Performance of Reed-Solomon codes with dependent symbol errors

T.A. Gulliver M.Jorgenson W.K. Moreland

Indexing terms: Reed-Solomon error-correcting codes, Burst errors, Voice communication

Abstract: Reed-Solomon (RS) error-correcting codes are often proposed for communication systems requiring burst-error-correction cap- abilities. The performance of RS codes is examined for the case when symbol errors are dependent and therefore not random. This situation arises when interleaving is not possible due to delay constraints, such as in most voice- communication systems. The performance is compared with that obtained using a random- error assumption.

1 Introduction

In communication systems, noise, fading and interfer- ence such as jamming are common factors which can degrade performance. Error-correcting (EC) codes are often used to overcome these difficulties. Therefore an important consideration in any system design is the choice of a suitable code. The class of Reed-Solomon (RS) EC codes [l] is examined in this paper. They are maximum-distance-separable (MDS), which means they achieve the maximum possible minimum distance (can correct the largest possible number of errors) for an EC code with the given code parameters.

In many instances, it is assumed that symbol errors occur randomly. Although this is not the case with fad- ing and interference channels, and when equalisation is employed, it can be achieved by interleaving to a suffi- cient depth. If interleaving is not possible, such as when the corresponding delay cannot be tolerated, this assumption is not valid and errors tend to occur in bursts. This case is considered in this paper.

The system model is shown in Fig. 1, and is described below. From a binary data source blocks of kq bits enter the encoder divided into k q-bit Q-ary symbols (Q = 29. The encoder appends to this n-k symbols to obtain an n-symbol codeword (nq bits long), assuming systematic encoding. These symbols 0 IEE, 1996 IEE Proceedings online no. 19960503 Paper first received 14th August 1995 and in revised form 2nd April 1996 T.A. Gulliver is with the Department of Systems & Computer Engineering, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario KlS 5B6, Canada M. Jorgenson and W.K. Moreland are with the Communication Research Centre, 3701 Carling Avenue, PO Box 11490, Station H, Ottawa, Ontario K2H 8S2. Canada

are then transmitted over the channel using K-bit M- ary modulation symbols (A4 =- 2 9 . The number of modulation symbols per RS symbol is C = log,(Q/M) and C need not be an integer. At the receiver, the demodulator attempts to determine the transmitted M- ary symbols, and these are formed into q-bit RS code- word symbols. The n codeword symbols are then decoded to obtain kq bits of data, which are passed to the data sink.

I I I I I I

n Q-ary nC M-ary bits symbols symbols symbols

k Q-ary kq

I I I Q-ary symbols Reed-Solomon Wary to Q-ary +

decoder symbols *

to bits *

Fig. 1 Block-coded communication-system model

In this paper, the performance is determined from the measured errors at the input to the RS decoder. A discussion of the effects of symbol matching between M-ary modulation symbols ancl Q-ary codeword sym- bols is given in [2, 31. The periformance can be deter- mined using the probability of an RS symbol error (random case), or using the probability o f j errors in an RS codeword (nonrandom case). The purpose of this paper is to compare the performance for these two cases.

2 Reed-Solomon codes

The (n, k) Q-ary Reed-Solomon (RS) block codes are symbol-error-correcting codes with symbol size q = log2Q bits and defined block length n = Q - I [I]. Since RS codes are MDS, with k information symbols per block the minimum distance is d = n ~ k + 1. The min- imum distance (Hamming distance) is defined as the smallest number of symbol positions in which two codewords differ. An RS code can correct up to t = L(d - 1)/21 = L(n - k)/21 symbol errors, where 1x1 is defined as the largest integer less than or equal to x.

The weight distribution of an MDS code is known exactly. If A, denotes the nuimber of codewords of weight j , then A, is [4]

with j > 0 and A , = 1. RS codes can be shortened, punctured or extended (within limits) to obtain other MDS codes, in which case eqn. 1 also holds.

