-decision decoding ofreed-solomon codes using successive error-and-ersure decod-ingrdquo in Proceedings of the 2008 IEEEGlobal TelecommunicationsConference GLOBECOM2008 pp 3065ndash3069USADecember2008

[23] E R Berlekamp ldquoReadable Erasures Improve the Performanceof Reed-Solomon Codesrdquo IEEE Transactions on InformationTheory vol 24 no 5 pp 632-633 1978

[24] J G Proakis andM SalehiEssentials of Communication SystemsEngineering 2005

[25] W G Phoel ldquoIterative demodulation and decoding offrequency-hopped PSK in partial-band jammingrdquo IEEEJournal on Selected Areas in Communications vol 23 no 5 pp1026ndash1033 2005

[26] S Lin and D J Costello Error Control Coding Pearson 2ndedition 2004

[27] J K Wolf A M Michelson and A H Levesque ldquoOn theProbability of Undetected Error for Linear Block Codesrdquo IEEETransactions on Communications vol 30 no 2 pp 317ndash3241982

[28] A Ramesh A Chockalingam and L B Milstein ldquoPerformanceof noncoherent turbo detection on Rayleigh fading channelsrdquoin Proceedings of the IEEE Global Telecommunicatins ConferenceGLOBECOMrsquo01 pp 1193ndash1198 usa November 2001

[29] M C Valenti and S Cheng ldquoIterative demodulation anddecoding of turbo-codedM-ary noncoherent orthogonal mod-ulationrdquo IEEE Journal on Selected Areas in Communications vol23 no 9 pp 1739ndash1747 2005

[30] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press BostonMassUSA 5th edition 1994

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Submit your manuscripts atwwwhindawicom

8 Security and Communication Networks

100 5 15 20minus5minus10

ΓR by simulationΓO by simulationΓS by simulation

ΓR by analysisΓO by analysisΓS by analysis

0

01

02

03

04

05Γ

06

07

08

09

1

EbNj (dB)

Figure 2 Trapped-error probabilities of ratio-based (ΓR) output-based (ΓO) and sum-based (ΓS) measures over partial-band jam-ming and AWGN channel

01

02

03

04

05

06

07

08

09

1

Γ

minus5 0 5 10 15 200

ΓR by simulationΓO by simulationΓS by simulation

ΓR by analysisΓO by analysisΓS by analysis

EbNj (dB)

Figure 3 Trapped-error probabilities of ratio-based (ΓR) output-based (ΓO) and sum-based (ΓS) measures over partial-band jam-ming and Rayleigh channels

sum-based ratio-based ratio-to-output-based and ratio-to-sum-based GMD decoding respectively In this figure wealso present the performance using log-likelihood ratio (LLR)based GMD decoding as the triangle-marked line This isthe performance of a system with perfect CSIR such asSNR SJR and the existence of a jamming signal This LLR-GMD decoding iteratively erases the least reliable symbolsbased on LLR as described in [19 22 26] Because it is wellknown that MO-RTT exhibits better performance than OTT

EbNj (dB)

10minus5

10minus4

10minus3

10minus2

10minus1

100

WER

5 10 15 20 250

Without EIMO-RTTO-GMDS-GMD

R-GMDRO-GMDRS-GMDLLR-GMD

Figure 4 Performance of various erasure insertion schemes overpartial-band jamming and AWGN channels with 120588 = 01 and1198641198871198730 = 5 dB

or RTT [5ndash7] the performances ofOTTandRTTare omittedBefore discussing the simulation result we note that RSO-GMD decoding which serially exploits the three measureshas almost the same performance as RS-GMD decodingtherefore we have also omitted it From Figure 4 we makethe following observations

(1) As we expect LLR-GMD decoding with CSIR showsthe best performance We will focus on the perfor-mance gap between LLR-GMD decoding with CSIRand the other erasure insertion schemes withoutCSIR To achieve WER = 10minus4 LLR-GMD decodingrequires 119864119887119873119895 = 19 dB The ratio-based GMDdecoding and its two-dimensional schemes achievethe target WER at 119864119887119873119895 = 21 dB which is muchcloser than that of MO-RTT Even if the jammingpower is decreased as 119864119887119873119895 = 25 dB the perfor-mance of MO-RTT is still far from WER = 10minus4 Inthe low SJR region only the two two-dimensionaliterative schemes can achieve WER = 10minus2 withoutCSIR We note that the one-dimensional iterativeschemes and MO-RTT require 119864119887119873119895 gt 9 dB toachieve WER = 10minus2

(2) The performance of RS-GMD decoding approachesthat of LLR-GMD decoding as SJR is increased At119864119887119873119895 = 20 dB LLR-GMD decoding with perfectCSIR has 35 less WER than RS-GMD decodingWe note that this gap cannot be closed even if thejamming power is decreased

(3) When the jamming power is decreased that isthe channel approaches the unjammed channel theperformance of R-GMD and RS-GMD decoding is

Security and Communication Networks 9

0

01

02

03

04

05

06

07

08

09

1

Γ

5 10 15 20 25 300

ΓRΓOΓS

EbNj (dB)

Figure 5 Trapped-error probability of various measures overpartial-band jamming and AWGN channels with 119901 = 03 120588 = 01and 1198641198871198730 = 5 dB

still better than the performance of ldquoWithout EIrdquoor MO-RTT This implies that the iterative erasureinsertion and decoding schemes do not decrease theperformance if there are no jamming signals Becauseof this we do not need to detect the jamming signal

In Figure 5 trapped-error probabilities of ratio outputand sum measures are represented which corresponds toFigure 4 To obtain the trapped-error probabilities SJR andSNR were scaled based on the code rate 2031 = 119873119870 whichis different from Section 51 We used 119901 = 03 and the otherparameters are the same as those used in Figure 4 In Figure 5ΓR is much larger than ΓO and ΓS The values of ΓR explain thegood performance of R-GMDdecodingWe find that ΓO goesto lt1 minus119870119873 asymp 03 at 119864119887119873119895 asymp 10 dB Any iterative schemethat has zero trapped-error probability at a given SJR has thesame performance as the performance of ldquoWithout EIrdquo Thisresult explains why O-GMD decoding has a performancedegradation region near 10 dB in Figure 4 Because theperformance of O-GMD decoding rapidly approaches theperformance of ldquoWithout EIrdquo its performance is degraded asSJR is increased We note that the performance of S-GMDdecoding slowly approaches the performance of ldquoWithoutEIrdquo therefore there is no performance degradation region

The performance gain of the iterative schemes in Fig-ure 4 cannot be obtained without drawbacks the iterativeschemes require more decoding trials To investigate howmany decoding trials are required we determine the averagenumber of decoding iterations of various iterative schemesvia simulations The results are shown in Figure 6 In factwe simulated iterative schemes that erase one symbol duringthe first loop and then erase two symbols for each remainingloop because this process does not decrease the performanceas described in [19] In general the average number of

1

11

12

13

14

15

16

17

18

Aver

age n

umbe

r of d

ecod

ing

tria

ls

2 4 6 8 10 12 14 16 18 200

O-GMDS-GMDR-GMD

RS-GMDLLR-GMD

EbNj (dB)

Figure 6 Average number of decoding iterations for various erasureinsertion schemes over partial-band jamming and AWGN channelswith 120588 = 01 and 1198641198871198730 = 5 dB

MO-RTTRS-SEI

10minus3

10minus2

10minus1

WER

02 03 04 05 06 07 08 0901

Figure 7 Performance comparison of MO-RTT and RS-GMDdecoding scheme over partial-band jamming and AWGN channelswith various values of 120588 1198641198871198730 = 5 dB and 119864119887119873119895 = 10 dB

decoding trials decreases rapidly as SJR increases At highSJR most of the received words are completely decoded by asingle trial of error-only decoding which holds true becausehigher SJR implies that the environment becomes less andless hostile For low SJR the average decoding trial of RS-GMD decoding is approximately 175 (at 119864119887119873119895 = 0 dB)which decreases as SJR increasesThis result indicates that theiterative erasure insertion and decoding scheme is practicallyimplementable We note that the trial number of O-GMDdecoding is increased at approximately 119864119887119873119895 asymp 10 dB

