phase noise suppression algorithm based on modified llr...

Research ArticlePhase Noise Suppression Algorithm Based on Modified LLRMetric in SC-FDMA System

Zijie Xu and Guangliang Ren

State Key Laboratory of Integrated Service Networks Xidian University No 2 South Taibai Road Xirsquoan 710071 China

Correspondence should be addressed to Guangliang Ren glrenmailxidianeducn

Received 26 January 2017 Accepted 4 April 2017 Published 9 May 2017

Academic Editor Jit S Mandeep

Copyright copy 2017 Zijie Xu and Guangliang RenThis is an open access article distributed under the Creative CommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

In single-carrier frequency-division multiple-access (SC-FDMA) receivers in the long-term evolution uplink phase noise is one ofthe major radio-frequency front-end impairments that degrade system performance To address this issue we propose a new log-likelihood ratio (LLR) computation algorithm A signal model with residual phase noise is considered in the algorithm Based onthis model we derive a closed-form expression of the likelihood function of the received symbol and calculate more accurate LLRinformation Thus the accuracy of the decoder is increased and the performance of the SC-FDMA system is improved Simulationresults show that our proposed algorithm achieves superior bit error rate (BER) performance compared with the existing LLRcalculation algorithms in high-order quadrature amplitude modulations (QAM)

1 Introduction

To achieve higher spectral efficiencies the current trend inthe long-term evolution (LTE) uplink is to employ high-ordermodulation schemes such as 256-ary quadrature ampli-tude modulation (256-QAM) or 1024-QAM [1] Howeverthe single-carrier frequency-division multiple-access (SC-FDMA) system with high-order modulations is very vulner-able to phase noise (PN) which will break the orthogonalityamong subcarriers and distort the received signals [2]There-fore the bit error rate (BER) performance of the SC-FDMAsystem is seriously degraded To cope with this impairmentmany algorithms have been proposed to suppress PN [3ndash6] A block-type pilot-based intercarrier interference (ICI)suppression algorithm was proposed in [3] The interferencecomponents were directly calculated and the ICI matrix wasreconstructed by exploiting block-type pilot However thealgorithm is impractical since the PN changing rapidly inevery SC-FDMA symbol In [4] a PN suppressing algorithmbased on the equalizer was discussed but this algorithmneeds to transmit some symbols twice which will cause highoverhead In [5] a PN estimation algorithmand away tomiti-gate the PNeffectwere proposedHowever due to the effect ofthe additive white Gaussian noise (AWGN) the estimation of

the PN becomes inaccurate which degrades the performanceof the algorithm In [6 7] iterative receivers were introducedto suppress the PN For such iterative receivers log-likelihoodratio (LLR) calculation at the demodulator is crucial forthe decoder to work properly A pilot-tone assisted LLRalgorithm based on amodified Bessel functionwas derived in[8 9] where only common phase error (CPE) is consideredin the signal model Another approach [10] was to employlinear transform (LT) on the phase noise term and derive anapproximate expression of the likelihood function By usingthe likelihood function a modified LLRmetric was obtainedThe performance improvement of the decoder is limited dueto the inaccurate LLR metric

In this paper we focus on the received signal equalizedby the pilot in the SC-FDMA receiver After carrier-phaserecovery the received signal is disturbed by the residual PNandAWGNBased on this received signalmodel we derive anaccurate closed-form formula of the likelihood function andpropose a modified LLR calculation algorithm Benefittingfrom the modified LLR the decoder works properly andthe BER performance of the system is improved The paperis organized as follows The phase noise model and thesignal model are described in Section 2 The proposed LLRcalculation algorithm is derived in Section 3 Simulation

HindawiJournal of Electrical and Computer EngineeringVolume 2017 Article ID 9410483 5 pageshttpsdoiorg10115520179410483