117 IEE Proc.-Cornmun., Vol. 143, No. 3, June 1996

2.7 Reed-Solomon-code performance evaluation Let c' be a codeword in an (n, k ) RS code. When c' is transmitted through a communication channel. noise and interference corrupt c so that

is received, with the error pattern being e. Note that only bounded distance decoding, i.e. the Berlekamp- Massey or Euclidean algorithms [5] . is considered here as maximum-likelihood decoding is not practical. Let the Hamming weight (number of nonzero elements) of e be w. If 1.1: 2 t , correct decoding will occur, i.e. if t or fewer errors occur in the received word; it falls within the decoding sphere of the transmitted codeword e. If I.V > t , the decoder cannot decode c correctly. In this case, the decoder will either ( a ) detect that errors have occurred (the received word lies in the intcrsticcs between decoding spheres): or (h) does not detect the errors and decodes i' into some other codeword, c" (the received word lies in the decod- ing sphere of some other codeword). The first of these two events is denoted as a decoding failure and the second as a decoding error. In most communication systems, event (h) is a much more seri- ous occurrence. For example, if ARQ is employed. a request for retransmission can be made when decoding fails. With voice communication where retransmission is not feasible due to delay requirements, the codekvord in error can be erased and techniques such as interpola- tion used to mask the missing data. With a decoding error, there is no knowledge of the errors in the code- word.

r = c + e

transmitted received symbol

Consider an M a y symmetric channel as shown in Fig. 2. For this channel, symbol errors are independent and occur with probability p . Define the probability of an RS symbol in error as P,. Then the probability of a correct RS symbol being received is 1 - P,. and the probability of a given incorrect symbol is P,i(Q - 1). Therefore the probability that a received word is cor- rectly decoded is given by

where Pde is the probability of- decoder error and Pdf is the probability of decoder failure.

The probability of decoder error Pde is [7] n

(4) J =@

mhere P i e is the probability that a received word falls within the decoding sphere of radius t about a weight j code*ord

The probability of a post-decoder symbol error is given [71

2.2 Performance with error statistics When the errors do not occur randomly, the perform- ance can be evaluated more accurately if error statistics are compiled. Let P, denote the probability of j errors in a codeword, and define

(7) p3

as the probability of receiving a particular j error n-tuple. Values for P, must be determined by experi- mental or other means. If the symbol errors occur randomly, i.e. the channel is memoryless, then

, - _ _ I

Pj = Lsl . Pi(1 - PJ-Z

The probability of correct decoding is t

J =o which is a generalisation of eqn. 2.

The probability of decoding error, and probability of symbol error when a decoding error occurs, are derived in the Appendix. The probability of decoding error is

and the corresponding probability of symbol error is

with h = v-2c+i.

3 Performance results

With BDD, the RS decoder codeword error rate is I - Pcii, which is given by [6]

This example considers the performance of a slow-fre- quency-hopped (SFH) spread-spectrum communication system. In this paper SFH means that the hop rate Rh is faster than the RS symbol rate R,, so there are N = RJR, Q-ary symbols, or h = qR,IRh bits, transmitted

IEE Proi -Commun Vol 143 No 3 June I996

on a hop. For this example, the number of bits per hop is 240. A QPSK waveform having a short preamble, followed by six alternating blocks of 20 unknown data and 20 known probe symbols per hop was simulated. With a code rate of kln = 0.6 and q = 8, a shortened (30, 18) RS code fits on one hop.

The channel model employed in this paper is a Wat- terson-channel model [8], the model most commonly used in simulation of ionospheric H F propagation. In general the Watterson model is composed of discrete multiplicative taps at specified delays, with each tap fading independently with a Rayleigh fading character- istic. In the simulations reported herein, we have simu- lated an equal-amplitude two-path Rayleigh-fading channel with 3ms of delay spread and varying degrees of Doppler spreading. In each case considered, the Doppler spread on each path is the same.

I d 0

i_i 1 0 ~ ~ \ ~ i 5 i o i 5 '\ io 35 Lo

EblN,, dB Fi 3 (30, I s ) , q = 8, RS code erformunce in a Ruyleigh fading chan- ne?ielay spread = 3ms and DoppLr spread = 2, 5, 10 and ISHz (i) measured, SHz (ii) measured, 2Hz (iii) calculated, SHz (iv) calculated, 2Hz

~ 100 a"

' 15 20 25 30 35 40

EdN,, dB Fi .4 neTdelay spread = 3ms and Doppler spread = 2, 5, 10 and 15Hz (i) measured, lOHz (ii) calculated, lOHr (iii) measured, 15Hz

(30, I8), q = 8, RS code performance in a Rayleigh fading chm-

(iv) cdlcukdted, 15 HZ

The received data were detected using a block-deci- sion-feedback equalisation scheme similar to the NDDE equaliser used in Harris HF modems [9]. This detection scheme provides two mechanisms for generat- ing correlated error events. Within a single block, there is the usual decision-feedback error-amplification effect; errors in early decisions make errors in decisions made later in the same block more probable. In addi-

IEE Proc.-Commun., Vol. 143, No. 3, June 1996

tion, there is a somewhat looser coupling of error sta- tistics between blocks. The block decision-feedback equaliser employed, like most modern equalisers for HF communication, uses deci,sions made on preceding blocks to update the channel estimate for the detection of subsequent blocks. Consequently, if there are a large number of errors in a block, it is likely that the quality of the channel estimate used to detect subsequent blocks will be poor, resulting in an increased probabil- ity of error in those blocks.

