MODELING OF MOBILE FREQUENCY HOPPED VHF CHANNELSANDERRORCORRECTIONONTHECHANNEL

ANDRE VAN HEERDEN HENDRIK C. FERREIRA, Member, IEEE

Cybernetics Laboratory, Rand Afrikaans University, PO Box 524, Johannesburg, 2000, Republic of South Africa

ABSTRACT: In this paper we present experimental and analytical results obtained when applying Fritchman [l] partitioned Markov chains to model the error sequences of mobile and stationary frequency hopped VHF radio channels. These results were obtained from real time measurements on the abovementioned channels. The mobile Markov chain channek models were investigated with respect to four modulation schemes namely - FSK, DPSK, QPSK and 8-ary PSK. By using error distributions obtained from both the experimental measurements and the channel models, BCH and Reed-Solomon error correcting codes of various block lengths and information rates, were evaluated. Interleaving is also investigated with regards to optimal depth and results are presented to show the effect of different interleaving depths on the block error rate.

1. INTRODUCTION

An engineer needs some statistical knowledge about the errors to be expected from a channel before he can intelligently choose an error correcting code. With this paper it is our intention to investigate channel models and error characteristics for mobile Frequency Hopped (FH) VHF digital communication systems. Our study involves the use of four modulation schemes namely FSK, DPSK, QPSK and 8- ary PSK, for the transmission of digital data from a vehicle moving (i) through a densely build urban area and (ii) along a freeway, to a stationary received station. At the receiving station the received data is compared to the transmitted data, errors identified and statistics pertaining to the nature of the occurrence of errors recorded. From the recorded statistics, discrete Markov chain models representing the statistical distribution of the errors, can be found. These models then allow studies of the channel to be undertaken without the need of having the channel available. The models are then used to evaluate BCH and Reed-Solomon error correcting codes on the FH channel. Interleaving in conjunction with the abovementioned error correcting codes are also evaluated for the different channel models. We shall also briefly present the methodology for determining the models as well as the P(m,n) distributions, ie the probability that a block of n bits will contain exactly m errors. Swarts et a1 [2] evaluated mobile VHF channels without frequency hopping for various

modulation schemes. Their work did however not include the evaluations of error correcting codes on the modeled channel. Finally, some of the most important conclusions to be drawn from the results will be presented.

11. CHANNEL MODELS

Before starting the discussion of the channel model itself we shall briefly discuss some of the concepts used to describe error statistics. To briefly present the relevant concepts for describing an error sequence, consider the following extract from an error sequence. 1's denote error bits, 0's denote correct bits. W denotes x successive correct bits:

. . . 1 0Ig0 1 o o 1 o o o o o o o 1 ; \ o ~ ~ ~ 1 ,301 o1Y- 1 1 1 . . . V' .4

BURST-IXTERVAL CLUSTER CAP BURST We now introduce the following terminology [3]: GAP: The region between any two consecutive errors. BURST: A region where the local bit error rate exceeds a certain threshold value, A. (0.1 for this study). A burst shall begin with an error (1) and end with an error (1). BURST - INTERVAL: The region between any two consecutive bursts. CLUSTER: A region where the errors occur consecutively with no correct bits in between.

The general structure of a partitioned Fritchman Markov chain model is depicted in fig.1 [l]. The partitioning in this model is done with respect to error and error-free states. In fig.1 the states l,Z, ..., N-I represent error-free bits while state N represents error bits.

?I4

P",

Fig. 1. The general structure of u Fritchman model

I

ICC '92

336.5.1 92CH3132-810000-1047 $3.00 0 1992 IEEE 1047

The various parameters shown in fig. 1 can be found by fitting an exponential curve to the error-free run distribution, P(O"ll), were P(O"l1) denotes the probability of m or more consecutive error-free bits following an error. The curve fit to the error-free run has the following form:

y ( m ) = A,B-"'~ + A ~ B - " ' ~ + . . . . + A ~ - ~ B - ' ~ - I ~ . (1)

In (0, m represents error-free runlengtli and A , Ab...., A,,,., as well as a , a, ...., a,,,.l represent the parameters of an optimal curve fit. From these parameters, the model parameters are given by the following equations:

pll = e-"' ,

pZ2 = e-aa ,

( 7 )

This then is the method for determining a Fritchman model with N-1 error-free states, I, 2, ............ N-1 and one error state designated by N . The state transition matrix, P for this type of model is given by:

0 PIN 1

111. BLOCK ERROR PROBABILITIES AND ERROR CORRECTING CODES

The block error probability distribution, or P(m,n), is a very important parameter to consider when deciding on the use of a particular block error correcting scheme [4]. For the Fritchman channel model, as presented in the previous section, P(m,n) is given by [4]:

where Pi is the probability of being in state i, also termed the stationary state probability of the state i. The first- return disixibutionA(m,n) is given by the following recurrent relationship:

f i (m,n) = pljfl (mn-1) + e p,,f,(m-i,n-i) ,

( 1 4 )

j -k+l

where imposed

i = 1,2, ....., N and the following conditions are

f,(m,n) = 0 J(m,n) = 0 A(0,O) = 1.

for m > n, for n < 0 or m < 0, and

The P(m,n) statistic is used to determine how effective an error correcting scheme is. P(mn) denotes to the code designer the probability that a block of n bits will contain exactly m errors.

In this study we used two error correcting coding schemes, namely the BCH codes because of their powerful random error-correcting capability and Reed-Solomon (RS) codes, which are a subclass of the q-ary BCH codes.

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a) BCH Codes [51 For any positive integers m (m 2 3) and t (t 5 7.' ) there exists a binary BCH code with the following parameters: Block length : Number of parity check digits : Minimum distance :

n =2" -1. n - k I mt. d,,,,.,, 2 2t + 1.

b) RS Codes [51 The binary BCH codes defined in (a) can be generalised to non-binary codes in a straightforward manner. For any choice of positive integers s and t, there exists a q-ary BCH code of length n = @ - I , which is capable of correcting any combination o f t or fewer errors and requires no more than 2st parity check digits. The special subclass of q-ary BCH codes for which s = 1 is the most important subclass of q- ary BCH codes. The codes of this subclass are usually called the Reed-Solomon codes. A t - error correcting RS code with symbols from GF(q) (i.e. q=Z) has the following parameters: Block length : n = 2 " - 1 . Number of parity-check digits : Minimum distance: d,,,,.,, = 2t +1.

n - k = 2t.

c) Interleaving There seems to be a general agreement that interleaving and short codes are an effective way to improve the error rate for many diverse applications [6] . As was shown by Tsai and Schmied [6] when raising the transition probability matrix P of the Fritchman channel model to the k'th power P, then P can be used to represent the channel, after interleaving of degree h.

IV EXPERIMENTAL SET-UP

Environment

city freeway

The experiments undertaken during this study involved the transmission of data from a motor vehicle travelling through an urban area as well as along a stretch of freeway. Various modulation schemes ( FSK (300 bids), DPSK (1200 bids), QPSK (2400 bits/s) and 8-ary PSK (4800 bit&)) were used to transmit the data, and at the stationary receiving end the error patterns that occurred, were recorded, using dedicated error recording equipment implemented with a microcomputer, as described in [7]. From these error recordings, gap-, burst-, burst interval- and cluster distributions were derived.

FSK DPSK QPSK 8PSK

(31,191 (31,211 (31.19) (31,19) (31,21) (31,211 (3121) (31,19)

Furthermore the error-free run distributions, or P(O"/l),were derived from the gap distributions [3]. Curves were fit to the error free runs obtained, enabling us to find the parameters of a partitioned Markov model, similar to those obtained by Fritchman [l] and Tsai 131.

A voice band frequency hopping F.M. transmitter was used to transmit the output of the modems at an R.F carrier frequency hopping between 30 MHz and 87 MHz with a hopping rate of 50 hops/s. The transmitter power was low, and at the transmitter and receiver site wideband monopoles were used. All the measurements were carried out in the Johannesburg metropolitan area.

Computer programs using P(m,n) and the gap distributions were used to evaluate the error correcting codes.

V RESULTS

The experiments were carried out as explained in section IV. The two different environments from which the data was transmitted was chosen as being representative of typical driving conditions encountered within a metropolitan area.

After obtaining the error-free run distribution for each experiment, three and four term curves, leading respectively to four and five state models, of the form in equation (l), were fitted to each error-free run distribution. The models obtained through curve fitting were investigated in terms of accuracy and it was found that there is very little difference between the four and five state models. For this reason, only four state model parameters are presented in tables 1. The error statistics for the models in the case of freeway driving are compared to the observed error statistics in fig.2 to fig. 5. Due to space limitations only the error statistics related to freeway driving conditions are presented. However, similar phenomena with regards to the accuracy of the models, were observed for other driving conditions.