2 Journal of Electrical and Computer Engineering

Datasource SP L-point

DFT

N-

pointIDFT

AddCPamp

PS

Channel

AWGNRemCPamp

SP

N-

pointDFT

PS FDeqDecoder point

IDFT

L-

s1

s2

sL

y1

y2

yL

s1

s2

sN

y1

y2

yN

ej120579119905

ej120579119903

Figure 1 Block diagram of the discrete baseband SC-FDMA transceiver

results are discussed in Section 4 and conclusions are drawnin Section 5

2 System Model

21 Phase Noise Model PN arises from imperfections in thelocal oscillator (LO) and results in random fluctuations in thephase of LOrsquos output sinusoidal signal PN is usually modeledin two ways one is Wiener model generated by the free-running oscillator [11] TheWiener-type PN is nonstationaryand its variance increases with time the other one is phaselocked loop (PLL) model which is a stationary zero-meanGaussian random process [12] In this paper we model theresidual PN as a zero-mean Gaussian random process withvariance 1205902120579 that is 120579(119899) sim N(0 1205902120579)22 Signal Model Figure 1 shows the block diagram of adiscrete baseband SC-FDMA transceiver Assume that thereare 119873 subcarriers in the SC-FDMA system where each useroccupies 119871 (119871 lt 119873) of them For simplicity and without lossof generality here we focus on a single user scenario An 119871times1data vector s is generated at the transmitter and F119871 denotesan 119871 times 119871 discrete Fourier transform (DFT) matrix with theelement in the119898th row and the 119899th column being given by(F119871)119898119899 = ( 1radic119871) 119890minus1198952120587(119898minus1)(119899minus1)119871 (1)

Similarly FH119873 denotes an 119873 times 119873 inverse DFT (IDFT) matrixwhere (sdot)H denotes the matrix Hermitian operation Thesubcarrier mapping of the user is defined by the 119873 times 119871matrix T whose elements are zeros except for a single ldquo1rdquoin each column The indexes of the rows containing theseldquo1rdquos correspond to the locations of the assigned subcarriersfor the user Then the 119873 times 1 SC-FDMA time-domain (TD)transmitted symbol vector s is given by

s = FH119873TF119871s (2)and the119873 times 1 TD received symbol vector y is given by

y = P119903HP119905s + z = P119903HP119905FH119873TF119871s + z (3)

where P119905 ≜ diag(119890119895120579119905(1) 119890119895120579119905(2) 119890119895120579119905(119873)) and P119903 ≜diag(119890119895120579119903(1) 119890119895120579119903(2) 119890119895120579119903(119873)) denote diagonal PNmatrices atthe transmitter and receiver respectively 120579119905(119899) and 120579119903(119899) arethe PN samples generated by the transmitter and receiverrespectively at the 119899th sample of the SC-FDMA symbol His the 119873 times 119873 circulant channel matrix whose first columncontains the zero-padded channel impulse response (CIR)between the user and the base station z is the119873times 1 complexAWGN vector After equalization the 119871times1 recovered symbolvector y is given by

y = FH119871 ETHF119873y (4)

where E is the frequency domain (FD) equalization matrixWhen we apply FD minimum mean square error (MMSE)equalization we have E ≜ diag(HH(119899 119899)(|H(119899 119899)|2 +120588minus1)119873119899=1) where H is the equivalent diagonal FD channelmatrix obtained from the pilot symbols and 120588 is the signal-to-noise ratio (SNR) of the transmitted signals Note thatH is theresult of the combination of the channel and the phase noiseat the pilot symbols we perform PN initial compensation inthe FD equalization process Substituting (3) back into (4) wehave

y = FH119871 ETHF119873P119903HP119905F

H119873TF119871s + FH119871 ET

HF119873z (5)

From [6] the recovered symbol vector y is approximated as

y = Ps + z (6)

where P ≜ diag(119890119895120579(1) 119890119895120579(2) 119890119895120579(119871)) is the residual PNmatrix 120579(119899) is the residual PN sample (see also in Section 21)and z = [119911(1) 119911(2) 119911(119871)]T is the complex AWGN vectorwith mean zero variance 1205902 that is 119911(119899) sim CN(0 1205902) Notethat residual PN 120579(119899) is independent of 119911(119899) For the sakeof simplicity of the following derivation we write (6) inelemental form and omit the time label as follows119910 = 119890119895120579119904 + 119911 (7)

Journal of Electrical and Computer Engineering 3

3 LLR Calculation Algorithm

31 Conventional LLR Calculation Algorithm ConventionalLLR algorithm considers a signal model where only AWGN119911 is presented [13] 119910 = 119904 + 119911 (8)

