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Performance of Parallel and Serial Concatenated Codes on Fading Channels Jinhong Yuan, Associate Member, IEEE, Wen Feng, and Branka Vucetic, Senior Member, IEEE

Abstract—The performance of parallel and serial concatenated codes on frequency-nonselective fading channels is considered. The analytical average upper bounds of the code performance over the Rician channels with independent fading are derived. Furthermore, the log-likelihood ratios and extrinsic information for maximum a posteriori (MAP) probability and soft-output Viterbi algorithm (SOVA) decoding methods on fading channels are developed. The derived upper bounds are evaluated and compared to the simulated bit-error rates over independent fading channels. The performance of parallel and serial codes with MAP and SOVA iterative decoding methods, with and without channel state information, is evaluated by simulation over independent and correlated fading channels. It is shown that, on correlated fading channels, the serial concatenated codes perform better than parallel concatenated codes. Furthermore, it has been demonstrated that the SOVA decoder has almost the same performance as the MAP decoder if ideal channel state information is used on correlated Rayleigh fading channels. Index Terms—Concatenated codes, fading channels, turbo codes.

I. INTRODUCTION

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N MANY REAL links such as cellular radio and satellite mobile channels, transmission errors are mainly caused by variations in received signal strength referred to as fading. This severely degrades the transmission performance, and powerful error-control coding techniques are needed to reduce the penalty in signal-to-noise ratio (SNR). Parallel concatenated convolutional codes, known as turbo codes, can achieve a remarkably low bit-error rate (BER) with iterative decoding at an SNR close to the Shannon capacity limit on additive white Gaussian noise (AWGN) channels [1]. On AWGN channels, turbo-code performance has been discussed and evaluated by using analytical average upper bounds [6]. The performance and design of turbo codes on Rayleigh fading channels have been considered in [5]. The upper bounds Paper approved by P. Hoeher, the Editor for Coding and Communication Theory of the IEEE Communications Society. Manuscript received July 19, 1999; revised January 9, 2001 and June 7, 2001. This paper was presented in part at the IEEE International Conference on Communications (ICC), Vancouver, BC, Canada, June, 1999, and in part at the IEEE International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC), Osaka, Japan, September, 1999. J. Yuan is with the School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia (e-mail: [email protected]). W. Feng is with the Telstra Research Laboratories, Locked Bag 6764, NSW 1100, Australia (e-mail: [email protected]). W. Feng and B. Vucetic are with the School of Electrical and Information Engineering, University of Sydney, Sydney, NSW 2006, Australia (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TCOMM.2002.803971

of turbo-code performance on fast and slow Rician fading channels with ideal channel state information (CSI) were presented in [15]. In this paper, we consider parallel and serial concatenated code error performance on frequency-nonselective Rician fading channels. Analytical upper limits for parallel and serial concatenated code performance were derived when no CSI is available at the receiver. The upper bounds can be used to evaluate code performance. For decoding, we consider the iterative maximum a posteriori (MAP) probability and soft-output Viterbi algorithm (SOVA) for concatenated codes. If no CSI is available, the MAP and SOVA methods with a standard Euclidean distance metric are used in the decoder. We modify the decoding metric for fading channels with CSI and derive the log-likelihood ratios and extrinsic information for MAP and SOVA decoding methods. The iterative decoding methods with the modified metric give an improvement of about 2 dB for SOVA and 1.5 dB for MAP at a BER of 10 . The derived upper bounds and the BER simulation results for both parallel and serial concatenated codes over independent and correlated fading channels are discussed. It is shown that, on an independent Rayleigh fading channel, the parallel codes have better performance than the serial codes at low SNR, but the serial codes outperform the parallel codes at high SNR. However, on a correlated fading channel, the serial codes perform better than parallel codes at both low and high SNRs. It is also shown that the SOVA decoder has almost the same performance as the MAP decoder if ideal CSI is used. The paper is organized as follows. Section II gives the description of the system and presents the channel model. The bit-error probability upper bounds for concatenated codes over independent Rician fading channels are derived in Section III. In Section IV, the modified MAP and SOVA decoding methods based on the new metric for fading channels are presented. In Section V, the derived upper bounds are evaluated and the code BER performance on the independent and correlated fading channels are simulated and discussed. Finally, Section VI presents the conclusions. II. SYSTEM MODEL A. Channel Encoder We consider parallel and serial concatenated convolutional codes. The encoders for the two schemes are shown in Fig. 1(a) and (b), respectively. For a parallel encoder, two identical component recursive systematic convolutional (RSC) encoders with ) are sepathe memory order (the number of states is rated by a random interleaver with size [1]. For a serial one,

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For , there is no specular component, and the Rician pdf becomes a Rayleigh pdf. For approaching infinity, there is no fading at all, resulting in an AWGN channel. III. AVERAGE UPPER BOUNDS ON THE FADING CHANNEL The bit-error probability of the coded system can be upper bounded by using a union bound that sums contributions from all the codewords with various Hamming weights [6]–[8]. As a parallel or serial concatenated code can be represented by an equivalent linear block code if the trellis termination is used to drive the component encoders to the all-zero state, the union upper bound for a block code can be applied to a concatenated code. We assume that a rate 1/3 parallel concatenated code with an ) interleaver size and memory order is equivalent to an ( block code, where

(a)

(b)

(5)

Fig. 1. (a) Turbo encoder. (b) Serial concatenated encoder.

the inner code is an RSC code, and the outer one is a nonrecursive convolutional code [12]. The inner encoder and the outer encoder are linked by a random interleaver. For an input message of length , the interleaver size for the serial code is 2 , provided that the rate of the outer encoder is 1/2. B. Channel Model The channel is modeled by frequency-nonselective Rician fading, whereby the fading amplitude is represented by a Rician random variable, which remains constant over each symbol. Let be the transmitted signal at time . The received signal at time is given by (1) is a random variable which represents the channel where fading variation, and is AWGN with single-sided power spec. In this model, the envelope amplitude of the tral density fading attenuation is a Rician random variable. By assuming to ensure that the received signal power is equal to the transmitted one, the probability density function (pdf) of the fading amplitude is given by

(2) where is the Rician factor denoting the ratio of the power of the specular signal component to the power of the diffuse fading is the zero-order modified Bessel signal component, and function of the first kind. The mean and variance of the Rician random variable are given by [10]

(3) (4) is the first-order modified Bessel function of the where first kind. Small values of indicate a severely faded channel.