10 Security and Communication Networks

and that is consistent with the performance degradation ofO-GMD decoding in Figures 4 and 5 Because O-GMDdecoding could not accurately erase the erroneous symbolsthe decoder has to try more decoding Because of this resultwemust use S-GMDdecoding instead of O-GMDdecodingfor the two-dimensional scheme with R-GMD decodingNow recall the analysis results of Section 51 In Figure 2we observed that the trapped-error probability of the ratiomeasure is increased as SJR is increased and that of the summeasure is decreased For the case in which the trapped-errorprobabilities cross at the point of SJR if we want to havefewer decodings at the higher SJR then we must use R-GMDdecoding first in RS-GMDdecoding If we want to have fewerdecodings at the lower SJR then S-GMD decoding should beapplied first In Figures 4 and 6 because we are consideringthe SJR region in which ΓR gt ΓS we have first exploited R-GMD decoding for RS-GMD decoding

In Figure 7 the performances of MO-RTT and RS-GMDdecoding are shown over a partial-band jamming and anAWGN channel with various jamming duty ratios 120588 withfixed 1198641198871198730 = 5 dB and 119864119887119873119895 = 10 dB For all values of120588 RS-GMD decoding works much better than MO-RTT Wenote that two erasure insertion schemes are more efficient forsmaller 120588 channels53 Performance over Rayleigh Fading Channel Figure 8shows the performance of various erasure insertion schemesover partial-band jamming and Rayleigh fading channel with120588 = 01 and 1198641198871198730 = 12 dB For each symbol transmission120572 in Figure 1 is realized by an iid Rayleigh distribution Weassume that fading coefficients are not known to the receiverexcept for the system of MO-RTT

In Figure 8 RS-GMD decoding exhibits better per-formance than MO-RTT which is the same result as wefound over the AWGN channel For the same WER of 10minus3the jammer should spend 14 dB more 119873119895 The WER ofratio-based schemes (including two-dimensional schemes)approach 10minus4 Unlike the results for an AWGN channelR-GMD decoding is dominant and the contribution of S-GMD decoding is marginal Therefore we conclude that R-GMD decoding is sufficient for Rayleigh fading channelsAdditionally we present the performance of LLR-GMDdecoding which knows perfect CSIR As we discussed inSection 52 the performance of R-SEI approaches that of LLR-SEI and the gap is maintained for every SJR

In Figure 9 the trapped-error probabilities of the mea-sures are displayed over a Rayleigh fading channel with1198641198871198730 = 12 dB The other parameters are the same as thosein Figure 5 In this figure the trapped-error probability of theratio measure is 1 in all SJR regions In other words R-GMDdecoding is the best one-dimensional iterative scheme for thetarget system parameters As we found for the other trapped-error probability results the trapped-error probabilities ofsum and output measures decrease as SJR increases Wherethe trapped-error probabilities are lower than lt1 minus119870119873 asymp03 that is 119864119887119873119895 = 10 dB and 16 dB for output andsum respectively the performances of O-GMD and S-GMDdecoding follow the performance of ldquoWithout EIrdquo As we

10minus5

10minus4

10minus3

10minus2

10minus1

100

WER

5 10 15 20 25 300

Without EIMO-RTTO-GMDS-GMD

R-GMDRO-GMDRS-GMDLLR-GMD

EbNj (dB)

Figure 8 Performance of various erasure insertion schemes over aRayleigh fading channel where 120588 = 01 and 1198641198871198730 = 12 dB

0

01

02

03

04

05

06

07

08

09

1

5 10 15 20 25 300

Γ

ΓRΓOΓS

EbNj (dB)

Figure 9 Trapped-error probability of various iterative schemesover partial-band jamming and Rayleigh fading channels with 119901 =03 120588 = 01 and 1198641198871198730 = 12 dB

discussed above the fast decreasing trapped-error probabilityof the output measure causes the performance degradation ofO-GMD decoding at approximately 10 dB in Figure 9

6 Concluding Remarks

In this paper we considered iterative erasure insertion anddecoding schemes that do not require any preoptimizedthresholds or any CSIR Additionally we proposed a new

Security and Communication Networks 11

analysis framework for the ratio output and sum mea-sures From the simulation results we confirmed that theratio-based GMD decoding scheme and its two-dimensionalschemes have the best performance Using the trapped-error probability the performances of the iterative erasureinsertion and decoding schemes are explained

Appendix

Error and Erasure Probability of SumThreshold Test

Let us denote the symbol error and erasure probabilities ofSTT as 119875err and 119875ers respectively The symbol error eventoccurs when a received symbol is erroneous but is not erasedby STT The symbol erasure event occurs when a receivedsymbol is erased by STTThe two probabilities can be dividedinto two cases as follows

119875err = (1 minus 120588) 119875119880err + 120588119875119869err119875ers = (1 minus 120588) 119875119880ers + 120588119875119869ers (A1)

where 119869 and 119880 refer to the jammed and the unjammed casesrespectively We denote the symbol erase event of STT by119867119904that is 119911 = sum119872minus1119895=0 119910119895 gt 120583 Then 119875119880err can be represented asfollows

119875119880err = Pr [1198670 119867119904 | 119880]= Pr [119867119904 | 119880] minus Pr [1198670 119867119904 | 119880] (A2)

The first term Pr[119867119904 | 119880] and the second term Pr[1198670 119867119904 |119880] are given as (A3) and (A4) respectively Similarly119875119880ers = Pr[119867119904 | 119880] = 1 minus Pr[119867119904 | 119880] can be obtained using(A3) By replacing 119892119880119895 with 119892119869119895 119895 = 0 119872 minus 1 we cancalculate the error and erase probabilities of the unjammedcase For the case in which a jamming signal exists 119875119869err and119875119869ers can be calculated by substituting 119892119869119895(119910119895) for 119892119880119895(119910119895)119895 = 0 119872 minus 1 The derivation of average error probability119875err and erasure probability 119875ers is complete

Pr [119867119904 | 119880] = Pr [119911 lt 120583 | 119880] = Pr[[119872minus1sum119895=0

119910119895 lt 120583]] = int1205830int120583minus11991000

sdot sdot sdot int120583minussum119872minus2119895=0 1199101198950

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 11988911991011198891199100 (A3)

Pr [1198670 119867119904 | 119880] = Pr[[1199100 gt 119910119895 forall119895 = 1 119872 minus 1 119872minus1sum119895=0

119910119895 lt 120583]]= int1205830intmin(1199100 120583minus1199100)

0sdot sdot sdot intmin(1199100 120583minussum

119872minus2119895=0 119910119895)

0

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 11988911991011198891199100(A4)

Disclosure

The present address of Jinsoo Park is the Agency for DefenseDevelopment Daejeon Republic of Korea

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors gratefully acknowledge the support from Elec-tronic Warfare Research Center at the Gwangju Institute ofScience and Technology (GIST) originally funded by theDefense Acquisition Program Administration (DAPA) andthe Agency for Defense Development (ADD)

References

[1] L Milstein and M Simon ldquoSpread Spectrum Communi-cationsrdquo in The Mobile Communications Handbook SecondEdition vol 19990749 of Electrical Engineering Handbook CRCPress 1999

[2] C W Baum and M B Pursley ldquoBayesian Methods for ErasureInsertion in Frequency-Hop Communication Systems withPartial-Band Interferencerdquo IEEE Transactions on Communica-tions vol 40 no 7 pp 1231ndash1238 1992

[3] L-L Yang and L Hanzo ldquoPerformance analysis of codedM-aryorthogonal signaling using errors-and-erasures decoding overfrequency-selective fading channelsrdquo IEEE Journal on SelectedAreas in Communications vol 19 no 2 pp 211ndash221 2001

[4] A J Viterbi ldquoA Robust Ratio-Threshold Technique to MitigateTone and Partial Band Jamming in Coded MFSK Systemsrdquo inProceedings of the IEEE Military Communications ConferenceMILCOM 1982 pp 224-1ndash224-5 Boston MA October 1982

[5] L-L Yang and L Hanzo ldquoLow complexity erasure insertion inRS-coded SFH spread-spectrum communications with partial-band interference and nakagami-m fadingrdquo IEEE Transactionson Communications vol 50 no 6 pp 914ndash925 2002

[6] S Ahmed L-L Yang and L Hanzo ldquoErasure insertion in RS-coded SFH MFSK subjected to tone jamming and Rayleighfadingrdquo IEEE Transactions on Vehicular Technology vol 56 no6 I pp 3563ndash3571 2007

[7] H A Ngo S Ahmed L-L Yang and L Hanzo ldquoNon-coherent cooperative communications dispensing with channelestimation relying on erasure insertion aided Reed-Solomon