The LLR of a particular received bit 119887 is defined as119871 = lnPr (119887 = 1 | 119910)Pr (119887 = 0 | 119910) (9)

where Pr(119887 = 1 | 119910) and Pr(119887 = 0 | 119910) denote the probabilitiesthat given the received symbol 119910 the received bit 119887 shouldbe decoded as ldquo1rdquo and the probability that bit 119887 should bedecoded as ldquo0rdquo respectively Assuming that ldquo1rdquo and ldquo0rdquo bitvalues occur at equal probabilities the conventional LLR (C-LLR) can be written as119871119862 = ln

Pr (119910 | 119887 = 1)Pr (119910 | 119887 = 0) = ln

sum119904isin1198781 119901 (119910 | 119904)sum119904isin1198780 119901 (119910 | 119904)= lnsum119904isin1198781 (11205871205902) exp (minus 1003816100381610038161003816119910 minus 11990410038161003816100381610038162 1205902)sum119904isin1198780 (11205871205902) exp (minus 1003816100381610038161003816119910 minus 11990410038161003816100381610038162 1205902)

(10)

where 1198781 and 1198780 denote subsets of the constellation symbolsin which the decoded bit 119887 equals ldquo1rdquo and ldquo0rdquo respectivelyIn order to reduce the computational complexity Max-Log-MAP algorithm [14] is used in (10) and (10) becomes119871119862 = minus 11205902 (max

119904isin1198781

1003816100381610038161003816119910 minus 11990410038161003816100381610038162 minusmax119904isin1198780

1003816100381610038161003816119910 minus 11990410038161003816100381610038162) (11)

From (11) we can see that for signal model described in(8) the LLR can be obtained by calculating the Euclideandistances between the received symbol 119910 and the nearestconstellation symbol in each of subsets 1198781 and 1198780 Whenit comes to a received signal model with residual PN asshown in (7) the C-LLR calculation algorithm in (11) willno longer be applicable Therefore we derive a modified LLRcalculation algorithm in the presence of residual PN

32 Proposed LLR Calculation Algorithm We focus on thereceived signal model as described in (7) Conditioned on theresidual PN 120579 and 119904 the probability density function (PDF) ofreceived symbol 119910 is119901 (119910 | 120579 119904) = 11205871205902 exp minus 11205902 10038161003816100381610038161003816119910 minus 119890119895120579119904100381610038161003816100381610038162 = 11205871205902sdot exp minus 11205902 (100381610038161003816100381611991010038161003816100381610038162 + |119904|2)sdot exp 21205902 10038161003816100381610038161003816119890minus119895120579119904lowast11991010038161003816100381610038161003816 cos (ang119910 minus ang119904 minus 120579) = 11205871205902sdot exp minus 11205902 (100381610038161003816100381611991010038161003816100381610038162 + |119904|2)sdot exp 21205902 10038161003816100381610038161199101003816100381610038161003816 |119904| cos (1205930 minus 120579)

(12)

where 1205930 ≜ ang119910 minus ang119904 is the angular difference betweenthe received symbol and the constellation symbol beingevaluatedThe likelihood function is evaluated by integrating(12) over residual PN 120579

119901 (119910 | 119904) = int+infinminusinfin

119901 (119910 120579 | 119904) 119889120579= int+infinminusinfin

119901 (119910 | 120579 119904) 119901 (120579 | 119904) 119889120579 (13)

Because the residual PN 120579 is independent of constellationsymbol 119904 that is 119901(120579 | 119904) = 119901(120579) we can rewrite (13) as


119901 (119910 | 120579 119904) 119901 (120579) 119889120579 (14)

where 119901(120579) = (1radic2120587120590120579) expminus120579221205902120579 is the PDF of theresidual PN Substituting (12) into (14) we have

119901 (119910 | 119904) = 11205871205902radic2120587120590120579 exp minus 11205902 (100381610038161003816100381611991010038161003816100381610038162 + |119904|2) sdot 119868 (15)

where

119868 ≜ int+infinminusinfin

exp 21205902 10038161003816100381610038161199101003816100381610038161003816 |119904| cos (1205930 minus 120579) minus 120579221205902120579

119889120579 (16)