In this section, we derive the upper bounds on the pairwise-error probability for block codes on an independent fading channel. Subsequently, we come up with the average upper bounds on the bit-error probability for parallel and serial concatenated codes on a fading channel. In the analysis, we assume that the channel interleaving depth is sufficiently large, resulting in independent and identically distributed (i.i.d.) fading amplitudes. A. Upper Bounds on the Pairwise-Error Probability 1) Decoding Without CSI: The performance upper bounds on Rician fading channels with ideal CSI in decoding were derived in [15], and we give a brief review of the results. , where Let a coded sequence be is the code length. This sequence is modulated by a binary phase-shift keying (BPSK) modulator. The modulated sequence is transmitted over the fading channel. is At the receiver, the received sequence observed. Furthermore, we assume that the fading attenuation can be perfectly recovered at the receiver. The decoder then performs soft decoding using the modified Euclidean distance metric (6) is the probThe pairwise-error probability ability that the decoder chooses as its estimate the sewhen the transmitted sequence . This occurs if quence was in fact . For a given realization of the fading variable , the conditional pairwise-error probability is given by

(7) , , represent the value of the fading ampliLet and tude in the th bit interval in which the two sequences are different, where is the Hamming distance between the

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and

. The pairwise probability conditioned

Using the bound , , the conditional pairwise-error probability can be upper bounded by

(8)

(14)

is the code rate, is the SNR per bit, where is the complementary error function. Introducing the and , , in (9), we obtain inequality

can then be deterThe pairwise-error probability , mined by averaging over the random fading amplitudes , and is given by

(9) can then be deterThe pairwise-error probability mined by averaging over the random fading amplitudes and is upper bounded by [15]

(15) In this analysis, we will introduce the random variable (16)

(10) 2) Decoding Without CSI: In deriving the above result, we assumed that the values of fading amplitudes can be recovered perfectly. Indeed, in some systems it may not be practical to attempt CSI recovery, so the pairwise-error bound in the absence of CSI becomes relevant. When no CSI is available at the receiver, the decoder performs soft decoding using the standard Euclidean distance metric (11) The pairwise-error probability

are i.i.d. random variAs the fading amplitudes ables, for a large Hamming distance (larger than five), according to the central limit theorem, the random variable is approximately Gaussian with the mean (17) and the variance (18) and are, respectively, the mean and variance of where the Rician random variable , and they are given by (3) and (4), respectively. Noting that

is given by

if Re (12) We will introduce a random variable equal to the term on the left-hand side of the inequality in (12). For a given realization is a Gaussian variable with zero of the fading variable , , where is the variance of mean and variance the Gaussian noise. The term on the right-hand side is a constant which depends on . Therefore, the conditional pairwise-error probability can be expressed as [5]

(13)

(19)

we get for the pairwise-error probability upper bound from (15)

(20) For the turbo codes, is typically larger than five, so the above bound holds in most cases. B. Average Upper Bound on the Bit-Error Probability ) linear systematic block code, let us denote by For an ( the error coefficient which determines the contribution of the codewords with the same weight to the bit-error probability. The bit-error probability of the code decoded by a maximum-likelihood algorithm over a fading channel can be upper bounded by [6] (21)

YUAN et al.: PERFORMANCE OF PARALLEL AND SERIAL CONCATENATED CODES ON FADING CHANNELS

Fig. 2. Bit-error probability upper bound for the four-state, rate 1/3 parallel = 100 on independent Rician fading channels code with interleaver size with ideal CSI. The curves are for Rician channels with = 0, 2, 5, 50, starting from the top, with the bottom one referring to an AWGN channel.

N

K

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Fig. 3. Bit-error probability upper bound for the four-state, rate 1/3 parallel code with interleaver size = 100 on independent Rician fading channels without CSI. The curves are for Rician channels with = 0, 2, 5, 50, starting from the top, with the bottom one referring to an AWGN channel.

N

K

where is the code minimum distance and is the pairwise-error probability when the Hamming distance between and is . the two sequences Therefore, the bit-error probability upper bound on an independent Rician fading channel is given by

(22) when ideal CSI is available and equal to

(23) when there is no CSI. The bounds (22) and (23) can be evaluated as functions of and if the code distance spectrum is known. Here, we assume the uniform interleavers are used in the concatenated encoders, so that the error coefficient can be obtained by averaging over all possible interleaver structures [6], [8]. In the evaluation, we consider a parallel and a serial code with and memory order . For fair the same code rate comparison of the parallel and the serial code, the interleaver size should be chosen in such a way that the decoding delay due to the interleaver is the same. In other words, the input message length for both codes should be the same. We chose a message in the performance evaluation. For the parlength of allel code, the generator matrix of the component code in the octal form is (1, 5/7). The bit-error probability upper bounds for are illustrated in Fig. 2 for the parallel code with variable the case of ideal CSI and in Fig. 3 for the case of no CSI. For the serial code, the generator matrix of the outer code is (7,5). The generator matrix of the inner code is

. The

Fig. 4. Bit-error probability upper bound for the four-state, rate 1/3 serial code with message length = 100 on independent Rician fading channels with ideal CSI. The curves are for Rician channels with = 0, 2, 5, 50, starting from the top, with the bottom one referring to an AWGN channel.

N

K

bit-error probability upper bounds for the serial code with variable are illustrated in Fig. 4 for the case of ideal CSI, and in Fig. 5 for the case of no CSI. From Figs. 4 and 5, we can see that ), the error performance for the Rayleigh fading channel ( can be improved by approximately 1.5 dB at the BER of 10 when ideal CSI is utilized, relative to the case with no CSI. This , improvement decreases as is getting larger. When the gain of CSI disappears. In order to compare the parallel and serial code performance on Rayleigh fading channels, we show the distance spectra and the bit-error probability upper bounds for both codes in Figs. 6 and 7, respectively. From Fig. 6, we can see that the serial code has smaller error coefficients than the parallel code at low-tomedium Hamming distances. Since these distance spectral lines dominate the code-error performance in the region of high SNR [11], the serial code can achieve better performance than the

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Fig. 5. Bit-error probability upper bound for the four-state, rate 1/3 serial code with message length = 100 on independent Rician fading channels without CSI. The curves are for Rician channels with = 0, 2, 5, 50, starting from the top, with the bottom one referring to an AWGN channel.

N

K

Fig. 6. Distance spectrum comparison of the four-state, rate 1/3 parallel and serial concatenated codes with message length = 100.

N

parallel code on fading channels at high SNR as illustrated in Fig. 7. It may also be observed from Fig. 6 that for mediumdistance values, the error coefficients of both codes are almost the same, which will result in almost the same error performance at low SNR, as the code performance in this region is determined by the medium-distance spectral lines [11]. In Fig. 8, we show the bit-error probability upper bounds for both codes on the Rayleigh fading channel with the ideal CSI. For each code, we consider the message length of 100 and 1000. It can be seen that when we increase the message length from 100 to 1000, the performance gain of the serial code relative to the parallel code is increased significantly. IV. ITERATIVE DECODING ON FADING CHANNELS On AWGN channels, the turbo decoding is performed by a suboptimal iterative algorithm. The decoder consists of two identical concatenated decoders of the component codes sepa-

Fig. 7. Bit-error probability upper bound comparison of the four-state, rate 1/3 parallel and serial concatenated codes with message length = 100 on independent Rayleigh fading channels.