12 Security and Communication Networks

coded SFH M-ary FSK subjected to partial-band interferenceand Rayleigh fadingrdquo IEEE Transactions on Communicationsvol 60 no 8 pp 2177ndash2186 2012

[8] T G Macdonald and M B Pursley ldquoStaggered interleavingand iterative errors-and-erasures decoding for frequency-hoppacket radiordquo IEEE Transactions on Wireless Communicationsvol 2 no 1 pp 92ndash98 2003

[9] C-M Lee Y T Su and L-D Jeng ldquoPerformance analysis ofblock codes in hidden Markov channelsrdquo IEEE Transactions onCommunications vol 56 no 1 pp 1ndash4 2008

[10] A Aung K C Teh and K H Li ldquoDetection and counter-measure of interference in slow FHMFSK systems over fadingchannelsrdquo Physical Communication vol 10 pp 11ndash23 2014

[11] J G Kim ldquoError and erasure decoding scheme for RS codes inMFSK based FHSSMA communicationrdquo International Journalof Multimedia and Ubiquitous Engineering vol 10 no 10 pp11ndash20 2015

[12] R Koetter and A Vardy ldquoAlgebraic soft-decision decodingof Reed-Solomon codesrdquo Institute of Electrical and ElectronicsEngineers Transactions on InformationTheory vol 49 no 11 pp2809ndash2825 2003

[13] J Jiang and K R Narayanan ldquoIterative soft decoding of Reed-Solomon codesrdquo IEEE Communications Letters vol 8 no 4 pp244ndash246 2004

[14] P Sehier P Brelivet and I Aubry ldquoNew jammer estimationmethod adapted to FH-PSK waveformsrdquo in Proceedings of the1994 IEEE MILCOM Part 3 (of 3) pp 1027ndash1031 October 1994

[15] M Pursley and J Skinner ldquoDecoding strategies for turboproduct codes in frequency-hop wireless communicationsrdquo inProceedings of the IEEE International Conference on Communi-cations pp 2963ndash2968 Anchorage AK USA

[16] M B Pursley and J S Skinner ldquoAdaptive coding for frequency-hop transmission in mobile ad hoc networks with partial-bandinterferencerdquo IEEETransactions onCommunications vol 57 no3 pp 801ndash811 2009

[17] M R Masse M B Pursley and J S Skinner ldquoAdaptive codingfor frequency-hop transmission over fading channels withpartial-band interferencerdquo IEEE Transactions on Communica-tions vol 59 no 3 pp 854ndash862 2011

[18] H El Gamal and E Geraniotis ldquoIterative channel estimationand decoding for convolutionally coded anti-jam FH signalsrdquoIEEE Transactions on Communications vol 50 no 2 pp 321ndash331 2002

[19] G D Forney ldquoGeneralized Minimum Distance DecodingrdquoIEEE Transactions on Information Theory vol IT-12 no 2 pp125ndash131 1966

[20] D Chase ldquoA Class of Algorithms for Decoding Block CodeswithChannelMeasurement Informationrdquo IEEETransactions onInformation Theory vol 18 no 1 pp 170ndash182 1972

[21] H Tang Y LiuM Fossorier and S Lin ldquoOn combining Chase-2 and GMD decoding algorithms for nonbinary block codesrdquoIEEE Communications Letters vol 5 no 5 pp 209ndash211 2001

[22] S-W Lee and B V K V Kumar ldquoSoft-decision decoding ofreed-solomon codes using successive error-and-ersure decod-ingrdquo in Proceedings of the 2008 IEEEGlobal TelecommunicationsConference GLOBECOM2008 pp 3065ndash3069USADecember2008

[23] E R Berlekamp ldquoReadable Erasures Improve the Performanceof Reed-Solomon Codesrdquo IEEE Transactions on InformationTheory vol 24 no 5 pp 632-633 1978

[24] J G Proakis andM SalehiEssentials of Communication SystemsEngineering 2005

[25] W G Phoel ldquoIterative demodulation and decoding offrequency-hopped PSK in partial-band jammingrdquo IEEEJournal on Selected Areas in Communications vol 23 no 5 pp1026ndash1033 2005

[26] S Lin and D J Costello Error Control Coding Pearson 2ndedition 2004

[27] J K Wolf A M Michelson and A H Levesque ldquoOn theProbability of Undetected Error for Linear Block Codesrdquo IEEETransactions on Communications vol 30 no 2 pp 317ndash3241982

[28] A Ramesh A Chockalingam and L B Milstein ldquoPerformanceof noncoherent turbo detection on Rayleigh fading channelsrdquoin Proceedings of the IEEE Global Telecommunicatins ConferenceGLOBECOMrsquo01 pp 1193ndash1198 usa November 2001

[29] M C Valenti and S Cheng ldquoIterative demodulation anddecoding of turbo-codedM-ary noncoherent orthogonal mod-ulationrdquo IEEE Journal on Selected Areas in Communications vol23 no 9 pp 1739ndash1747 2005

[30] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press BostonMassUSA 5th edition 1994

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Security and Communication Networks 9

0

01

02

03

04

05

06

07

08

09

1

Γ

5 10 15 20 25 300

ΓRΓOΓS

EbNj (dB)

Figure 5 Trapped-error probability of various measures overpartial-band jamming and AWGN channels with 119901 = 03 120588 = 01and 1198641198871198730 = 5 dB

still better than the performance of ldquoWithout EIrdquoor MO-RTT This implies that the iterative erasureinsertion and decoding schemes do not decrease theperformance if there are no jamming signals Becauseof this we do not need to detect the jamming signal

In Figure 5 trapped-error probabilities of ratio outputand sum measures are represented which corresponds toFigure 4 To obtain the trapped-error probabilities SJR andSNR were scaled based on the code rate 2031 = 119873119870 whichis different from Section 51 We used 119901 = 03 and the otherparameters are the same as those used in Figure 4 In Figure 5ΓR is much larger than ΓO and ΓS The values of ΓR explain thegood performance of R-GMDdecodingWe find that ΓO goesto lt1 minus119870119873 asymp 03 at 119864119887119873119895 asymp 10 dB Any iterative schemethat has zero trapped-error probability at a given SJR has thesame performance as the performance of ldquoWithout EIrdquo Thisresult explains why O-GMD decoding has a performancedegradation region near 10 dB in Figure 4 Because theperformance of O-GMD decoding rapidly approaches theperformance of ldquoWithout EIrdquo its performance is degraded asSJR is increased We note that the performance of S-GMDdecoding slowly approaches the performance of ldquoWithoutEIrdquo therefore there is no performance degradation region

The performance gain of the iterative schemes in Fig-ure 4 cannot be obtained without drawbacks the iterativeschemes require more decoding trials To investigate howmany decoding trials are required we determine the averagenumber of decoding iterations of various iterative schemesvia simulations The results are shown in Figure 6 In factwe simulated iterative schemes that erase one symbol duringthe first loop and then erase two symbols for each remainingloop because this process does not decrease the performanceas described in [19] In general the average number of

1

11

12

13

14

15

16

17

18

Aver

age n

umbe

r of d

ecod

ing

tria

ls

2 4 6 8 10 12 14 16 18 200

O-GMDS-GMDR-GMD

RS-GMDLLR-GMD

EbNj (dB)

Figure 6 Average number of decoding iterations for various erasureinsertion schemes over partial-band jamming and AWGN channelswith 120588 = 01 and 1198641198871198730 = 5 dB

MO-RTTRS-SEI

10minus3

10minus2

10minus1

WER

02 03 04 05 06 07 08 0901

Figure 7 Performance comparison of MO-RTT and RS-GMDdecoding scheme over partial-band jamming and AWGN channelswith various values of 120588 1198641198871198730 = 5 dB and 119864119887119873119895 = 10 dB

decoding trials decreases rapidly as SJR increases At highSJR most of the received words are completely decoded by asingle trial of error-only decoding which holds true becausehigher SJR implies that the environment becomes less andless hostile For low SJR the average decoding trial of RS-GMD decoding is approximately 175 (at 119864119887119873119895 = 0 dB)which decreases as SJR increasesThis result indicates that theiterative erasure insertion and decoding scheme is practicallyimplementable We note that the trial number of O-GMDdecoding is increased at approximately 119864119887119873119895 asymp 10 dB