To obtain a closed-form solution of the integral 119868 weintroduce the Taylormean value theoremThe correspondingfirst-order Taylor expansion for cos(1205930 minus 120579) in integral 119868 at120579 = 0 is given by

cos (1205930 minus 120579) = cos1205930 + sin1205930 sdot 120579 minus 05sdot cos (1205930 minus 120585) sdot 1205792 (17)

where 119877(120579) ≜ minus05 sdotcos(1205930minus120585) sdot1205792 is the Lagrange remainderand the value of 120585 ranges from 0 to 120579 Considering 1205930 minus 120585approximates to 0 in high SNR scenarios we can rewrite (17)as

cos (1205930 minus 120579) = cos1205930 + sin1205930 sdot 120579 minus 051205792 (18)

Substituting (16) and (18) into (15) we have119901 (119910 | 119904)= 1radic2120587120590120590120579 sdot 1radic10038161003816100381610038161199101003816100381610038161003816 |119904| + 120590221205902120579times exp

11205902 [[[[[

(10038161003816100381610038161199101003816100381610038161003816 |119904| sin1205930)210038161003816100381610038161199101003816100381610038161003816 |119904| + 120590221205902120579

minus 1003816100381610038161003816119910 minus 11990410038161003816100381610038162]]]]] (19)


C-LLRLLR in [10]P-LLR

22 24 26 28 30 32 34 3620SNR (dB)

10minus4

10minus3

10minus2

10minus1

BER

Figure 2 BER performance of various LLR algorithms for 256-QAM

Similar to (10) and (11) we useMax-Log-MAP algorithm andobtain the proposed LLR (P-LLR) as119871119875 = minus 11205902

sdot((((((((

max119904isin1198781

((10038161003816100381610038161199101003816100381610038161003816 |119904| sin1205930)210038161003816100381610038161199101003816100381610038161003816 |119904| + 120590221205902120579

minus 1003816100381610038161003816119910 minus 11990410038161003816100381610038162)minusmax119904isin1198780

((10038161003816100381610038161199101003816100381610038161003816 |119904| sin1205930)210038161003816100381610038161199101003816100381610038161003816 |119904| + 120590221205902120579

minus 1003816100381610038161003816119910 minus 11990410038161003816100381610038162)))))))))

(20)

4 Simulation Results

Numerical results for the proposed LLR (P-LLR) algorithmare presented in this section Conventional LLR (C-LLR) andLLR algorithm in [10] are also experimented for comparisonWe model SC-FDMA systems with 119873 = 512 119871 = 96and the sampling rate is 768MHz as specified in the LTEuplink standard [15] for 5MHz bandwidth The data bits arechannel coded with the convolutional code of code rate 12The 256-QAM and 1024-QAM modulations are consideredThe standard deviation of the residual PN is set to 120590120579 = 7∘

Figures 2 and 3 depict the systemrsquos BER versus SNRperformance for 256-QAM and 1024-QAM with 120590120579 = 7∘respectively The BER of various LLR algorithms decreases asSNR grows It can be seen from the two figures that the BERperformance of our P-LLR algorithm is superior to the othertwo algorithms for both cases The BER of P-LLR algorithmis reduced to 10minus3 at SNR = 28 dB (256-QAM) and SNR =36 dB (1024-QAM) The P-LLR algorithm has about 15 dB

22 24 26 28 30 32 34 36 3820SNR (dB)

10minus3

10minus2

10minus1

BER



and 3 dB gain over algorithm in [10] at BER = 10minus3 (256-QAM) and BER = 2 times 10minus3 (1024-QAM) respectively TheLLR algorithm in [10] has a BER error floor of 4 times 10minus4 (256-QAM) at SNRs above 34 dB and 15 times 10minus3 (1024-QAM) atSNRs above 36 dBThe C-LLR algorithm is much worse thanour P-LLR algorithm and has a BER error floor of 16 times 10minus2(256-QAM) at SNRs above 36 dB and 25times10minus2 (1024-QAM)at SNRs above 38 dB The superiority of the proposed P-LLRalgorithm in BER performance over C-LLR is due to the factthat C-LLR only considers the AWGN while P-LLR takescare of both aspects of the AWGN and the residual PN Theperformance gain of the P-LLR algorithm compared withLLR algorithm in [10] comes from the application of accuratelikelihood function in (19) By applying more accurate LLRthe decoding performance of the decoder is improved andthus the BER of the system is decreased