Through simulation of the system, the number of RS symbol errors before decoding was determined for each hop. From these values, the probability o f j errors in a codeword Pi was calculated, aind used in eqns. 7 , 9 and 10 to determine the performance. Figs. 3 and 4 show Pd, and Ped, respectively, against Eb/No, for Doppler spreads of 2, 5 , 10 and 15Hz. The dotted lines corre- spond to the measured values of Pj, and the solid lines to the values of Pi, calculated using eqn. 8. From these results, it is clear that performance degrades as the Doppler spread increases, so the worst results corre- spond to the highest Doppler spread value of 15Hz. For a Doppler spread of 2Hz, Pd, using the measured values is much higher (up to several orders of magni- tude). In addition, for a Doppler spread of 15Hz, PCLl using the measured va.lues is ,at least double that with the calculated values,, Thus, the probability Pdf of decoding failure is found to be: significantly lower using the measured error rates. These results can be explained by examination of .the corresponding values of Pj.

100

H - \ i

> "

5 " 10 1;5 '' 20 25 1

30

number of errors j

Fi .5 &?No = 10, 20 dB, Doppler spread = 5 Hz (i) calculated, 20dB (ii) calculated, lOdB (iii) measured, 20dB (iv) measured, lOdB

4 with q = 8 in a Rayleigh fading channel, delay spread = 3ms

The measured values of PI and those calculated using the average RS symbol error rate and eqn. 8, are plot- ted in Fig. 5 for q = 8 and a Doppler spread of 5Hz, with Eb/No = 1OdB and 20dB. Fig. 6 shows PI for q = 8 and a Doppler spread of lOHz, with EbINo = lOdB and 20dB, respectively. What is noticeable in both these Figures is that P, has a parabolic shape when eqn. 8 is used, while the measured values have a much more linear shape. This is a common characteristic of all the cases that were simulated. Because of this, for EbINo = 20dB, Fde is actually much higher using the measured values of P, with a Doppler spread of 5Hz, and slightly higher with a Doppler spread of 10Hz. Recall that t = 6, so that Pde is determined only by those values of P, f o r j > 6 . For EbINo = 20dB, the two

119

Pde curves are much closer because the parabolas for the calculated values of Pj have shifted to the right, so that larger values of Pj occur after j = t = 6. The corre- sponding measured values of Pj change much less, which results in a much smaller change in Pde.

A I 0 5 10 15 20 25 30

10-41 j

number of errors j Fi .6 with 8 in a Rayleigh fading channel, delay spread = 3 m

(1) calculated, 20dB (ii) calculated, lOdB (iii) measured, lOdB (iv) measured, 20dB

= IO, 2O%<Doppler spread = lOHz

Pcd is given by the values of Pj for j = t = 6 or less errors. With EbINo = 10dB, the calculated values of Ped are much lower than the measured values. However, when &/No is increased to 20dB, the measured Pcd val- ues are lower. These results can also be explained by the shift in the parabolas. When EdN, is increased, a substantial portion of the parabolas move to the left of j = t = 6. If t were lower (say, t = 1 or 2), the measured value of Pcd would remain higher than the calculated value for EbINo = 20dB.

4 Summary

Expressions for the performance of Reed-Solomon codes using measured error rates at the decoder input have been derived. Results were given which show that the performance based on an average RS symbol-error rate can differ from that based on measured symbol- error rates by several orders of magnitude.

5 Acknowledgment

This research was supported by Communications Research Centre, 3701 Carling Avenue, Box 11490, Station H, Ottawa, Ontario K2H 8S2, and the Natural Sciences and Engineering Research Council of Canada.