10 Security and Communication Networks

and that is consistent with the performance degradation ofO-GMD decoding in Figures 4 and 5 Because O-GMDdecoding could not accurately erase the erroneous symbolsthe decoder has to try more decoding Because of this resultwemust use S-GMDdecoding instead of O-GMDdecodingfor the two-dimensional scheme with R-GMD decodingNow recall the analysis results of Section 51 In Figure 2we observed that the trapped-error probability of the ratiomeasure is increased as SJR is increased and that of the summeasure is decreased For the case in which the trapped-errorprobabilities cross at the point of SJR if we want to havefewer decodings at the higher SJR then we must use R-GMDdecoding first in RS-GMDdecoding If we want to have fewerdecodings at the lower SJR then S-GMD decoding should beapplied first In Figures 4 and 6 because we are consideringthe SJR region in which ΓR gt ΓS we have first exploited R-GMD decoding for RS-GMD decoding

In Figure 7 the performances of MO-RTT and RS-GMDdecoding are shown over a partial-band jamming and anAWGN channel with various jamming duty ratios 120588 withfixed 1198641198871198730 = 5 dB and 119864119887119873119895 = 10 dB For all values of120588 RS-GMD decoding works much better than MO-RTT Wenote that two erasure insertion schemes are more efficient forsmaller 120588 channels53 Performance over Rayleigh Fading Channel Figure 8shows the performance of various erasure insertion schemesover partial-band jamming and Rayleigh fading channel with120588 = 01 and 1198641198871198730 = 12 dB For each symbol transmission120572 in Figure 1 is realized by an iid Rayleigh distribution Weassume that fading coefficients are not known to the receiverexcept for the system of MO-RTT

In Figure 8 RS-GMD decoding exhibits better per-formance than MO-RTT which is the same result as wefound over the AWGN channel For the same WER of 10minus3the jammer should spend 14 dB more 119873119895 The WER ofratio-based schemes (including two-dimensional schemes)approach 10minus4 Unlike the results for an AWGN channelR-GMD decoding is dominant and the contribution of S-GMD decoding is marginal Therefore we conclude that R-GMD decoding is sufficient for Rayleigh fading channelsAdditionally we present the performance of LLR-GMDdecoding which knows perfect CSIR As we discussed inSection 52 the performance of R-SEI approaches that of LLR-SEI and the gap is maintained for every SJR

In Figure 9 the trapped-error probabilities of the mea-sures are displayed over a Rayleigh fading channel with1198641198871198730 = 12 dB The other parameters are the same as thosein Figure 5 In this figure the trapped-error probability of theratio measure is 1 in all SJR regions In other words R-GMDdecoding is the best one-dimensional iterative scheme for thetarget system parameters As we found for the other trapped-error probability results the trapped-error probabilities ofsum and output measures decrease as SJR increases Wherethe trapped-error probabilities are lower than lt1 minus119870119873 asymp03 that is 119864119887119873119895 = 10 dB and 16 dB for output andsum respectively the performances of O-GMD and S-GMDdecoding follow the performance of ldquoWithout EIrdquo As we

10minus5

10minus4

10minus3

10minus2

10minus1

100

WER

5 10 15 20 25 300

Without EIMO-RTTO-GMDS-GMD

R-GMDRO-GMDRS-GMDLLR-GMD

EbNj (dB)

Figure 8 Performance of various erasure insertion schemes over aRayleigh fading channel where 120588 = 01 and 1198641198871198730 = 12 dB

0

01

02

03

04

05

06

07

08

09

1

5 10 15 20 25 300

Γ

ΓRΓOΓS

EbNj (dB)

Figure 9 Trapped-error probability of various iterative schemesover partial-band jamming and Rayleigh fading channels with 119901 =03 120588 = 01 and 1198641198871198730 = 12 dB

discussed above the fast decreasing trapped-error probabilityof the output measure causes the performance degradation ofO-GMD decoding at approximately 10 dB in Figure 9

6 Concluding Remarks

In this paper we considered iterative erasure insertion anddecoding schemes that do not require any preoptimizedthresholds or any CSIR Additionally we proposed a new

Security and Communication Networks 11

analysis framework for the ratio output and sum mea-sures From the simulation results we confirmed that theratio-based GMD decoding scheme and its two-dimensionalschemes have the best performance Using the trapped-error probability the performances of the iterative erasureinsertion and decoding schemes are explained

Appendix

Error and Erasure Probability of SumThreshold Test

Let us denote the symbol error and erasure probabilities ofSTT as 119875err and 119875ers respectively The symbol error eventoccurs when a received symbol is erroneous but is not erasedby STT The symbol erasure event occurs when a receivedsymbol is erased by STTThe two probabilities can be dividedinto two cases as follows

119875err = (1 minus 120588) 119875119880err + 120588119875119869err119875ers = (1 minus 120588) 119875119880ers + 120588119875119869ers (A1)

where 119869 and 119880 refer to the jammed and the unjammed casesrespectively We denote the symbol erase event of STT by119867119904that is 119911 = sum119872minus1119895=0 119910119895 gt 120583 Then 119875119880err can be represented asfollows

119875119880err = Pr [1198670 119867119904 | 119880]= Pr [119867119904 | 119880] minus Pr [1198670 119867119904 | 119880] (A2)

The first term Pr[119867119904 | 119880] and the second term Pr[1198670 119867119904 |119880] are given as (A3) and (A4) respectively Similarly119875119880ers = Pr[119867119904 | 119880] = 1 minus Pr[119867119904 | 119880] can be obtained using(A3) By replacing 119892119880119895 with 119892119869119895 119895 = 0 119872 minus 1 we cancalculate the error and erase probabilities of the unjammedcase For the case in which a jamming signal exists 119875119869err and119875119869ers can be calculated by substituting 119892119869119895(119910119895) for 119892119880119895(119910119895)119895 = 0 119872 minus 1 The derivation of average error probability119875err and erasure probability 119875ers is complete

Pr [119867119904 | 119880] = Pr [119911 lt 120583 | 119880] = Pr[[119872minus1sum119895=0

119910119895 lt 120583]] = int1205830int120583minus11991000

sdot sdot sdot int120583minussum119872minus2119895=0 1199101198950

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 11988911991011198891199100 (A3)

Pr [1198670 119867119904 | 119880] = Pr[[1199100 gt 119910119895 forall119895 = 1 119872 minus 1 119872minus1sum119895=0

119910119895 lt 120583]]= int1205830intmin(1199100 120583minus1199100)

0sdot sdot sdot intmin(1199100 120583minussum

119872minus2119895=0 119910119895)

0

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 11988911991011198891199100(A4)

Disclosure

The present address of Jinsoo Park is the Agency for DefenseDevelopment Daejeon Republic of Korea

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors gratefully acknowledge the support from Elec-tronic Warfare Research Center at the Gwangju Institute ofScience and Technology (GIST) originally funded by theDefense Acquisition Program Administration (DAPA) andthe Agency for Defense Development (ADD)

References

[1] L Milstein and M Simon ldquoSpread Spectrum Communi-cationsrdquo in The Mobile Communications Handbook SecondEdition vol 19990749 of Electrical Engineering Handbook CRCPress 1999

[2] C W Baum and M B Pursley ldquoBayesian Methods for ErasureInsertion in Frequency-Hop Communication Systems withPartial-Band Interferencerdquo IEEE Transactions on Communica-tions vol 40 no 7 pp 1231ndash1238 1992

[3] L-L Yang and L Hanzo ldquoPerformance analysis of codedM-aryorthogonal signaling using errors-and-erasures decoding overfrequency-selective fading channelsrdquo IEEE Journal on SelectedAreas in Communications vol 19 no 2 pp 211ndash221 2001

[4] A J Viterbi ldquoA Robust Ratio-Threshold Technique to MitigateTone and Partial Band Jamming in Coded MFSK Systemsrdquo inProceedings of the IEEE Military Communications ConferenceMILCOM 1982 pp 224-1ndash224-5 Boston MA October 1982

[5] L-L Yang and L Hanzo ldquoLow complexity erasure insertion inRS-coded SFH spread-spectrum communications with partial-band interference and nakagami-m fadingrdquo IEEE Transactionson Communications vol 50 no 6 pp 914ndash925 2002

[6] S Ahmed L-L Yang and L Hanzo ldquoErasure insertion in RS-coded SFH MFSK subjected to tone jamming and Rayleighfadingrdquo IEEE Transactions on Vehicular Technology vol 56 no6 I pp 3563ndash3571 2007