5 Conclusion

We propose a modified LLR calculation algorithm for thereceived signal model with residual PN in LTE uplink SC-FDMA system whose performance is easily impaired by thePN By using Taylor mean value theorem the closed-formformula of the likelihood function is derived Based on thelikelihood function a modified LLR calculation algorithmis obtained Compared with existing LLR algorithms ourproposed LLR algorithm achieves superior BER perfor-mance The proposed algorithm can be used in high-ordermodulation system which meets the increasing demand forultra-high bit rate in 5G communication systems

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper


Acknowledgments

This work was supported by the National Natural Sci-ence Foundation of China under Grant 91538105 and theNational Basic Research Program of China under Grant2014CB340206

References

[1] P Banelli S Buzzi G Colavolpe A Modenini F Rusek and AUgolini ldquoModulation formats and waveforms for 5G networkswho will be the heir of OFDM an overview of alternativemodulation schemes for improved spectral efficiencyrdquo IEEESignal Processing Magazine vol 31 no 6 pp 80ndash93 2014

[2] G Sridharan and T J Lim ldquoPerformance analysis of SC-FDMAin the presence of receiver phase noiserdquo IEEE Transactions onCommunications vol 60 no 12 pp 3876ndash3885 2012

[3] X Zhang and H-G Ryu ldquoJoint estimation and suppres-sion of phase noise and carrier frequency offset in multiple-inputmultiple-output single carrier frequency divisionmultipleaccess with single-carrier space frequency block codingrdquo IETCommunications vol 4 no 16 pp 1998ndash2007 2010

[4] B R Sang K Jangsu R Heung-Gyoon and L Yingshan ldquoPNSalgorithm for the SC-FDMAcommunication systemwith phasenoiserdquo in Proceedings of the 2009 IFIP International Conferenceon Wireless and Optical Communications Networks (WOCNrsquo09) pp 1ndash5 Cairo Egypt April 2009

[5] V Syrjala ldquoModelling and practical iterative mitigation ofphase noise in SC-FDMArdquo in Proceedings of the 2012 IEEE23rd International Symposium on Personal Indoor and MobileRadio Communications (PIMRC rsquo12) pp 2395ndash2400 AustraliaSeptember 2012

[6] A Gomaa andN Al-Dhahir ldquoPhase noise in asynchronous SC-FDMA systems performance analysis and data-aided compen-sationrdquo IEEE Transactions on Vehicular Technology vol 63 no6 pp 2642ndash2652 2014

[7] S Suyama H Suzuki K Fukawa and J Izumi ldquoIterativereceiver employing phase noise compensation and channelestimation for millimeter-wave OFDM systemsrdquo IEEE Journalon Selected Areas in Communications vol 27 no 8 pp 1358ndash1366 2009

[8] S Cao P-Y Kam and C Yu ldquoPilot-aided log-likelihood ratiofor LDPC codedM-QAMCO-OFDM systemrdquo in Proceedings oftheConference onOptical Fiber Communication TechnicalDigestSeries San Francisco Calif USA March 2014

[9] S Cao P-Y Kam C Yu and X Cheng ldquoPilot-tone assistedlog-likelihood ratio for LDPC coded CO-OFDM systemrdquo IEEEPhotonics Technology Letters vol 26 no 15 pp 1577ndash1580 2014

[10] T Koike-Akino D S Millar K Kojima and K Parsons ldquoPhasenoise robust LLR calculation with linearbilinear transformfor LDPC-coded coherent communicationsrdquo in Proceedings oftheConference on Lasers and Electro-Optics Europe pp 1ndash2 SanJose Calif USA June 2015

[11] L Tomba ldquoOn the effect of wiener phase noise in OFDMsystemsrdquo IEEE Transactions on Communications vol 46 no 5pp 580ndash583 1998

[12] Z Chen and F F Dai ldquoEffects of LOphase and amplitude imbal-ances and phase noise on M-QAM transceiver performancerdquoIEEE Transactions on Industrial Electronics vol 57 no 5 pp1505ndash1517 2010

[13] XWang andHV Poor ldquoIterative (Turbo) soft interference can-cellation and decoding for codedCDMArdquo IEEE Transactions onCommunications vol 47 no 7 pp 1046ndash1061 1999