N

Fig. 8. Bit-error probability upper bound comparison of the four-state, rate 1/3 parallel and serial concatenated codes with message lengths of 100 and 1000 on independent Rayleigh fading channels with ideal CSI.

rated by an interleaver. The component decoders are based on a MAP algorithm or a SOVA generating a weighted soft estimate of the input sequence [2]–[4], [9]. In the turbo decoding for AWGN channels, a branch metric based on the standard Euclidean distance between the received and transmitted signal sequences is used. This metric is optimum for Gaussian channels. For fading channels with no CSI, we apply the metric for AWGN channels in turbo decoding. If ideal CSI is used to decode turbo codes for fading channels, the branch metric should optimize the conditional prob. This is equivalent to replacing the transability by in the branch metric [14]. Based on mitted signal this branch metric, we modified the log-likelihood ratio and extrinsic information for MAP and SOVA decoding methods. Note that the decoding methods described here are for parallel concatenated codes. However, the algorithms can be directly extended to serial codes.

YUAN et al.: PERFORMANCE OF PARALLEL AND SERIAL CONCATENATED CODES ON FADING CHANNELS

Without loss of generality, we assume that the bit sequence is encoded by an ( , 1, ) RSC endiscoder. The code trellis has a total number of . The tinct states, indexed by the integer , , where RSC encoder output sequence is are defined by encoder generators. The corresponding modulated sequence and received sequence are given by

(24)

(25)

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information generated by the code parity check bit and is given by

(29) is interleaved (or deinterThis extrinsic information leaved) and then used as the a priori information for the other decoder. 2) Modified SOVA Decoding Method With CSI: In the SOVA method, the path metric corresponding to path at time , denoted by , is calculated by [4]

1) Modified MAP Decoding Method With CSI: In the MAP decoding method, the log-likelihood ratio for the input symbol , is given by [2], [3] at time , denoted by

(26)

is the branch-transition probability which where can be determined from the transition probabilities of the and are the channel and the encoder trellis. forward and backward recursive probability functions obtained . For an independent fading channel with ideal from , based on the CSI, the branch transition probability modified branch metric, can be computed as

(30)

is the th modulated symbol on path at time , where is the smallest metric of the path connected to path at time , is the logarithm of the a priori probability at time and obtained from the previous decoder for the information bit . The SOVA decoder provides a soft output in the form of an approximate log-likelihood ratio of the a posteriori probabilities of the information bits 1 and 0. The soft output of the SOVA decoder for the input symbol at time can be approximately expressed as the metric difference of the maximum-likelihood path and its strongest competitor at time . The strongest competitor of the maximum-likelihood path is the path which has the minimum path metric among all paths obtained by replacing the trellis symbol on the maximum-likelihood path at time by its at complementary symbol. The SOVA output, denoted by time , can be expressed as [4]

(27)

(31)

is the a priori probability of and is the where variance of the Gaussian noise. In a derivation similar to [2], [3], and [9], the log-likelihood ratio for the input symbol at time , , can be decomposed into

is the minimum path metric corresponding to where and is the minimum path metric corresponding to . Following the derivation in MAP, we can split the soft output , and the extrinsic into two parts, the intrinsic information , as information (32)

(28) The intrinsic information, where the first term is the a priori information obtained from the other decoder. The second term is the systematic information is the extrinsic generated by the code information bit.

, is given by (33)

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Fig. 9. Performance comparison of MAP and SOVA, with and without CSI, for the 16-state, rate 1/3 parallel code on an independent Rayleigh fading channel, the message length N = 1024, the number of iterations I = 8.

Fig. 10. Performance comparison for the four-state, rate 1/3 parallel and serial codes on an independent Rayleigh fading channel, the message length N = 1024 and 4096.

Therefore, the extrinsic information from (32) and (33) as

the parallel code, a BER error floor appears at 10 . A similar result has been obtained for AWGN channels [4]. In order to compare the code performance against the Shannon limit, we computed the capacity for the Rician fading channel [5], [13]. The Shannon capacity limits for coherent BPSK on the independent Rician fading channel for various values and code rates are shown in Table I, where the values correspond to an arbitrarily small error probability. These values can serve as references in indicating how far away the performance of a parallel or serial code is from the theoretical channel capacity limits. The BER performance of on the the rate 1/3 parallel code with memory order factors is plotted in independent Rician channel for various Fig. 11. The message length is 65 536. The iterative decoding with MAP algorithm with ideal CSI is used in the simulation. The number of iterations is 20. Comparing the curves of Fig. 11 and the theoretical channel capacity limits in Table I, we can observe that the BER performance at 10 for the parallel code is within 0.5 0.6 dB of the capacity limit for channel models ranging from AWGN to Rayleigh. In some applications, the channel-interleaver size is severely constrained by the maximum delay requirements. This results in a correlated fading channel. However, the performance of the communication systems with correlated fades is not amenable to analytical evaluation. In this section, we investigate the BER performance of the coding schemes on correlated Rayleigh fading channels by simulation. In the channel model, we assume that the power spectral density of the faded amplitude due to the Doppler shift, denoted by , is given by

can be obtained

(34) and it will be used to calculate the a priori probabilities ratio in the next step of decoding after interleaving or deinterleaving. V. SIMULATION RESULTS The BER performance of the parallel concatenated coding scheme on fading channels is estimated by simulation. In the simulation, we considered the rate 1/3 parallel code and message length . with memory order The generator polynomial of the parallel code in the octal form is (1, 21/37). In the receiver, iterative decoding with symbol-by-symbol MAP and SOVA methods, with and without CSI, is employed. The number of iterations is eight for the interleaver size of 1024. For the independent Rayleigh fading channels, the performance simulation results are shown in Fig. 9. At a BER of 10 , both MAP and SOVA decoders with CSI give an improvement of about 1 dB relative to the respective decoders without CSI. It can also be observed that MAP decoding is superior to SOVA decoding by 0.5 dB at the BER of 10 , for both decoders with and without CSI. The BER performance comparison of the parallel and serial coding schemes on fading channels was also carried out by simulation. We simulated the rate 1/3 parallel and serial codes with 2 and message length of 1024 and memory order 4096. In the receiver, iterative SOVA decoding is employed. The number of iterations was eight for message length 1024 and 18 for message length 4096. For the independent Rayleigh fading channels, the performance simulation results for the parallel and serial codes are shown in Fig. 10. It can be observed that the parallel code has a better performance at low SNR than the serial one. However, the serial code outperforms the parallel code at high SNR. For

if

(35)

otherwise is the fade where is the frequency and is the fade rate. rate normalized by the symbol rate. It serves as a measure of the channel memory. For correlated fading channels, this parameter

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TABLE I CHANNEL CAPACITY LIMITS FOR COHERENT BPSK ON FADING CHANNELS WITH IDEAL CSI

Fig. 11. Performance of the 16-state, rate 1/3 parallel code on an AWGN and independent Rician fading channels with ideal CSI, the message length N = 65 536, the number of iterations I = 20.