[7] H A Ngo S Ahmed L-L Yang and L Hanzo ldquoNon-coherent cooperative communications dispensing with channelestimation relying on erasure insertion aided Reed-Solomon

12 Security and Communication Networks

coded SFH M-ary FSK subjected to partial-band interferenceand Rayleigh fadingrdquo IEEE Transactions on Communicationsvol 60 no 8 pp 2177ndash2186 2012

[8] T G Macdonald and M B Pursley ldquoStaggered interleavingand iterative errors-and-erasures decoding for frequency-hoppacket radiordquo IEEE Transactions on Wireless Communicationsvol 2 no 1 pp 92ndash98 2003

[9] C-M Lee Y T Su and L-D Jeng ldquoPerformance analysis ofblock codes in hidden Markov channelsrdquo IEEE Transactions onCommunications vol 56 no 1 pp 1ndash4 2008

[10] A Aung K C Teh and K H Li ldquoDetection and counter-measure of interference in slow FHMFSK systems over fadingchannelsrdquo Physical Communication vol 10 pp 11ndash23 2014

[11] J G Kim ldquoError and erasure decoding scheme for RS codes inMFSK based FHSSMA communicationrdquo International Journalof Multimedia and Ubiquitous Engineering vol 10 no 10 pp11ndash20 2015

[12] R Koetter and A Vardy ldquoAlgebraic soft-decision decodingof Reed-Solomon codesrdquo Institute of Electrical and ElectronicsEngineers Transactions on InformationTheory vol 49 no 11 pp2809ndash2825 2003

[13] J Jiang and K R Narayanan ldquoIterative soft decoding of Reed-Solomon codesrdquo IEEE Communications Letters vol 8 no 4 pp244ndash246 2004

[14] P Sehier P Brelivet and I Aubry ldquoNew jammer estimationmethod adapted to FH-PSK waveformsrdquo in Proceedings of the1994 IEEE MILCOM Part 3 (of 3) pp 1027ndash1031 October 1994

[15] M Pursley and J Skinner ldquoDecoding strategies for turboproduct codes in frequency-hop wireless communicationsrdquo inProceedings of the IEEE International Conference on Communi-cations pp 2963ndash2968 Anchorage AK USA

[16] M B Pursley and J S Skinner ldquoAdaptive coding for frequency-hop transmission in mobile ad hoc networks with partial-bandinterferencerdquo IEEETransactions onCommunications vol 57 no3 pp 801ndash811 2009

[17] M R Masse M B Pursley and J S Skinner ldquoAdaptive codingfor frequency-hop transmission over fading channels withpartial-band interferencerdquo IEEE Transactions on Communica-tions vol 59 no 3 pp 854ndash862 2011

[18] H El Gamal and E Geraniotis ldquoIterative channel estimationand decoding for convolutionally coded anti-jam FH signalsrdquoIEEE Transactions on Communications vol 50 no 2 pp 321ndash331 2002

[19] G D Forney ldquoGeneralized Minimum Distance DecodingrdquoIEEE Transactions on Information Theory vol IT-12 no 2 pp125ndash131 1966

[20] D Chase ldquoA Class of Algorithms for Decoding Block CodeswithChannelMeasurement Informationrdquo IEEETransactions onInformation Theory vol 18 no 1 pp 170ndash182 1972

[21] H Tang Y LiuM Fossorier and S Lin ldquoOn combining Chase-2 and GMD decoding algorithms for nonbinary block codesrdquoIEEE Communications Letters vol 5 no 5 pp 209ndash211 2001

[22] S-W Lee and B V K V Kumar ldquoSoft-decision decoding ofreed-solomon codes using successive error-and-ersure decod-ingrdquo in Proceedings of the 2008 IEEEGlobal TelecommunicationsConference GLOBECOM2008 pp 3065ndash3069USADecember2008

[23] E R Berlekamp ldquoReadable Erasures Improve the Performanceof Reed-Solomon Codesrdquo IEEE Transactions on InformationTheory vol 24 no 5 pp 632-633 1978

[24] J G Proakis andM SalehiEssentials of Communication SystemsEngineering 2005

[25] W G Phoel ldquoIterative demodulation and decoding offrequency-hopped PSK in partial-band jammingrdquo IEEEJournal on Selected Areas in Communications vol 23 no 5 pp1026ndash1033 2005

[26] S Lin and D J Costello Error Control Coding Pearson 2ndedition 2004

[27] J K Wolf A M Michelson and A H Levesque ldquoOn theProbability of Undetected Error for Linear Block Codesrdquo IEEETransactions on Communications vol 30 no 2 pp 317ndash3241982

[28] A Ramesh A Chockalingam and L B Milstein ldquoPerformanceof noncoherent turbo detection on Rayleigh fading channelsrdquoin Proceedings of the IEEE Global Telecommunicatins ConferenceGLOBECOMrsquo01 pp 1193ndash1198 usa November 2001

[29] M C Valenti and S Cheng ldquoIterative demodulation anddecoding of turbo-codedM-ary noncoherent orthogonal mod-ulationrdquo IEEE Journal on Selected Areas in Communications vol23 no 9 pp 1739ndash1747 2005

[30] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press BostonMassUSA 5th edition 1994

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

10 Security and Communication Networks

and that is consistent with the performance degradation ofO-GMD decoding in Figures 4 and 5 Because O-GMDdecoding could not accurately erase the erroneous symbolsthe decoder has to try more decoding Because of this resultwemust use S-GMDdecoding instead of O-GMDdecodingfor the two-dimensional scheme with R-GMD decodingNow recall the analysis results of Section 51 In Figure 2we observed that the trapped-error probability of the ratiomeasure is increased as SJR is increased and that of the summeasure is decreased For the case in which the trapped-errorprobabilities cross at the point of SJR if we want to havefewer decodings at the higher SJR then we must use R-GMDdecoding first in RS-GMDdecoding If we want to have fewerdecodings at the lower SJR then S-GMD decoding should beapplied first In Figures 4 and 6 because we are consideringthe SJR region in which ΓR gt ΓS we have first exploited R-GMD decoding for RS-GMD decoding

In Figure 7 the performances of MO-RTT and RS-GMDdecoding are shown over a partial-band jamming and anAWGN channel with various jamming duty ratios 120588 withfixed 1198641198871198730 = 5 dB and 119864119887119873119895 = 10 dB For all values of120588 RS-GMD decoding works much better than MO-RTT Wenote that two erasure insertion schemes are more efficient forsmaller 120588 channels53 Performance over Rayleigh Fading Channel Figure 8shows the performance of various erasure insertion schemesover partial-band jamming and Rayleigh fading channel with120588 = 01 and 1198641198871198730 = 12 dB For each symbol transmission120572 in Figure 1 is realized by an iid Rayleigh distribution Weassume that fading coefficients are not known to the receiverexcept for the system of MO-RTT

In Figure 8 RS-GMD decoding exhibits better per-formance than MO-RTT which is the same result as wefound over the AWGN channel For the same WER of 10minus3the jammer should spend 14 dB more 119873119895 The WER ofratio-based schemes (including two-dimensional schemes)approach 10minus4 Unlike the results for an AWGN channelR-GMD decoding is dominant and the contribution of S-GMD decoding is marginal Therefore we conclude that R-GMD decoding is sufficient for Rayleigh fading channelsAdditionally we present the performance of LLR-GMDdecoding which knows perfect CSIR As we discussed inSection 52 the performance of R-SEI approaches that of LLR-SEI and the gap is maintained for every SJR

In Figure 9 the trapped-error probabilities of the mea-sures are displayed over a Rayleigh fading channel with1198641198871198730 = 12 dB The other parameters are the same as thosein Figure 5 In this figure the trapped-error probability of theratio measure is 1 in all SJR regions In other words R-GMDdecoding is the best one-dimensional iterative scheme for thetarget system parameters As we found for the other trapped-error probability results the trapped-error probabilities ofsum and output measures decrease as SJR increases Wherethe trapped-error probabilities are lower than lt1 minus119870119873 asymp03 that is 119864119887119873119895 = 10 dB and 16 dB for output andsum respectively the performances of O-GMD and S-GMDdecoding follow the performance of ldquoWithout EIrdquo As we

10minus5

10minus4

10minus3

10minus2

10minus1

100

WER

5 10 15 20 25 300

Without EIMO-RTTO-GMDS-GMD

R-GMDRO-GMDRS-GMDLLR-GMD

EbNj (dB)