[14] A J Viterbi ldquoAn intuitive justification and a simplified imple-mentation of the MAP decoder for convolutional codesrdquo IEEEJournal on Selected Areas in Communications vol 16 no 2 pp260ndash264 1998

[15] I T S Sesia and M Baker LTEmdashThe UMTS Long TermEvolution From Theory to Practice Wiley Hoboken NJ USA2009

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Submit your manuscripts athttpswwwhindawicom

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Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



Datasource SP L-point

DFT

N-

pointIDFT

AddCPamp

PS

Channel

AWGNRemCPamp

SP

N-

pointDFT

PS FDeqDecoder point

IDFT

L-

s1

s2

sL

y1

y2

yL

s1

s2

sN

y1

y2

yN

ej120579119905

ej120579119903

Figure 1 Block diagram of the discrete baseband SC-FDMA transceiver

results are discussed in Section 4 and conclusions are drawnin Section 5

2 System Model

21 Phase Noise Model PN arises from imperfections in thelocal oscillator (LO) and results in random fluctuations in thephase of LOrsquos output sinusoidal signal PN is usually modeledin two ways one is Wiener model generated by the free-running oscillator [11] TheWiener-type PN is nonstationaryand its variance increases with time the other one is phaselocked loop (PLL) model which is a stationary zero-meanGaussian random process [12] In this paper we model theresidual PN as a zero-mean Gaussian random process withvariance 1205902120579 that is 120579(119899) sim N(0 1205902120579)22 Signal Model Figure 1 shows the block diagram of adiscrete baseband SC-FDMA transceiver Assume that thereare 119873 subcarriers in the SC-FDMA system where each useroccupies 119871 (119871 lt 119873) of them For simplicity and without lossof generality here we focus on a single user scenario An 119871times1data vector s is generated at the transmitter and F119871 denotesan 119871 times 119871 discrete Fourier transform (DFT) matrix with theelement in the119898th row and the 119899th column being given by(F119871)119898119899 = ( 1radic119871) 119890minus1198952120587(119898minus1)(119899minus1)119871 (1)

Similarly FH119873 denotes an 119873 times 119873 inverse DFT (IDFT) matrixwhere (sdot)H denotes the matrix Hermitian operation Thesubcarrier mapping of the user is defined by the 119873 times 119871matrix T whose elements are zeros except for a single ldquo1rdquoin each column The indexes of the rows containing theseldquo1rdquos correspond to the locations of the assigned subcarriersfor the user Then the 119873 times 1 SC-FDMA time-domain (TD)transmitted symbol vector s is given by

s = FH119873TF119871s (2)and the119873 times 1 TD received symbol vector y is given by

y = P119903HP119905s + z = P119903HP119905FH119873TF119871s + z (3)

where P119905 ≜ diag(119890119895120579119905(1) 119890119895120579119905(2) 119890119895120579119905(119873)) and P119903 ≜diag(119890119895120579119903(1) 119890119895120579119903(2) 119890119895120579119903(119873)) denote diagonal PNmatrices atthe transmitter and receiver respectively 120579119905(119899) and 120579119903(119899) arethe PN samples generated by the transmitter and receiverrespectively at the 119899th sample of the SC-FDMA symbol His the 119873 times 119873 circulant channel matrix whose first columncontains the zero-padded channel impulse response (CIR)between the user and the base station z is the119873times 1 complexAWGN vector After equalization the 119871times1 recovered symbolvector y is given by

y = FH119871 ETHF119873y (4)

where E is the frequency domain (FD) equalization matrixWhen we apply FD minimum mean square error (MMSE)equalization we have E ≜ diag(HH(119899 119899)(|H(119899 119899)|2 +120588minus1)119873119899=1) where H is the equivalent diagonal FD channelmatrix obtained from the pilot symbols and 120588 is the signal-to-noise ratio (SNR) of the transmitted signals Note thatH is theresult of the combination of the channel and the phase noiseat the pilot symbols we perform PN initial compensation inthe FD equalization process Substituting (3) back into (4) wehave

y = FH119871 ETHF119873P119903HP119905F

H119873TF119871s + FH119871 ET

HF119873z (5)