Fig. 12. Performance comparison of MAP and SOVA, with and without CSI, for the 16–state, rate 1/3 parallel code on a correlated Rayleigh fading channel, the fading rate normalized by the symbol rate is 10 , the message length N = 1024, the number of iterations I = 8.

is in the range , indicating finite channel memory. The autocorrelation function of the fading process is given by (36) is the zeroth-order Bessel function of the first kind. where In the simulations, the fade rate normalized by the symbol rate was 0.01. The performance comparison of MAP and SOVA, with and without CSI, for the rate 1/3 parallel code with memory and message length on a correlated order Rayleigh fading channel is shown in Fig. 12. We can see from the figure that the MAP decoder with CSI achieves an improvement of about 1.2 dB at a BER of 10 relative to the MAP decoder without CSI. The corresponding improvement for the SOVA decoder with ideal CSI is about 1.8 dB. It can be also observed that without CSI, the performance degradation of the SOVA decoding relative to the MAP decoding is approximately 0.7 dB. However, the SOVA CSI decoder has almost the same . By comperformance as the MAP CSI decoder at high paring Fig. 9 with Fig. 12, we can see that the parallel code performance degrades significantly due to the correlated fades in the channel. When the fade rate normalized by the symbol rate is 0.01, the turbo code performance is about 1 1.8 dB worse relative to an independent fading channel. Performance comparison of the rate 1/3 parallel and serial and various message lengths codes with memory order on a correlated Rayleigh fading channel is shown in Fig. 13. In the simulation, iterative SOVA decoding is employed and the

Fig. 13. Performance comparison for the four-state, rate 1/3 parallel and serial codes on a correlated Rayleigh fading channel, the fading rate normalized by the symbol rate is 10 , the message length N = 1024, 4096, and 8192.

message lengths are 1024, 4096, and 8192. The number of iterations is eight for message length 1024 and 18 for message lengths 4096 and 8192. We can see from the figure that the serial code outperforms the parallel code on the correlated fading channels. At a BER of 10 , the corresponding improvements achieved by the serial code with message lengths of 1024, 4096, and 8192 relative to the parallel code are 1.25, 0.8, and 1.25 dB, respectively.

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VI. CONCLUSIONS In this paper, we consider the performance of parallel and serial concatenated convolutional codes on frequency-nonselective Rician fading channels. The analytical average upper limits of the code performance based on the union bound and code weight distributions are derived and discussed. The parallel and serial concatenated code performance based on the derived analytical bounds are compared over independent fading channels. The log-likelihood ratios and extrinsic information for MAP and SOVA decoding methods on fading channels are developed. The performance of MAP and SOVA iterative decoding methods, with and without CSI, is evaluated by simulation over independent and correlated fading channels. It is shown that the serial concatenated codes perform better than parallel concatenated codes on correlated fading channels. Furthermore, it is demonstrated that the SOVA decoder has almost the same performance as the MAP decoder if ideal CSI is used on correlated Rayleigh fading channels. REFERENCES [1] C. Berrou, A. Glavieux, and P. Thitimajshima, “Near Shannon limit error-correcting coding and decoding: Turbo-codes (1),” in Proc. Int. Conf. Communications, May 1993, pp. 1064–1070. [2] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear code for minimizing symbol error rate,” IEEE Trans. Inform. Theory, vol. IT–20, pp. 284–287, Mar. 1974. [3] J. Hagenauer, P. Robertson, and L. Papke, “Iterative (“Turbo”) decoding of systematic convolutional codes with the MAP and SOVA algorithms,” in Proc. ITG Conf. Source, Channel Coding, Munich, Germany, Oct. 1994, pp. 1–9. [4] B. Vucetic, “Iterative decoding algorithm,” in Proc. PIMRC, Helsinki, Finland, Sept. 1997, pp. 99–120. [5] E. K. Hall and S. G. Wilson, “Design and analysis of turbo codes on Rayleigh fading channels,” IEEE J. Select. Areas Commun., vol. 16, pp. 160–174, Feb. 1998. [6] S. Benedetto and G. Montorsi, “Unveiling turbo codes: Some results on parallel concatenated coding schemes,” IEEE Trans. Inform. Theory, vol. 42, pp. 409–428, Mar. 1996. , “Design of parallel concatenated convolutional codes,” IEEE [7] Trans. Commun., vol. 44, pp. 591–600, May 1996. [8] D. Divsalar, S. Dolinar, F. Pollara, and R. J. McEliece, “Transfer function bounds on the performance of turbo codes,”, JPL TDA Progress Rep. 42-122, 1995. [9] P. Robertson, “Illuminating the structure of parallel concatenated recursive systematic (TURBO) codes,” in Proc. Globecom, San Francisco, CA, Nov. 1994, pp. 1298–1303. [10] B. Vucetic, “Bandwidth efficient concatenated coding schemes for fading channels,” IEEE Trans. Commun., vol. 41, pp. 50–61, Jan. 1993. [11] J. Yuan, B. Vucetic, and W. Feng, “Combined turbo codes with interleaver design,” IEEE Trans. Commun., vol. 47, pp. 484–487, Apr. 1999.

[12] S. Benedetto, G. Montorsi, D. Divsalar, and F. Pollara, “Serial concatenation of interleaved codes: performance analysis, design and iterative decoding,”, JPL TDA Progress Rep. 42-126, 1996. [13] E. Biglieri, J. Proakis, and S. Shamai (Shitz), “Fading channels: Information-theoretic and communications aspects,” IEEE Trans. Inform. Theory, vol. 44, pp. 2619–2692, Oct. 1998. [14] D. Divsalar and M. K. Simon, “Trellis-coded modulation for 4800 to 9600 bps transmission over a fading satellite channel,” IEEE J. Select. Areas Commun., vol. 5, pp. 162–175, Feb. 1987. [15] F. Babich, G. Montorsi, and F. Vatta, “Performance bounds of continuous and blockwise decoded turbo codes in Rician fading channel,” IEE Electron. Lett., vol. 34, no. 17, pp. 1646–1648, Aug. 1998.

Jinhong Yuan (S’96–A’97) received the B.E. and Ph.D. degrees in electrical engineering from the Beijing Institute of Technology, Beijing, China, in 1991 and 1996, respectively. In 1997, he joined the School of Electrical and Information Engineering, University of Sydney, Sydney, Australia, as a Research Fellow. He is currently with School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, Australia. His research interests include wireless communications, communication theory, error control coding, and digital modulation.

Wen Feng received the B.E. degree in electrical engineering from Chongqi Institute of Architecture, Chongqi, China, in 1990 and the Ph.D. degree in electrical and information engineering from the University of Sydney, Sydney, Australia, in 2001. She joined Telstra Research Laboratories in 1999, where she is currently Technology Specialist. Her research interests include wireless communication, wireless LAN, and error-control coding.

Branka Vucetic (M’83–SM’00) received the B.S.E.E., M.S.E.E., and Ph.D. degrees in 1972, 1978, and 1982, respectively, from the University of Belgrade, Belgrade, Yugoslavia. She is the Director of Telecommunications Laboratory and Professor in Telecommunications at the University of Sydney, Sydney, Australia. Her research interests include wireless communications, digital communication theory, coding, and multiuser detection.