BCH and Reed-Solomon error correcting codes with block length 15 and 31 were investigated. When evaluating the error correcting codes it is important to use the simplest code and the code that gives the highest throughput and complies with the block error rate (BLER) criteria (BLER I lo"). -

PARA.

PI I

PZZ P33 P44

P42 P43

P24 P34

-

P41

PI4

-

FSK - 0.996971 0.519513 0.999791 0.455070 0.049369 0.453460 0.042101 0.003029 0.480487 0.000209

DPSK

0.878271 0.407002 0.999393 0.401301 0.162808 0.390619 0.045272 0.121729 0.592998 0.000607

QPSK

0.967649 0.560886 0.999848 0.462447 0.026676 0.477272 0.033605 0.032351 0.4391 14 0.000152

8PSK

0.963876 0.549516 0.999845 0.456641 0.035874 0.481271 0.026214 0.036124 0.450484 0.000155

Table 1. Four state Fritchman model parameters for freeway environment using FSK, DPSK, QPSK and 8 - ary PSK as modulation scheme.

Without interleaving the only error correcting code that gave a BLER I lo" for R 2 0.5 was the Reed-Solomon error correcting codes.

In Table 2 a list of Reed-Solomon error correcting codes for the different modulation and environmental conditions are presented that comply with a BLER I 10" and which have the highest data throughput for a BLER I

In Table 3 and 4 a list of BCH and RS error correcting codes with interleaving for the different modulation and environmental conditions are presented. These coding schemes comply with a BLER I lo5 and have the highest data throughput for a BLER S and a delay time I 5s.

Environment I FSK

In fig. 6 - 13 we compare the measured and calculated block error probabilities for 15 and 31 bit blocks.

Table 3. The BCH error correcting codes (n, k), h that give th.e highest throughput for a BLER I IO5 and a delay time I 5s. (Note: x denotes that there k no code that satkfles the criteria.)

1049

1 .o

0.8

0.6

0.4

0.2

0.0 i oo 1 0 ’ t oz 10’ l o4 l o 5 io6

LENGTH

Fig. 2. Error statktics for FSK under conditions of freeway driving.Four state model statktics are compared to the observed error statktics.

1 .o

5 0.8

0.6

>- 0

3

LL

4 0 4 [r

5 0.2

0.0 3 U

loo 1 0 ’ l o 2 l o 3 lo4 lo5 lo6

LENGTH

Fig. 3. Error statktics for DPSK under conditions of freeway driving.Four state model statktics are compared to the observed error statktics.

1 .o > 0

3 6 0.8

0.6

4 0.4 LL

[r

5 0.2 3 0

0.0 ioo 1 0 % io2 lo3 l o 4 lo5 lo6

LENGTH

Fig. 4. Error statktics for QPSK under conditions of freeway driving. Four state model statistics are compared to the observed error statktics.

1 0

6 0.8

0.6

0.4

0.2

0.0

>- U

3

LL

IY

3 U

i o o 1 0 ’ i o 2 lo3 lo4 lo5 lo6

LENGTH

Fig. 5. Error stutktics for 8 -ay PSK under conditions of freeway driving.Foirr state model statktics are compared to the observed error statistics.

1050

7

10-2

io-‘

10-6 IO-’ 10-8 10-9

10-’0 10-10 -1 0 2 4 6 8 10 12 14 16

NUMBER OF EITS IN ERROR NUMBER Of BITS IN ERROR

Fig. 6. and Fig. 7. Cornparkon between the measured and calculated P(m15) for FSK and DPSK as modulation schemes under conditions of freeway driving.

0 * 9wm 4 0 2 4 6 8 IO 12 14 16

10-1 4 I 10-3 10-4

IO-’ 10-6

IO-’ 10-8 10-9 10-10

0 2 I 6 8 1 0 12 14 16

NUMBER OF BITS IN ERROR NUMBER OF BITS IN ERROR

Fig 8. and Fig 9. Comparkon between the measured and calculated P(k,15) for QPSK and 8PSK as modulation schemes under conditions offreeway driving.

0 4 0 12 16 20 24 28 32

NUMBER OF BITS IN ERROR NUMBER OF BITS IN ERROR

Fig. 10. and Fig. 11. Comparison between the measured and calculated P(m,31) for FSK and DPSK as modulation schemes under conditions offreeway driving.