From [6] the recovered symbol vector y is approximated as

y = Ps + z (6)

where P ≜ diag(119890119895120579(1) 119890119895120579(2) 119890119895120579(119871)) is the residual PNmatrix 120579(119899) is the residual PN sample (see also in Section 21)and z = [119911(1) 119911(2) 119911(119871)]T is the complex AWGN vectorwith mean zero variance 1205902 that is 119911(119899) sim CN(0 1205902) Notethat residual PN 120579(119899) is independent of 119911(119899) For the sakeof simplicity of the following derivation we write (6) inelemental form and omit the time label as follows119910 = 119890119895120579119904 + 119911 (7)






Pr (119910 | 119887 = 1)Pr (119910 | 119887 = 0) = ln


(10)


119904isin1198781


1003816100381610038161003816119910 minus 11990410038161003816100381610038162) (11)



(12)




119901 (119910 | 120579 119904) 119901 (120579 | 119904) 119889120579 (13)



119901 (119910 | 120579 119904) 119901 (120579) 119889120579 (14)



where



119889120579 (16)






11205902 [[[[[

(10038161003816100381610038161199101003816100381610038161003816 |119904| sin1205930)210038161003816100381610038161199101003816100381610038161003816 |119904| + 120590221205902120579

minus 1003816100381610038161003816119910 minus 11990410038161003816100381610038162]]]]] (19)



22 24 26 28 30 32 34 3620SNR (dB)

10minus4

10minus3

10minus2

10minus1

BER



sdot((((((((

max119904isin1198781

((10038161003816100381610038161199101003816100381610038161003816 |119904| sin1205930)210038161003816100381610038161199101003816100381610038161003816 |119904| + 120590221205902120579


((10038161003816100381610038161199101003816100381610038161003816 |119904| sin1205930)210038161003816100381610038161199101003816100381610038161003816 |119904| + 120590221205902120579

minus 1003816100381610038161003816119910 minus 11990410038161003816100381610038162)))))))))

(20)




22 24 26 28 30 32 34 36 3820SNR (dB)

10minus3

10minus2

10minus1

BER




5 Conclusion





Acknowledgments


References
















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














Pr (119910 | 119887 = 1)Pr (119910 | 119887 = 0) = ln


(10)


119904isin1198781


1003816100381610038161003816119910 minus 11990410038161003816100381610038162) (11)



(12)




119901 (119910 | 120579 119904) 119901 (120579 | 119904) 119889120579 (13)



119901 (119910 | 120579 119904) 119901 (120579) 119889120579 (14)



where



119889120579 (16)






11205902 [[[[[

(10038161003816100381610038161199101003816100381610038161003816 |119904| sin1205930)210038161003816100381610038161199101003816100381610038161003816 |119904| + 120590221205902120579

minus 1003816100381610038161003816119910 minus 11990410038161003816100381610038162]]]]] (19)



22 24 26 28 30 32 34 3620SNR (dB)

10minus4

10minus3

10minus2

10minus1

BER



sdot((((((((

max119904isin1198781

((10038161003816100381610038161199101003816100381610038161003816 |119904| sin1205930)210038161003816100381610038161199101003816100381610038161003816 |119904| + 120590221205902120579


((10038161003816100381610038161199101003816100381610038161003816 |119904| sin1205930)210038161003816100381610038161199101003816100381610038161003816 |119904| + 120590221205902120579

minus 1003816100381610038161003816119910 minus 11990410038161003816100381610038162)))))))))

(20)




22 24 26 28 30 32 34 36 3820SNR (dB)

10minus3

10minus2

10minus1

BER




5 Conclusion





Acknowledgments


References
















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











22 24 26 28 30 32 34 3620SNR (dB)

10minus4

10minus3

10minus2

10minus1

BER



sdot((((((((

max119904isin1198781

((10038161003816100381610038161199101003816100381610038161003816 |119904| sin1205930)210038161003816100381610038161199101003816100381610038161003816 |119904| + 120590221205902120579


((10038161003816100381610038161199101003816100381610038161003816 |119904| sin1205930)210038161003816100381610038161199101003816100381610038161003816 |119904| + 120590221205902120579

minus 1003816100381610038161003816119910 minus 11990410038161003816100381610038162)))))))))

(20)




22 24 26 28 30 32 34 36 3820SNR (dB)

10minus3

10minus2

10minus1

BER




5 Conclusion





Acknowledgments


References
















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Acknowledgments


References
















RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









phase noise suppression algorithm based on modified llr...

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