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 50, NO. 10, OCTOBER 2002

Performance of Parallel and Serial Concatenated Codes on Fading Channels Jinhong Yuan, Associate Member, IEEE, Wen Feng, and Branka Vucetic, Senior Member, IEEE

Abstract—The performance of parallel and serial concatenated codes on frequency-nonselective fading channels is considered. The analytical average upper bounds of the code performance over the Rician channels with independent fading are derived. Furthermore, the log-likelihood ratios and extrinsic information for maximum a posteriori (MAP) probability and soft-output Viterbi algorithm (SOVA) decoding methods on fading channels are developed. The derived upper bounds are evaluated and compared to the simulated bit-error rates over independent fading channels. The performance of parallel and serial codes with MAP and SOVA iterative decoding methods, with and without channel state information, is evaluated by simulation over independent and correlated fading channels. It is shown that, on correlated fading channels, the serial concatenated codes perform better than parallel concatenated codes. Furthermore, it has been demonstrated that the SOVA decoder has almost the same performance as the MAP decoder if ideal channel state information is used on correlated Rayleigh fading channels. Index Terms—Concatenated codes, fading channels, turbo codes.

I. INTRODUCTION

O

N MANY REAL links such as cellular radio and satellite mobile channels, transmission errors are mainly caused by variations in received signal strength referred to as fading. This severely degrades the transmission performance, and powerful error-control coding techniques are needed to reduce the penalty in signal-to-noise ratio (SNR). Parallel concatenated convolutional codes, known as turbo codes, can achieve a remarkably low bit-error rate (BER) with iterative decoding at an SNR close to the Shannon capacity limit on additive white Gaussian noise (AWGN) channels [1]. On AWGN channels, turbo-code performance has been discussed and evaluated by using analytical average upper bounds [6]. The performance and design of turbo codes on Rayleigh fading channels have been considered in [5]. The upper bounds Paper approved by P. Hoeher, the Editor for Coding and Communication Theory of the IEEE Communications Society. Manuscript received July 19, 1999; revised January 9, 2001 and June 7, 2001. This paper was presented in part at the IEEE International Conference on Communications (ICC), Vancouver, BC, Canada, June, 1999, and in part at the IEEE International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC), Osaka, Japan, September, 1999. J. Yuan is with the School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia (e-mail: [email protected]). W. Feng is with the Telstra Research Laboratories, Locked Bag 6764, NSW 1100, Australia (e-mail: [email protected]). W. Feng and B. Vucetic are with the School of Electrical and Information Engineering, University of Sydney, Sydney, NSW 2006, Australia (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TCOMM.2002.803971

of turbo-code performance on fast and slow Rician fading channels with ideal channel state information (CSI) were presented in [15]. In this paper, we consider parallel and serial concatenated code error performance on frequency-nonselective Rician fading channels. Analytical upper limits for parallel and serial concatenated code performance were derived when no CSI is available at the receiver. The upper bounds can be used to evaluate code performance. For decoding, we consider the iterative maximum a posteriori (MAP) probability and soft-output Viterbi algorithm (SOVA) for concatenated codes. If no CSI is available, the MAP and SOVA methods with a standard Euclidean distance metric are used in the decoder. We modify the decoding metric for fading channels with CSI and derive the log-likelihood ratios and extrinsic information for MAP and SOVA decoding methods. The iterative decoding methods with the modified metric give an improvement of about 2 dB for SOVA and 1.5 dB for MAP at a BER of 10 . The derived upper bounds and the BER simulation results for both parallel and serial concatenated codes over independent and correlated fading channels are discussed. It is shown that, on an independent Rayleigh fading channel, the parallel codes have better performance than the serial codes at low SNR, but the serial codes outperform the parallel codes at high SNR. However, on a correlated fading channel, the serial codes perform better than parallel codes at both low and high SNRs. It is also shown that the SOVA decoder has almost the same performance as the MAP decoder if ideal CSI is used. The paper is organized as follows. Section II gives the description of the system and presents the channel model. The bit-error probability upper bounds for concatenated codes over independent Rician fading channels are derived in Section III. In Section IV, the modified MAP and SOVA decoding methods based on the new metric for fading channels are presented. In Section V, the derived upper bounds are evaluated and the code BER performance on the independent and correlated fading channels are simulated and discussed. Finally, Section VI presents the conclusions. II. SYSTEM MODEL A. Channel Encoder We consider parallel and serial concatenated convolutional codes. The encoders for the two schemes are shown in Fig. 1(a) and (b), respectively. For a parallel encoder, two identical component recursive systematic convolutional (RSC) encoders with ) are sepathe memory order (the number of states is rated by a random interleaver with size [1]. For a serial one,

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For , there is no specular component, and the Rician pdf becomes a Rayleigh pdf. For approaching infinity, there is no fading at all, resulting in an AWGN channel. III. AVERAGE UPPER BOUNDS ON THE FADING CHANNEL The bit-error probability of the coded system can be upper bounded by using a union bound that sums contributions from all the codewords with various Hamming weights [6]–[8]. As a parallel or serial concatenated code can be represented by an equivalent linear block code if the trellis termination is used to drive the component encoders to the all-zero state, the union upper bound for a block code can be applied to a concatenated code. We assume that a rate 1/3 parallel concatenated code with an ) interleaver size and memory order is equivalent to an ( block code, where

(a)

(b)

(5)

Fig. 1. (a) Turbo encoder. (b) Serial concatenated encoder.

the inner code is an RSC code, and the outer one is a nonrecursive convolutional code [12]. The inner encoder and the outer encoder are linked by a random interleaver. For an input message of length , the interleaver size for the serial code is 2 , provided that the rate of the outer encoder is 1/2. B. Channel Model The channel is modeled by frequency-nonselective Rician fading, whereby the fading amplitude is represented by a Rician random variable, which remains constant over each symbol. Let be the transmitted signal at time . The received signal at time is given by (1) is a random variable which represents the channel where fading variation, and is AWGN with single-sided power spec. In this model, the envelope amplitude of the tral density fading attenuation is a Rician random variable. By assuming to ensure that the received signal power is equal to the transmitted one, the probability density function (pdf) of the fading amplitude is given by

(2) where is the Rician factor denoting the ratio of the power of the specular signal component to the power of the diffuse fading is the zero-order modified Bessel signal component, and function of the first kind. The mean and variance of the Rician random variable are given by [10]

(3) (4) is the first-order modified Bessel function of the where first kind. Small values of indicate a severely faded channel.