:%: I

0 1 (I I2 16 20 24 28 32 o II 11 16 m 14 28 U

NUMBER OF BITS IN ERROR NUMBER OF BITS IN ERROR

Fig 12. and Fig 13. Comparkon between the measured and calculated P(m,31) for QPSK and 8PSK as modulation schemes under conditions of freewuy driving.

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Table 4. The RS error correcting codes (n,k), A that give the highest throughput for a BLER I IO' and a deluy time I 5s.

VI. CONCLUSIONS

In all cases the cluster- and error-free run distributions of the models compared extremely favourably to those of the actual observations on the real channels.

As can be seen in fig. 6-13 the measured block error probabilities are very close to the block error probabilities derived from the Fritchman model. The discrepancies in the high order distributions (burst and burst interval) in fig. 2-5 are thus of less importance when evaluating error correcting codes. When there is a deviation between the measured and calculated block error probability, most of the time the calculated block error probabilities tend to be pessimistic and show the probability of an error occurring in a block to be slightly higher for the model than for the measured data. This is a favourable phenomenon because the designer knows that if he uses the model's P(m,n) in designing error correcting codes, the chance that the real channel will exceed the error correcting code's ability is slight because of the model's pessimistic behaviour. The values of P(m,n) obtained from the model form a tight upperbound.

Thus the Fritchman partitioned Markov chain, models tlie occurrence of errors accurately enough in all the cases investigated. Furthermore the code designer can use the model's values to evaluate error correcting codes with the assurance that the code's ability to correct errors will not be exceeded because of the model's pessimistic view of events. The optimum class of codes to use on the FH VHF channel, is long Reed-Solomon block codes with n 2 31 and rates of approximately 0.68 and 0.61. BCH error correcting codes can be used on the channel if they are used with interleaving. The interleaving depths vary but a interleaving depth of 51 ( ie. smallest prime larger than 48 ) will in general be sufficient. For the optimum length of the interleaver, the length must be some prime number, so that periodic errors introduced into the data stream will be non periodic when taken out of the receiver's deinterleaver unit [8]. Further investigations showed that with deeper interleaving depths, say greater than 131, the code rate will increase for the BCH codes and a higher data throughput can be achieved.

With the Fritchman model it is possible to investigate each individual case for the optimal interleaving depth. The interleaver design must include the capability to randomize periodic emors, otherwise the performance can be

considerably poorer than error correction without interleaving [8]. The interleaver unit introduces time delay. This time delay is the shortest for BCH error correcting codes. When using the same data rate and code rate the time delay for say interleaving depth of 400 is 7 seconds when using BCH codes and 70 seconds when using Reed-Solomon codes (due to the multiple bits per symbol), so if a time delay is going to be a problem in the system, deep interleaving of Reed- Solomon error correcting codes must not be attempted. The final conclusion that can be drawn is that with the aid of the Fritchman model any error correcting code and interleaving depth can be investigated so that the optimum system can be constructed.

REFERENCES

B. D. Fritchman, "A binary channel characterization using partitioned Markov chains," IEEE Trans. Injorm. Theory, vol. IT-13, pp. 221 - 227, April 1967. F. Swam and H.C. Ferreira, "Modeling and performance evaluation of mobile VHF radio channels employing FSK, DPSK, QPSK and 8-ary PSK as modulation scheme", Proc. IEEE GLOBECOM '91 ,pp. 1935 - 1939, December 1991. S. Tsai,"Markov characterization of the H.F.channe1" IEEE Trans. Commun. Technol.,vol COM - 17, pp. 24 - 32, Feb.1969. L.N. Kana1 and A.R.K. Sastry, "Models for channels with memory and their applications to error control, "Proc. IEEEvol. 66, pp. 724 - 744, July 1978. S . Lin and DJ. CostelloJr, "Error Control Coding: Fundamentals and Applications " Prentice-Hall, 1983. S. Tsai and P.S. Schmied, "Interleaving and Error- Burst Distribution, "IEEE Trans. Commun ,vol. COM- 20.No.3, pp. 291-296June 1972. F. Swarts, D.R. Oosthuizen and H.C. Ferreira,"On the Real Time Measurement and theoretical modeling of error distributions on digital communication channels," Proc. IEEE COMSIG '90, pp.78 - 82, 1990 K. Brayer and 0. Cardinale, "Evaluation of Error Correction Block Encoding for High-speed FH data", IEEE Trans. Commun. Technol. ,VOL COM-15, N0.3, pp.371-382, June 1967.

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