6 References

1 REED, I.S., and SOLOMON, G.: ‘Polynomial codes over certain finite fields’, J. Soc. Ind. Appl. Math., 1960, 8, pp. 300-304

2 GULLIVER, T.A.: ‘Performance evaluation of Reed-Solomon codes’. Technical report SCE-93-91, Department of Systems & Computer Engineering, Carleton University, Canada, December 1993

3 GULLIVER, T.A.: ‘Matching Reed-Solomon codes with M-ary modulation’, IEEE Trans., COM-44, (to be published)

4 MacWILLIAMS, F.J., and SLOANE, N.J.A.: ‘The theory of error-correcting codes’ (North-Holland Publishing Co., 1977)

5 CLARK, G.C. Jr., and CAIN, J.B.: ‘Error-correcting coding for digital communications’ (Plenum Press, 1981)

6 TORRIERI, D.: ‘Information-bit, information-symbol, and decoded-symbol error rates for linear block codes’, IEEE Trans., 1988, COM-36, pp. 613-617

120

WICKER, S.B.: ‘Reed-Solomon error control coding for Rayleigh fading channels with feedback‘, IEEE Trans., 1992, VT- 41, pp. 124-133 WATTERSON, C.C., JUROSHEK, J.R., and BENSE- MA, W.D.: ‘Experimental confirmation of an HF channel model’, IEEE Trans., 1970, COM-18, (6) McRAE, et al.: ‘Technique for high rate digital transmission over a dynamic dispersive channel’. US patent 4 365 338, issued 21st December 1982, assigned to Harris Corporation, Melbourne, Florida

10 BLAHUT, R.E.: ‘Theory and practice of error control codes’ (Addison-Wesley, 1983)

7 Appendix

In this Appendix, the probability of decoding error with error statistics, and the corresponding probability of symbol error, are derived. This is an extension of the results given in chapter 14 of [lo].

Since RS codes are linear, any codeword is equally likely to be decoded erroneously. Thus the all-zero codeword can be chosen as the transmitted codeword with no loss of generality. The decoder error probabil- ity is calculated by computing the probability of all error patterns which are within t or fewer symbols of any nonzero codeword.

Let cj be a codeword of weight j , and let rj+; be a received word containing j + i errors. A decoding error occurs when rl+i falls within the decoding sphere of radius t about a nonzero codeword. There are five val- ues which must be considered when deriving an expres- sion for Pde, the probability of decoding error. These are: (i) a = the number of co-ordinates in which both rjCi and cj have zero symbols. In this case there is one pos- sible value for these co-ordinates of rj+;; (ii) b = the number of nonzero co-ordinates in rj+; which correspond to different nonzero co-ordinates in cj. In this case there are Q - 2 possible values for these co-ordinates of vi+;; (iii) c = the number of nonzero co-ordinates in ri+; which correspond to zeros in cj. In this case there are Q - 1 possible values for these co-ordinates of rj+i; (iv) d = the number of co-ordinates in rj+i which corre- spond to the same nonzero co-ordinate in cj. In this case there is one possible value for these co-ordinates

(v) f = the number of zero co-ordinates in rj+i which correspond to nonzero co-ordinates in cj. In this case there is one possible value for these co-ordinates of rjii. Using these variables, the decoding-error events can be enumerated by summing over all possible error pat- terns which cause successful decoding to a nonzero codeword. Let j be the codeword weight, j + i the received-word weight, and v the distance between the codeword and the received word. From eqn. 7, 4Li is the probability of receiving a particular word with j + i errors. The probability of decoding error is then

of rj+;;

x [’Jh] ( Q - I)‘(& -

From the above variables, the number of zero co-ordi- nates in cj is n - J , so that

and the number of zero co-ordinates in rj+; is n - j - i, a + c = n - j (13)

IEE Proc.-Commun., Vol. 143, No. 3, June 1996

so that

From eqns. 13 and 14

which can be substituted into eqn. 12 to eliminate f. Using similar arguments, it can be shown that

Combining eqns. 15 and 16 gives b = v - 2c + i, and substitution in eqn. 12 results in

a+ f =n-j-i (14)

f = c - 2 (15)

b + c + e = w (16)

which is the desired result (eqn. 10). The symbol-error rate when a decoding error occurs

can be obtained by modifying eqn. 10. Each decoding error event must be multiplied by the number of errors present after decoding, then the result divided by the number of symbols in the codeword. Decoding to a weight-j codeword cj results in j symbol errors. Thus the symbol-error rate is

x [ 7 :] (Q -- l)"(Q - 2 ) b

which is the same as eqn. 11.

IEE Proc-Commun., Vol. 143, No. 3, June 1996 121