In this section, we derive the upper bounds on the pairwise-error probability for block codes on an independent fading channel. Subsequently, we come up with the average upper bounds on the bit-error probability for parallel and serial concatenated codes on a fading channel. In the analysis, we assume that the channel interleaving depth is sufficiently large, resulting in independent and identically distributed (i.i.d.) fading amplitudes. A. Upper Bounds on the Pairwise-Error Probability 1) Decoding Without CSI: The performance upper bounds on Rician fading channels with ideal CSI in decoding were derived in [15], and we give a brief review of the results. , where Let a coded sequence be is the code length. This sequence is modulated by a binary phase-shift keying (BPSK) modulator. The modulated sequence is transmitted over the fading channel. is At the receiver, the received sequence observed. Furthermore, we assume that the fading attenuation can be perfectly recovered at the receiver. The decoder then performs soft decoding using the modified Euclidean distance metric (6) is the probThe pairwise-error probability ability that the decoder chooses as its estimate the sewhen the transmitted sequence . This occurs if quence was in fact . For a given realization of the fading variable , the conditional pairwise-error probability is given by

(7) , , represent the value of the fading ampliLet and tude in the th bit interval in which the two sequences are different, where is the Hamming distance between the

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and

. The pairwise probability conditioned

Using the bound , , the conditional pairwise-error probability can be upper bounded by

(8)

(14)

is the code rate, is the SNR per bit, where is the complementary error function. Introducing the and , , in (9), we obtain inequality

can then be deterThe pairwise-error probability , mined by averaging over the random fading amplitudes , and is given by

(9) can then be deterThe pairwise-error probability mined by averaging over the random fading amplitudes and is upper bounded by [15]

(15) In this analysis, we will introduce the random variable (16)

(10) 2) Decoding Without CSI: In deriving the above result, we assumed that the values of fading amplitudes can be recovered perfectly. Indeed, in some systems it may not be practical to attempt CSI recovery, so the pairwise-error bound in the absence of CSI becomes relevant. When no CSI is available at the receiver, the decoder performs soft decoding using the standard Euclidean distance metric (11) The pairwise-error probability

are i.i.d. random variAs the fading amplitudes ables, for a large Hamming distance (larger than five), according to the central limit theorem, the random variable is approximately Gaussian with the mean (17) and the variance (18) and are, respectively, the mean and variance of where the Rician random variable , and they are given by (3) and (4), respectively. Noting that

is given by

if Re (12) We will introduce a random variable equal to the term on the left-hand side of the inequality in (12). For a given realization is a Gaussian variable with zero of the fading variable , , where is the variance of mean and variance the Gaussian noise. The term on the right-hand side is a constant which depends on . Therefore, the conditional pairwise-error probability can be expressed as [5]

(13)

(19)

we get for the pairwise-error probability upper bound from (15)

(20) For the turbo codes, is typically larger than five, so the above bound holds in most cases. B. Average Upper Bound on the Bit-Error Probability ) linear systematic block code, let us denote by For an ( the error coefficient which determines the contribution of the codewords with the same weight to the bit-error probability. The bit-error probability of the code decoded by a maximum-likelihood algorithm over a fading channel can be upper bounded by [6] (21)

YUAN et al.: PERFORMANCE OF PARALLEL AND SERIAL CONCATENATED CODES ON FADING CHANNELS

Fig. 2. Bit-error probability upper bound for the four-state, rate 1/3 parallel = 100 on independent Rician fading channels code with interleaver size with ideal CSI. The curves are for Rician channels with = 0, 2, 5, 50, starting from the top, with the bottom one referring to an AWGN channel.

N

K

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Fig. 3. Bit-error probability upper bound for the four-state, rate 1/3 parallel code with interleaver size = 100 on independent Rician fading channels without CSI. The curves are for Rician channels with = 0, 2, 5, 50, starting from the top, with the bottom one referring to an AWGN channel.

N

K

where is the code minimum distance and is the pairwise-error probability when the Hamming distance between and is . the two sequences Therefore, the bit-error probability upper bound on an independent Rician fading channel is given by

(22) when ideal CSI is available and equal to

(23) when there is no CSI. The bounds (22) and (23) can be evaluated as functions of and if the code distance spectrum is known. Here, we assume the uniform interleavers are used in the concatenated encoders, so that the error coefficient can be obtained by averaging over all possible interleaver structures [6], [8]. In the evaluation, we consider a parallel and a serial code with and memory order . For fair the same code rate comparison of the parallel and the serial code, the interleaver size should be chosen in such a way that the decoding delay due to the interleaver is the same. In other words, the input message length for both codes should be the same. We chose a message in the performance evaluation. For the parlength of allel code, the generator matrix of the component code in the octal form is (1, 5/7). The bit-error probability upper bounds for are illustrated in Fig. 2 for the parallel code with variable the case of ideal CSI and in Fig. 3 for the case of no CSI. For the serial code, the generator matrix of the outer code is (7,5). The generator matrix of the inner code is

. The

Fig. 4. Bit-error probability upper bound for the four-state, rate 1/3 serial code with message length = 100 on independent Rician fading channels with ideal CSI. The curves are for Rician channels with = 0, 2, 5, 50, starting from the top, with the bottom one referring to an AWGN channel.

N

K

bit-error probability upper bounds for the serial code with variable are illustrated in Fig. 4 for the case of ideal CSI, and in Fig. 5 for the case of no CSI. From Figs. 4 and 5, we can see that ), the error performance for the Rayleigh fading channel ( can be improved by approximately 1.5 dB at the BER of 10 when ideal CSI is utilized, relative to the case with no CSI. This , improvement decreases as is getting larger. When the gain of CSI disappears. In order to compare the parallel and serial code performance on Rayleigh fading channels, we show the distance spectra and the bit-error probability upper bounds for both codes in Figs. 6 and 7, respectively. From Fig. 6, we can see that the serial code has smaller error coefficients than the parallel code at low-tomedium Hamming distances. Since these distance spectral lines dominate the code-error performance in the region of high SNR [11], the serial code can achieve better performance than the

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Fig. 5. Bit-error probability upper bound for the four-state, rate 1/3 serial code with message length = 100 on independent Rician fading channels without CSI. The curves are for Rician channels with = 0, 2, 5, 50, starting from the top, with the bottom one referring to an AWGN channel.

N

K

Fig. 6. Distance spectrum comparison of the four-state, rate 1/3 parallel and serial concatenated codes with message length = 100.

N

parallel code on fading channels at high SNR as illustrated in Fig. 7. It may also be observed from Fig. 6 that for mediumdistance values, the error coefficients of both codes are almost the same, which will result in almost the same error performance at low SNR, as the code performance in this region is determined by the medium-distance spectral lines [11]. In Fig. 8, we show the bit-error probability upper bounds for both codes on the Rayleigh fading channel with the ideal CSI. For each code, we consider the message length of 100 and 1000. It can be seen that when we increase the message length from 100 to 1000, the performance gain of the serial code relative to the parallel code is increased significantly. IV. ITERATIVE DECODING ON FADING CHANNELS On AWGN channels, the turbo decoding is performed by a suboptimal iterative algorithm. The decoder consists of two identical concatenated decoders of the component codes sepa-

Fig. 7. Bit-error probability upper bound comparison of the four-state, rate 1/3 parallel and serial concatenated codes with message length = 100 on independent Rayleigh fading channels.