Figure 8 Performance of various erasure insertion schemes over aRayleigh fading channel where 120588 = 01 and 1198641198871198730 = 12 dB

0

01

02

03

04

05

06

07

08

09

1

5 10 15 20 25 300

Γ

ΓRΓOΓS

EbNj (dB)

Figure 9 Trapped-error probability of various iterative schemesover partial-band jamming and Rayleigh fading channels with 119901 =03 120588 = 01 and 1198641198871198730 = 12 dB

discussed above the fast decreasing trapped-error probabilityof the output measure causes the performance degradation ofO-GMD decoding at approximately 10 dB in Figure 9

6 Concluding Remarks

In this paper we considered iterative erasure insertion anddecoding schemes that do not require any preoptimizedthresholds or any CSIR Additionally we proposed a new

Security and Communication Networks 11

analysis framework for the ratio output and sum mea-sures From the simulation results we confirmed that theratio-based GMD decoding scheme and its two-dimensionalschemes have the best performance Using the trapped-error probability the performances of the iterative erasureinsertion and decoding schemes are explained

Appendix

Error and Erasure Probability of SumThreshold Test

Let us denote the symbol error and erasure probabilities ofSTT as 119875err and 119875ers respectively The symbol error eventoccurs when a received symbol is erroneous but is not erasedby STT The symbol erasure event occurs when a receivedsymbol is erased by STTThe two probabilities can be dividedinto two cases as follows

119875err = (1 minus 120588) 119875119880err + 120588119875119869err119875ers = (1 minus 120588) 119875119880ers + 120588119875119869ers (A1)

where 119869 and 119880 refer to the jammed and the unjammed casesrespectively We denote the symbol erase event of STT by119867119904that is 119911 = sum119872minus1119895=0 119910119895 gt 120583 Then 119875119880err can be represented asfollows

119875119880err = Pr [1198670 119867119904 | 119880]= Pr [119867119904 | 119880] minus Pr [1198670 119867119904 | 119880] (A2)

The first term Pr[119867119904 | 119880] and the second term Pr[1198670 119867119904 |119880] are given as (A3) and (A4) respectively Similarly119875119880ers = Pr[119867119904 | 119880] = 1 minus Pr[119867119904 | 119880] can be obtained using(A3) By replacing 119892119880119895 with 119892119869119895 119895 = 0 119872 minus 1 we cancalculate the error and erase probabilities of the unjammedcase For the case in which a jamming signal exists 119875119869err and119875119869ers can be calculated by substituting 119892119869119895(119910119895) for 119892119880119895(119910119895)119895 = 0 119872 minus 1 The derivation of average error probability119875err and erasure probability 119875ers is complete

Pr [119867119904 | 119880] = Pr [119911 lt 120583 | 119880] = Pr[[119872minus1sum119895=0

119910119895 lt 120583]] = int1205830int120583minus11991000

sdot sdot sdot int120583minussum119872minus2119895=0 1199101198950

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 11988911991011198891199100 (A3)

Pr [1198670 119867119904 | 119880] = Pr[[1199100 gt 119910119895 forall119895 = 1 119872 minus 1 119872minus1sum119895=0

119910119895 lt 120583]]= int1205830intmin(1199100 120583minus1199100)

0sdot sdot sdot intmin(1199100 120583minussum

119872minus2119895=0 119910119895)

0

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 11988911991011198891199100(A4)

Disclosure

The present address of Jinsoo Park is the Agency for DefenseDevelopment Daejeon Republic of Korea

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors gratefully acknowledge the support from Elec-tronic Warfare Research Center at the Gwangju Institute ofScience and Technology (GIST) originally funded by theDefense Acquisition Program Administration (DAPA) andthe Agency for Defense Development (ADD)

References

[1] L Milstein and M Simon ldquoSpread Spectrum Communi-cationsrdquo in The Mobile Communications Handbook SecondEdition vol 19990749 of Electrical Engineering Handbook CRCPress 1999

[2] C W Baum and M B Pursley ldquoBayesian Methods for ErasureInsertion in Frequency-Hop Communication Systems withPartial-Band Interferencerdquo IEEE Transactions on Communica-tions vol 40 no 7 pp 1231ndash1238 1992

[3] L-L Yang and L Hanzo ldquoPerformance analysis of codedM-aryorthogonal signaling using errors-and-erasures decoding overfrequency-selective fading channelsrdquo IEEE Journal on SelectedAreas in Communications vol 19 no 2 pp 211ndash221 2001

[4] A J Viterbi ldquoA Robust Ratio-Threshold Technique to MitigateTone and Partial Band Jamming in Coded MFSK Systemsrdquo inProceedings of the IEEE Military Communications ConferenceMILCOM 1982 pp 224-1ndash224-5 Boston MA October 1982

[5] L-L Yang and L Hanzo ldquoLow complexity erasure insertion inRS-coded SFH spread-spectrum communications with partial-band interference and nakagami-m fadingrdquo IEEE Transactionson Communications vol 50 no 6 pp 914ndash925 2002

[6] S Ahmed L-L Yang and L Hanzo ldquoErasure insertion in RS-coded SFH MFSK subjected to tone jamming and Rayleighfadingrdquo IEEE Transactions on Vehicular Technology vol 56 no6 I pp 3563ndash3571 2007

[7] H A Ngo S Ahmed L-L Yang and L Hanzo ldquoNon-coherent cooperative communications dispensing with channelestimation relying on erasure insertion aided Reed-Solomon

12 Security and Communication Networks

coded SFH M-ary FSK subjected to partial-band interferenceand Rayleigh fadingrdquo IEEE Transactions on Communicationsvol 60 no 8 pp 2177ndash2186 2012

[8] T G Macdonald and M B Pursley ldquoStaggered interleavingand iterative errors-and-erasures decoding for frequency-hoppacket radiordquo IEEE Transactions on Wireless Communicationsvol 2 no 1 pp 92ndash98 2003

[9] C-M Lee Y T Su and L-D Jeng ldquoPerformance analysis ofblock codes in hidden Markov channelsrdquo IEEE Transactions onCommunications vol 56 no 1 pp 1ndash4 2008

[10] A Aung K C Teh and K H Li ldquoDetection and counter-measure of interference in slow FHMFSK systems over fadingchannelsrdquo Physical Communication vol 10 pp 11ndash23 2014

[11] J G Kim ldquoError and erasure decoding scheme for RS codes inMFSK based FHSSMA communicationrdquo International Journalof Multimedia and Ubiquitous Engineering vol 10 no 10 pp11ndash20 2015

[12] R Koetter and A Vardy ldquoAlgebraic soft-decision decodingof Reed-Solomon codesrdquo Institute of Electrical and ElectronicsEngineers Transactions on InformationTheory vol 49 no 11 pp2809ndash2825 2003

[13] J Jiang and K R Narayanan ldquoIterative soft decoding of Reed-Solomon codesrdquo IEEE Communications Letters vol 8 no 4 pp244ndash246 2004

[14] P Sehier P Brelivet and I Aubry ldquoNew jammer estimationmethod adapted to FH-PSK waveformsrdquo in Proceedings of the1994 IEEE MILCOM Part 3 (of 3) pp 1027ndash1031 October 1994

[15] M Pursley and J Skinner ldquoDecoding strategies for turboproduct codes in frequency-hop wireless communicationsrdquo inProceedings of the IEEE International Conference on Communi-cations pp 2963ndash2968 Anchorage AK USA

[16] M B Pursley and J S Skinner ldquoAdaptive coding for frequency-hop transmission in mobile ad hoc networks with partial-bandinterferencerdquo IEEETransactions onCommunications vol 57 no3 pp 801ndash811 2009

[17] M R Masse M B Pursley and J S Skinner ldquoAdaptive codingfor frequency-hop transmission over fading channels withpartial-band interferencerdquo IEEE Transactions on Communica-tions vol 59 no 3 pp 854ndash862 2011

[18] H El Gamal and E Geraniotis ldquoIterative channel estimationand decoding for convolutionally coded anti-jam FH signalsrdquoIEEE Transactions on Communications vol 50 no 2 pp 321ndash331 2002

[19] G D Forney ldquoGeneralized Minimum Distance DecodingrdquoIEEE Transactions on Information Theory vol IT-12 no 2 pp125ndash131 1966

[20] D Chase ldquoA Class of Algorithms for Decoding Block CodeswithChannelMeasurement Informationrdquo IEEETransactions onInformation Theory vol 18 no 1 pp 170ndash182 1972

[21] H Tang Y LiuM Fossorier and S Lin ldquoOn combining Chase-2 and GMD decoding algorithms for nonbinary block codesrdquoIEEE Communications Letters vol 5 no 5 pp 209ndash211 2001

[22] S-W Lee and B V K V Kumar ldquoSoft-decision decoding ofreed-solomon codes using successive error-and-ersure decod-ingrdquo in Proceedings of the 2008 IEEEGlobal TelecommunicationsConference GLOBECOM2008 pp 3065ndash3069USADecember2008

[23] E R Berlekamp ldquoReadable Erasures Improve the Performanceof Reed-Solomon Codesrdquo IEEE Transactions on InformationTheory vol 24 no 5 pp 632-633 1978

[24] J G Proakis andM SalehiEssentials of Communication SystemsEngineering 2005

[25] W G Phoel ldquoIterative demodulation and decoding offrequency-hopped PSK in partial-band jammingrdquo IEEEJournal on Selected Areas in Communications vol 23 no 5 pp1026ndash1033 2005

[26] S Lin and D J Costello Error Control Coding Pearson 2ndedition 2004

[27] J K Wolf A M Michelson and A H Levesque ldquoOn theProbability of Undetected Error for Linear Block Codesrdquo IEEETransactions on Communications vol 30 no 2 pp 317ndash3241982

[28] A Ramesh A Chockalingam and L B Milstein ldquoPerformanceof noncoherent turbo detection on Rayleigh fading channelsrdquoin Proceedings of the IEEE Global Telecommunicatins ConferenceGLOBECOMrsquo01 pp 1193ndash1198 usa November 2001

[29] M C Valenti and S Cheng ldquoIterative demodulation anddecoding of turbo-codedM-ary noncoherent orthogonal mod-ulationrdquo IEEE Journal on Selected Areas in Communications vol23 no 9 pp 1739ndash1747 2005

[30] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press BostonMassUSA 5th edition 1994

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

Security and Communication Networks 11

analysis framework for the ratio output and sum mea-sures From the simulation results we confirmed that theratio-based GMD decoding scheme and its two-dimensionalschemes have the best performance Using the trapped-error probability the performances of the iterative erasureinsertion and decoding schemes are explained

Appendix

Error and Erasure Probability of SumThreshold Test

Let us denote the symbol error and erasure probabilities ofSTT as 119875err and 119875ers respectively The symbol error eventoccurs when a received symbol is erroneous but is not erasedby STT The symbol erasure event occurs when a receivedsymbol is erased by STTThe two probabilities can be dividedinto two cases as follows

119875err = (1 minus 120588) 119875119880err + 120588119875119869err119875ers = (1 minus 120588) 119875119880ers + 120588119875119869ers (A1)

where 119869 and 119880 refer to the jammed and the unjammed casesrespectively We denote the symbol erase event of STT by119867119904that is 119911 = sum119872minus1119895=0 119910119895 gt 120583 Then 119875119880err can be represented asfollows

119875119880err = Pr [1198670 119867119904 | 119880]= Pr [119867119904 | 119880] minus Pr [1198670 119867119904 | 119880] (A2)

The first term Pr[119867119904 | 119880] and the second term Pr[1198670 119867119904 |119880] are given as (A3) and (A4) respectively Similarly119875119880ers = Pr[119867119904 | 119880] = 1 minus Pr[119867119904 | 119880] can be obtained using(A3) By replacing 119892119880119895 with 119892119869119895 119895 = 0 119872 minus 1 we cancalculate the error and erase probabilities of the unjammedcase For the case in which a jamming signal exists 119875119869err and119875119869ers can be calculated by substituting 119892119869119895(119910119895) for 119892119880119895(119910119895)119895 = 0 119872 minus 1 The derivation of average error probability119875err and erasure probability 119875ers is complete

Pr [119867119904 | 119880] = Pr [119911 lt 120583 | 119880] = Pr[[119872minus1sum119895=0

119910119895 lt 120583]] = int1205830int120583minus11991000

sdot sdot sdot int120583minussum119872minus2119895=0 1199101198950

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 11988911991011198891199100 (A3)

Pr [1198670 119867119904 | 119880] = Pr[[1199100 gt 119910119895 forall119895 = 1 119872 minus 1 119872minus1sum119895=0

119910119895 lt 120583]]= int1205830intmin(1199100 120583minus1199100)

0sdot sdot sdot intmin(1199100 120583minussum

119872minus2119895=0 119910119895)

0

119872minus1prod119895=0

119892119880119895 (119910119895) 119889119910119872minus1 sdot sdot sdot 11988911991011198891199100(A4)

Disclosure

The present address of Jinsoo Park is the Agency for DefenseDevelopment Daejeon Republic of Korea

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors gratefully acknowledge the support from Elec-tronic Warfare Research Center at the Gwangju Institute ofScience and Technology (GIST) originally funded by theDefense Acquisition Program Administration (DAPA) andthe Agency for Defense Development (ADD)

References

[1] L Milstein and M Simon ldquoSpread Spectrum Communi-cationsrdquo in The Mobile Communications Handbook SecondEdition vol 19990749 of Electrical Engineering Handbook CRCPress 1999

[2] C W Baum and M B Pursley ldquoBayesian Methods for ErasureInsertion in Frequency-Hop Communication Systems withPartial-Band Interferencerdquo IEEE Transactions on Communica-tions vol 40 no 7 pp 1231ndash1238 1992

[3] L-L Yang and L Hanzo ldquoPerformance analysis of codedM-aryorthogonal signaling using errors-and-erasures decoding overfrequency-selective fading channelsrdquo IEEE Journal on SelectedAreas in Communications vol 19 no 2 pp 211ndash221 2001

[4] A J Viterbi ldquoA Robust Ratio-Threshold Technique to MitigateTone and Partial Band Jamming in Coded MFSK Systemsrdquo inProceedings of the IEEE Military Communications ConferenceMILCOM 1982 pp 224-1ndash224-5 Boston MA October 1982

[5] L-L Yang and L Hanzo ldquoLow complexity erasure insertion inRS-coded SFH spread-spectrum communications with partial-band interference and nakagami-m fadingrdquo IEEE Transactionson Communications vol 50 no 6 pp 914ndash925 2002

[6] S Ahmed L-L Yang and L Hanzo ldquoErasure insertion in RS-coded SFH MFSK subjected to tone jamming and Rayleighfadingrdquo IEEE Transactions on Vehicular Technology vol 56 no6 I pp 3563ndash3571 2007

[7] H A Ngo S Ahmed L-L Yang and L Hanzo ldquoNon-coherent cooperative communications dispensing with channelestimation relying on erasure insertion aided Reed-Solomon

12 Security and Communication Networks

coded SFH M-ary FSK subjected to partial-band interferenceand Rayleigh fadingrdquo IEEE Transactions on Communicationsvol 60 no 8 pp 2177ndash2186 2012

[8] T G Macdonald and M B Pursley ldquoStaggered interleavingand iterative errors-and-erasures decoding for frequency-hoppacket radiordquo IEEE Transactions on Wireless Communicationsvol 2 no 1 pp 92ndash98 2003

[9] C-M Lee Y T Su and L-D Jeng ldquoPerformance analysis ofblock codes in hidden Markov channelsrdquo IEEE Transactions onCommunications vol 56 no 1 pp 1ndash4 2008

[10] A Aung K C Teh and K H Li ldquoDetection and counter-measure of interference in slow FHMFSK systems over fadingchannelsrdquo Physical Communication vol 10 pp 11ndash23 2014

[11] J G Kim ldquoError and erasure decoding scheme for RS codes inMFSK based FHSSMA communicationrdquo International Journalof Multimedia and Ubiquitous Engineering vol 10 no 10 pp11ndash20 2015

[12] R Koetter and A Vardy ldquoAlgebraic soft-decision decodingof Reed-Solomon codesrdquo Institute of Electrical and ElectronicsEngineers Transactions on InformationTheory vol 49 no 11 pp2809ndash2825 2003

[13] J Jiang and K R Narayanan ldquoIterative soft decoding of Reed-Solomon codesrdquo IEEE Communications Letters vol 8 no 4 pp244ndash246 2004