N

Fig. 8. Bit-error probability upper bound comparison of the four-state, rate 1/3 parallel and serial concatenated codes with message lengths of 100 and 1000 on independent Rayleigh fading channels with ideal CSI.

rated by an interleaver. The component decoders are based on a MAP algorithm or a SOVA generating a weighted soft estimate of the input sequence [2]–[4], [9]. In the turbo decoding for AWGN channels, a branch metric based on the standard Euclidean distance between the received and transmitted signal sequences is used. This metric is optimum for Gaussian channels. For fading channels with no CSI, we apply the metric for AWGN channels in turbo decoding. If ideal CSI is used to decode turbo codes for fading channels, the branch metric should optimize the conditional prob. This is equivalent to replacing the transability by in the branch metric [14]. Based on mitted signal this branch metric, we modified the log-likelihood ratio and extrinsic information for MAP and SOVA decoding methods. Note that the decoding methods described here are for parallel concatenated codes. However, the algorithms can be directly extended to serial codes.

YUAN et al.: PERFORMANCE OF PARALLEL AND SERIAL CONCATENATED CODES ON FADING CHANNELS

Without loss of generality, we assume that the bit sequence is encoded by an ( , 1, ) RSC endiscoder. The code trellis has a total number of . The tinct states, indexed by the integer , , where RSC encoder output sequence is are defined by encoder generators. The corresponding modulated sequence and received sequence are given by

(24)

(25)

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information generated by the code parity check bit and is given by

(29) is interleaved (or deinterThis extrinsic information leaved) and then used as the a priori information for the other decoder. 2) Modified SOVA Decoding Method With CSI: In the SOVA method, the path metric corresponding to path at time , denoted by , is calculated by [4]

1) Modified MAP Decoding Method With CSI: In the MAP decoding method, the log-likelihood ratio for the input symbol , is given by [2], [3] at time , denoted by

(26)

is the branch-transition probability which where can be determined from the transition probabilities of the and are the channel and the encoder trellis. forward and backward recursive probability functions obtained . For an independent fading channel with ideal from , based on the CSI, the branch transition probability modified branch metric, can be computed as

(30)

is the th modulated symbol on path at time , where is the smallest metric of the path connected to path at time , is the logarithm of the a priori probability at time and obtained from the previous decoder for the information bit . The SOVA decoder provides a soft output in the form of an approximate log-likelihood ratio of the a posteriori probabilities of the information bits 1 and 0. The soft output of the SOVA decoder for the input symbol at time can be approximately expressed as the metric difference of the maximum-likelihood path and its strongest competitor at time . The strongest competitor of the maximum-likelihood path is the path which has the minimum path metric among all paths obtained by replacing the trellis symbol on the maximum-likelihood path at time by its at complementary symbol. The SOVA output, denoted by time , can be expressed as [4]

(27)

(31)

is the a priori probability of and is the where variance of the Gaussian noise. In a derivation similar to [2], [3], and [9], the log-likelihood ratio for the input symbol at time , , can be decomposed into

is the minimum path metric corresponding to where and is the minimum path metric corresponding to . Following the derivation in MAP, we can split the soft output , and the extrinsic into two parts, the intrinsic information , as information (32)

(28) The intrinsic information, where the first term is the a priori information obtained from the other decoder. The second term is the systematic information is the extrinsic generated by the code information bit.

, is given by (33)

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Fig. 9. Performance comparison of MAP and SOVA, with and without CSI, for the 16-state, rate 1/3 parallel code on an independent Rayleigh fading channel, the message length N = 1024, the number of iterations I = 8.

Fig. 10. Performance comparison for the four-state, rate 1/3 parallel and serial codes on an independent Rayleigh fading channel, the message length N = 1024 and 4096.

Therefore, the extrinsic information from (32) and (33) as

the parallel code, a BER error floor appears at 10 . A similar result has been obtained for AWGN channels [4]. In order to compare the code performance against the Shannon limit, we computed the capacity for the Rician fading channel [5], [13]. The Shannon capacity limits for coherent BPSK on the independent Rician fading channel for various values and code rates are shown in Table I, where the values correspond to an arbitrarily small error probability. These values can serve as references in indicating how far away the performance of a parallel or serial code is from the theoretical channel capacity limits. The BER performance of on the the rate 1/3 parallel code with memory order factors is plotted in independent Rician channel for various Fig. 11. The message length is 65 536. The iterative decoding with MAP algorithm with ideal CSI is used in the simulation. The number of iterations is 20. Comparing the curves of Fig. 11 and the theoretical channel capacity limits in Table I, we can observe that the BER performance at 10 for the parallel code is within 0.5 0.6 dB of the capacity limit for channel models ranging from AWGN to Rayleigh. In some applications, the channel-interleaver size is severely constrained by the maximum delay requirements. This results in a correlated fading channel. However, the performance of the communication systems with correlated fades is not amenable to analytical evaluation. In this section, we investigate the BER performance of the coding schemes on correlated Rayleigh fading channels by simulation. In the channel model, we assume that the power spectral density of the faded amplitude due to the Doppler shift, denoted by , is given by

can be obtained

(34) and it will be used to calculate the a priori probabilities ratio in the next step of decoding after interleaving or deinterleaving. V. SIMULATION RESULTS The BER performance of the parallel concatenated coding scheme on fading channels is estimated by simulation. In the simulation, we considered the rate 1/3 parallel code and message length . with memory order The generator polynomial of the parallel code in the octal form is (1, 21/37). In the receiver, iterative decoding with symbol-by-symbol MAP and SOVA methods, with and without CSI, is employed. The number of iterations is eight for the interleaver size of 1024. For the independent Rayleigh fading channels, the performance simulation results are shown in Fig. 9. At a BER of 10 , both MAP and SOVA decoders with CSI give an improvement of about 1 dB relative to the respective decoders without CSI. It can also be observed that MAP decoding is superior to SOVA decoding by 0.5 dB at the BER of 10 , for both decoders with and without CSI. The BER performance comparison of the parallel and serial coding schemes on fading channels was also carried out by simulation. We simulated the rate 1/3 parallel and serial codes with 2 and message length of 1024 and memory order 4096. In the receiver, iterative SOVA decoding is employed. The number of iterations was eight for message length 1024 and 18 for message length 4096. For the independent Rayleigh fading channels, the performance simulation results for the parallel and serial codes are shown in Fig. 10. It can be observed that the parallel code has a better performance at low SNR than the serial one. However, the serial code outperforms the parallel code at high SNR. For

if

(35)

otherwise is the fade where is the frequency and is the fade rate. rate normalized by the symbol rate. It serves as a measure of the channel memory. For correlated fading channels, this parameter

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TABLE I CHANNEL CAPACITY LIMITS FOR COHERENT BPSK ON FADING CHANNELS WITH IDEAL CSI

Fig. 11. Performance of the 16-state, rate 1/3 parallel code on an AWGN and independent Rician fading channels with ideal CSI, the message length N = 65 536, the number of iterations I = 20.

Fig. 12. Performance comparison of MAP and SOVA, with and without CSI, for the 16–state, rate 1/3 parallel code on a correlated Rayleigh fading channel, the fading rate normalized by the symbol rate is 10 , the message length N = 1024, the number of iterations I = 8.

is in the range , indicating finite channel memory. The autocorrelation function of the fading process is given by (36) is the zeroth-order Bessel function of the first kind. where In the simulations, the fade rate normalized by the symbol rate was 0.01. The performance comparison of MAP and SOVA, with and without CSI, for the rate 1/3 parallel code with memory and message length on a correlated order Rayleigh fading channel is shown in Fig. 12. We can see from the figure that the MAP decoder with CSI achieves an improvement of about 1.2 dB at a BER of 10 relative to the MAP decoder without CSI. The corresponding improvement for the SOVA decoder with ideal CSI is about 1.8 dB. It can be also observed that without CSI, the performance degradation of the SOVA decoding relative to the MAP decoding is approximately 0.7 dB. However, the SOVA CSI decoder has almost the same . By comperformance as the MAP CSI decoder at high paring Fig. 9 with Fig. 12, we can see that the parallel code performance degrades significantly due to the correlated fades in the channel. When the fade rate normalized by the symbol rate is 0.01, the turbo code performance is about 1 1.8 dB worse relative to an independent fading channel. Performance comparison of the rate 1/3 parallel and serial and various message lengths codes with memory order on a correlated Rayleigh fading channel is shown in Fig. 13. In the simulation, iterative SOVA decoding is employed and the

Fig. 13. Performance comparison for the four-state, rate 1/3 parallel and serial codes on a correlated Rayleigh fading channel, the fading rate normalized by the symbol rate is 10 , the message length N = 1024, 4096, and 8192.

message lengths are 1024, 4096, and 8192. The number of iterations is eight for message length 1024 and 18 for message lengths 4096 and 8192. We can see from the figure that the serial code outperforms the parallel code on the correlated fading channels. At a BER of 10 , the corresponding improvements achieved by the serial code with message lengths of 1024, 4096, and 8192 relative to the parallel code are 1.25, 0.8, and 1.25 dB, respectively.

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VI. CONCLUSIONS In this paper, we consider the performance of parallel and serial concatenated convolutional codes on frequency-nonselective Rician fading channels. The analytical average upper limits of the code performance based on the union bound and code weight distributions are derived and discussed. The parallel and serial concatenated code performance based on the derived analytical bounds are compared over independent fading channels. The log-likelihood ratios and extrinsic information for MAP and SOVA decoding methods on fading channels are developed. The performance of MAP and SOVA iterative decoding methods, with and without CSI, is evaluated by simulation over independent and correlated fading channels. It is shown that the serial concatenated codes perform better than parallel concatenated codes on correlated fading channels. Furthermore, it is demonstrated that the SOVA decoder has almost the same performance as the MAP decoder if ideal CSI is used on correlated Rayleigh fading channels. REFERENCES [1] C. Berrou, A. Glavieux, and P. Thitimajshima, “Near Shannon limit error-correcting coding and decoding: Turbo-codes (1),” in Proc. Int. Conf. Communications, May 1993, pp. 1064–1070. [2] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear code for minimizing symbol error rate,” IEEE Trans. Inform. Theory, vol. IT–20, pp. 284–287, Mar. 1974. [3] J. Hagenauer, P. Robertson, and L. Papke, “Iterative (“Turbo”) decoding of systematic convolutional codes with the MAP and SOVA algorithms,” in Proc. ITG Conf. Source, Channel Coding, Munich, Germany, Oct. 1994, pp. 1–9. [4] B. Vucetic, “Iterative decoding algorithm,” in Proc. PIMRC, Helsinki, Finland, Sept. 1997, pp. 99–120. [5] E. K. Hall and S. G. Wilson, “Design and analysis of turbo codes on Rayleigh fading channels,” IEEE J. Select. Areas Commun., vol. 16, pp. 160–174, Feb. 1998. [6] S. Benedetto and G. Montorsi, “Unveiling turbo codes: Some results on parallel concatenated coding schemes,” IEEE Trans. Inform. Theory, vol. 42, pp. 409–428, Mar. 1996. , “Design of parallel concatenated convolutional codes,” IEEE [7] Trans. Commun., vol. 44, pp. 591–600, May 1996. [8] D. Divsalar, S. Dolinar, F. Pollara, and R. J. McEliece, “Transfer function bounds on the performance of turbo codes,”, JPL TDA Progress Rep. 42-122, 1995. [9] P. Robertson, “Illuminating the structure of parallel concatenated recursive systematic (TURBO) codes,” in Proc. Globecom, San Francisco, CA, Nov. 1994, pp. 1298–1303. [10] B. Vucetic, “Bandwidth efficient concatenated coding schemes for fading channels,” IEEE Trans. Commun., vol. 41, pp. 50–61, Jan. 1993. [11] J. Yuan, B. Vucetic, and W. Feng, “Combined turbo codes with interleaver design,” IEEE Trans. Commun., vol. 47, pp. 484–487, Apr. 1999.

[12] S. Benedetto, G. Montorsi, D. Divsalar, and F. Pollara, “Serial concatenation of interleaved codes: performance analysis, design and iterative decoding,”, JPL TDA Progress Rep. 42-126, 1996. [13] E. Biglieri, J. Proakis, and S. Shamai (Shitz), “Fading channels: Information-theoretic and communications aspects,” IEEE Trans. Inform. Theory, vol. 44, pp. 2619–2692, Oct. 1998. [14] D. Divsalar and M. K. Simon, “Trellis-coded modulation for 4800 to 9600 bps transmission over a fading satellite channel,” IEEE J. Select. Areas Commun., vol. 5, pp. 162–175, Feb. 1987. [15] F. Babich, G. Montorsi, and F. Vatta, “Performance bounds of continuous and blockwise decoded turbo codes in Rician fading channel,” IEE Electron. Lett., vol. 34, no. 17, pp. 1646–1648, Aug. 1998.

Jinhong Yuan (S’96–A’97) received the B.E. and Ph.D. degrees in electrical engineering from the Beijing Institute of Technology, Beijing, China, in 1991 and 1996, respectively. In 1997, he joined the School of Electrical and Information Engineering, University of Sydney, Sydney, Australia, as a Research Fellow. He is currently with School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, Australia. His research interests include wireless communications, communication theory, error control coding, and digital modulation.

Wen Feng received the B.E. degree in electrical engineering from Chongqi Institute of Architecture, Chongqi, China, in 1990 and the Ph.D. degree in electrical and information engineering from the University of Sydney, Sydney, Australia, in 2001. She joined Telstra Research Laboratories in 1999, where she is currently Technology Specialist. Her research interests include wireless communication, wireless LAN, and error-control coding.

Branka Vucetic (M’83–SM’00) received the B.S.E.E., M.S.E.E., and Ph.D. degrees in 1972, 1978, and 1982, respectively, from the University of Belgrade, Belgrade, Yugoslavia. She is the Director of Telecommunications Laboratory and Professor in Telecommunications at the University of Sydney, Sydney, Australia. Her research interests include wireless communications, digital communication theory, coding, and multiuser detection.