[14] P Sehier P Brelivet and I Aubry ldquoNew jammer estimationmethod adapted to FH-PSK waveformsrdquo in Proceedings of the1994 IEEE MILCOM Part 3 (of 3) pp 1027ndash1031 October 1994

[15] M Pursley and J Skinner ldquoDecoding strategies for turboproduct codes in frequency-hop wireless communicationsrdquo inProceedings of the IEEE International Conference on Communi-cations pp 2963ndash2968 Anchorage AK USA

[16] M B Pursley and J S Skinner ldquoAdaptive coding for frequency-hop transmission in mobile ad hoc networks with partial-bandinterferencerdquo IEEETransactions onCommunications vol 57 no3 pp 801ndash811 2009

[17] M R Masse M B Pursley and J S Skinner ldquoAdaptive codingfor frequency-hop transmission over fading channels withpartial-band interferencerdquo IEEE Transactions on Communica-tions vol 59 no 3 pp 854ndash862 2011

[18] H El Gamal and E Geraniotis ldquoIterative channel estimationand decoding for convolutionally coded anti-jam FH signalsrdquoIEEE Transactions on Communications vol 50 no 2 pp 321ndash331 2002

[19] G D Forney ldquoGeneralized Minimum Distance DecodingrdquoIEEE Transactions on Information Theory vol IT-12 no 2 pp125ndash131 1966

[20] D Chase ldquoA Class of Algorithms for Decoding Block CodeswithChannelMeasurement Informationrdquo IEEETransactions onInformation Theory vol 18 no 1 pp 170ndash182 1972

[21] H Tang Y LiuM Fossorier and S Lin ldquoOn combining Chase-2 and GMD decoding algorithms for nonbinary block codesrdquoIEEE Communications Letters vol 5 no 5 pp 209ndash211 2001

[22] S-W Lee and B V K V Kumar ldquoSoft-decision decoding ofreed-solomon codes using successive error-and-ersure decod-ingrdquo in Proceedings of the 2008 IEEEGlobal TelecommunicationsConference GLOBECOM2008 pp 3065ndash3069USADecember2008

[23] E R Berlekamp ldquoReadable Erasures Improve the Performanceof Reed-Solomon Codesrdquo IEEE Transactions on InformationTheory vol 24 no 5 pp 632-633 1978

[24] J G Proakis andM SalehiEssentials of Communication SystemsEngineering 2005

[25] W G Phoel ldquoIterative demodulation and decoding offrequency-hopped PSK in partial-band jammingrdquo IEEEJournal on Selected Areas in Communications vol 23 no 5 pp1026ndash1033 2005

[26] S Lin and D J Costello Error Control Coding Pearson 2ndedition 2004

[27] J K Wolf A M Michelson and A H Levesque ldquoOn theProbability of Undetected Error for Linear Block Codesrdquo IEEETransactions on Communications vol 30 no 2 pp 317ndash3241982

[28] A Ramesh A Chockalingam and L B Milstein ldquoPerformanceof noncoherent turbo detection on Rayleigh fading channelsrdquoin Proceedings of the IEEE Global Telecommunicatins ConferenceGLOBECOMrsquo01 pp 1193ndash1198 usa November 2001

[29] M C Valenti and S Cheng ldquoIterative demodulation anddecoding of turbo-codedM-ary noncoherent orthogonal mod-ulationrdquo IEEE Journal on Selected Areas in Communications vol23 no 9 pp 1739ndash1747 2005

[30] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press BostonMassUSA 5th edition 1994

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

12 Security and Communication Networks

coded SFH M-ary FSK subjected to partial-band interferenceand Rayleigh fadingrdquo IEEE Transactions on Communicationsvol 60 no 8 pp 2177ndash2186 2012

[8] T G Macdonald and M B Pursley ldquoStaggered interleavingand iterative errors-and-erasures decoding for frequency-hoppacket radiordquo IEEE Transactions on Wireless Communicationsvol 2 no 1 pp 92ndash98 2003

[9] C-M Lee Y T Su and L-D Jeng ldquoPerformance analysis ofblock codes in hidden Markov channelsrdquo IEEE Transactions onCommunications vol 56 no 1 pp 1ndash4 2008

[10] A Aung K C Teh and K H Li ldquoDetection and counter-measure of interference in slow FHMFSK systems over fadingchannelsrdquo Physical Communication vol 10 pp 11ndash23 2014

[11] J G Kim ldquoError and erasure decoding scheme for RS codes inMFSK based FHSSMA communicationrdquo International Journalof Multimedia and Ubiquitous Engineering vol 10 no 10 pp11ndash20 2015

[12] R Koetter and A Vardy ldquoAlgebraic soft-decision decodingof Reed-Solomon codesrdquo Institute of Electrical and ElectronicsEngineers Transactions on InformationTheory vol 49 no 11 pp2809ndash2825 2003

[13] J Jiang and K R Narayanan ldquoIterative soft decoding of Reed-Solomon codesrdquo IEEE Communications Letters vol 8 no 4 pp244ndash246 2004

[14] P Sehier P Brelivet and I Aubry ldquoNew jammer estimationmethod adapted to FH-PSK waveformsrdquo in Proceedings of the1994 IEEE MILCOM Part 3 (of 3) pp 1027ndash1031 October 1994

[15] M Pursley and J Skinner ldquoDecoding strategies for turboproduct codes in frequency-hop wireless communicationsrdquo inProceedings of the IEEE International Conference on Communi-cations pp 2963ndash2968 Anchorage AK USA

[16] M B Pursley and J S Skinner ldquoAdaptive coding for frequency-hop transmission in mobile ad hoc networks with partial-bandinterferencerdquo IEEETransactions onCommunications vol 57 no3 pp 801ndash811 2009

[17] M R Masse M B Pursley and J S Skinner ldquoAdaptive codingfor frequency-hop transmission over fading channels withpartial-band interferencerdquo IEEE Transactions on Communica-tions vol 59 no 3 pp 854ndash862 2011

[18] H El Gamal and E Geraniotis ldquoIterative channel estimationand decoding for convolutionally coded anti-jam FH signalsrdquoIEEE Transactions on Communications vol 50 no 2 pp 321ndash331 2002

[19] G D Forney ldquoGeneralized Minimum Distance DecodingrdquoIEEE Transactions on Information Theory vol IT-12 no 2 pp125ndash131 1966

[20] D Chase ldquoA Class of Algorithms for Decoding Block CodeswithChannelMeasurement Informationrdquo IEEETransactions onInformation Theory vol 18 no 1 pp 170ndash182 1972

[21] H Tang Y LiuM Fossorier and S Lin ldquoOn combining Chase-2 and GMD decoding algorithms for nonbinary block codesrdquoIEEE Communications Letters vol 5 no 5 pp 209ndash211 2001

[22] S-W Lee and B V K V Kumar ldquoSoft-decision decoding ofreed-solomon codes using successive error-and-ersure decod-ingrdquo in Proceedings of the 2008 IEEEGlobal TelecommunicationsConference GLOBECOM2008 pp 3065ndash3069USADecember2008

[23] E R Berlekamp ldquoReadable Erasures Improve the Performanceof Reed-Solomon Codesrdquo IEEE Transactions on InformationTheory vol 24 no 5 pp 632-633 1978

[24] J G Proakis andM SalehiEssentials of Communication SystemsEngineering 2005

[25] W G Phoel ldquoIterative demodulation and decoding offrequency-hopped PSK in partial-band jammingrdquo IEEEJournal on Selected Areas in Communications vol 23 no 5 pp1026ndash1033 2005

[26] S Lin and D J Costello Error Control Coding Pearson 2ndedition 2004

[27] J K Wolf A M Michelson and A H Levesque ldquoOn theProbability of Undetected Error for Linear Block Codesrdquo IEEETransactions on Communications vol 30 no 2 pp 317ndash3241982

[28] A Ramesh A Chockalingam and L B Milstein ldquoPerformanceof noncoherent turbo detection on Rayleigh fading channelsrdquoin Proceedings of the IEEE Global Telecommunicatins ConferenceGLOBECOMrsquo01 pp 1193ndash1198 usa November 2001

[29] M C Valenti and S Cheng ldquoIterative demodulation anddecoding of turbo-codedM-ary noncoherent orthogonal mod-ulationrdquo IEEE Journal on Selected Areas in Communications vol23 no 9 pp 1739ndash1747 2005

[30] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press BostonMassUSA 5th edition 1994

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wwwhindawicom Volume 2018

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Submit your manuscripts atwwwhindawicom

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom