Serial Concatenation Of Interleaved Codes - Telecommunications

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put information bits followed by the parity check bits of the first and second ... by an interleaver of length N . We call the obtained concatenated codes SCBC.
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Serial concatenation of interleaved codes: analytical performance bounds Sergio Benedetto, Guido Montorsi Dipartimento di Elettronica, Politecnico di Torino C.so Duca degli Abruzzi 24, 10129 Torino, Italy Ph: $39-1 1-5644031, FAX: $39-1 1-5644099 e-mail: authorname@polito. it Abstract - - Parallel concatenated coding schemes employing convolutional codes as constituent codes linked by an interleaver have been proposed in the literature as "turbo codes". They yield very good performance in connection with simple suboptimum decoding algorithms. In this paper, we propose an alternative scheme consisting in the serial concatenation of block or convolutional codes and evaluate its average performance in terms of analytical bounds on the bit error probability. Comparisons with parallel concatenated coding schemes show in some cases sensible advantages.

I. INTRODUCTION Since their appearance in [l],"turbo codes" have been the object of a great interest, and consequently of wide investigation, in the coding community. Schematically, a turbo code consists of two (or more) constituent systematic recursive convolutional encoders separated by an interleaver which performs a scrambling of the input sequence of the first encoder. The encoded sequence is formed by the input information bits followed by the parity check bits of the first and second encoders. Their importance stems from the fact that they achieve considerable coding gains, yet admitting iterative soft decoding schemes whose complexity is not significantly higher than that of the decoders for the single constituent codes. Astonishingly good results within 0.6 dB of the Shannon limit have been demonstrated with low rate codes (less than l / 2 ) [2;3] and high rate codes in connection with trellis-coded modulation [4;5;6]. In [7;8] we have shown how to evaluate the performance of parallel concatenated coding schemes using as constituent codes both block and convolutional codes, and how t o design the constituent codes in order t o optimize the turbo code performance. In this paper, we analyze an alternative t o the parallel concatenation, which consists in the serial concatenation of two constituent codes (CCs) separated by an interleaver of length N . We call the obtained concatenated codes SCBC (serial concatenated block codes) or SCCC (serial concatenated convolutional codes) according to the nature of the CCs. We derive upper bounds to the bit error probability obtained with maximum-likelihood (ML) decoding of SCBC and SCCC codes and show with examples that, in some cases, This work was supported by Italian National Research Council (CNR) under "Progetto Finalizzato Trasporti (Prometheus)", by the Italian Ministry of University and Research (MURST) under " P r o g e t t o M U R S T 40% "Comunicazioni con Mezzi Mobili", and by NATO under t h e Research Grant CRG 951208.

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Fag. 1. Serially concatenated block code. the new scheme outperforms turbo codes. In [9] we deal with the algorithms for iterative decoding, whose preliminary performance confirms the analytical findings of this paper. For simplicity of the exposition, we will present the methodology in connection with SCBC, and then extend it to the more complicated case of SCCC.

11. SERIALLYCONCATENATED

BLOCK CODES

The scheme of two serially concatenated block codes is shown in Fig. 1. It is composed of two cascaded CCs, the outer ( k , N ) code and the znner ( N ,n ) code, linked by an interleaver of length N . The overall SCBC is then a ( k , n ) code. Serially concatenated codes were proposed by Forney in [lo], and have found many practical applications since then. In particular, the concatenation of a Reed Solomon code with an inner 64-state convolutional code has become a standard for space communications. Serial concatenation of convolutional codes was studied in [ll]. Recently, the serial concatenation of two convolutional codes separated by an interleaver has been considered by [la]. The proposed scheme used the interleaver to break the burst of errors of the inner code in a standard two-step decoding algorithm. Here, we will estimate the ML performance of the overall code including the interleaver. As in [13;14], a crucial step in the analysis consists in replacing the actual interleaver that performs a permutation of the N input bits with an abstract interleaver called uniform znterleauer, defined as a probabilistic device that maps a given input word of weight 1 into all distinct

(y).

( y ) per-

mutations of it with equal probability 1/ Use of the uniform interleaver leads t o a much easier computation of the average performance of SCBC, intended as the expectation over the ensemble of all interleavers of a given length of the performance of any SCBC with the same CCs. Moreover,

0-7803-3336-5/96 $5.00 0 1996 IEEE

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107 the aveFage performance obtained through the use of the uniform inherleaver are highly significant, in the sense that there will al4ays be, for each value of the signal-to-noise ratio, at least ohe particular interleaver yielding performance better or equal to those of the uniform interleaver. Let s define the Input-Output Weight Enumerating Function (I WEF) of the SCBC as

Example I Consider the parity check code (4,3) concatenated through an interleaver of length 4 t o a Hamming code (7,4). The IOWEF Aco(W,L ) and ACs(L,H ) of the outer and inner code are

8

Aco(W,L ) = 1

is the number of codewords of the SCBC with associated t o an input word of weight w. define the Condataonal Wezght Enumeratang Functaon ( C V E F ) ACs(w, H ) of the SCBC as the weight distribution of the codewords of the SCBC conditioned t o a given weight b of the input word. It is related to the IOWEF by

+ W ( 3 L 2 )+ W 2 ( 3 L 2 )+ w ~ ( L ~ )

where

so that

Aco(W,O)

Aco(W,l) Ac0(W,2) ACO(W,3) Ac0(W,4)

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The knowledge of the CWEF permits t o obtain an upper bound to the bit error probability of the SCBC in the form

= = = = =

1 0 3W+3W2 0 W3

and ACg(O,H) = 1 AC'(l,H) = 3 H 3 + H 4

~ 3 1

= 3H3+3H4 AC'(3,H) = H3+3H4 A C z ( 4 , H ) = H7 . AC'(2,H)

where & = k / n is the code rate, and Eb/No is the signalto-noise; ratio per bit. The problem thus consists in the evaluation of the CWEF of the SFBC from the knowledge of the CWEFs of the outer and innler CCs, which we call Aco(w,L) and A C t ( I , H ) . To do this,lwe exploit the properties of the uniform interleaver, which tFansforms a codeword of weight 1 at the output of the first' encoder into all its distinct permutations. As a consequence, each codeword of the outer code COof weight 1 generdtes codewords of the inner code C, so that the

(r)

Thus, using ( 5 ) , we obtain 4

ACs(W,H) =

(">

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number~Azyhof codewords of the SCBC of weight h associated wifh an input word of weight w is given by I

x Act(/,H )

1.1 0.(3H3+H4)+ -+ 1 4 ( 3 W + 3 W 2 ) . ( 3 H 3 + 3H4) + + 6 + 0 . ( H 34+ 3 H 4 ) + W 31.H7

0

Previous results (5) and (6) can be easily generalized t o the case of an interleaver with length N being an integer multiple (by a factor m) of the length of the outer codewords. Denoting by AcT(W,L) the IOWEF of the new ( N , m k ) outer code, and similarly by A C r ( L , H ) the IOWEF of the new ( m n ,N ) inner code, it is straightforward t o obtain AC? (W,L )

W,l ) is the conditional weight distribution of the input w rds where ( that generate codewords of the outer code of weight 11

I)

= 1 + W ( 1 . 5 H 3+ 1.5H4)+ +W2(1.5H3 + 1.5H4)+ W3H7

(4) From~(4)we easily derive the expressions of the IOWEF and C y E F of the SCBC

Aco(W,

=

[Aco(W,L)lm

A C r ( L , H ) = [Acz(L,H ) l m .

(7)

co

From the IOWEFs (7), through (2) we obtain the CWEFs ACr(W,l)and ACr((l,H ) of the new CCs, and, finally, the

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108

Rate 113 SCCC 10.' ,

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-~ -~ Fig. 3. Serially Concatenated convolutional code.

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CWEF and IOWEF of the new (mn,mk, N ) SCBC

Cy

upper bound to the bit error probability can be obtained through the standard transfer function technique. As explained in [7], an approximation valid when the length of the interleaver is much greater than the constraint length of the CCs consists in retaining only the branch of the hypertrellis joining the hyperstates S00,Soo.In the following, we will always use this approximation.

Example 3 Consider a rate 1/3 SCCC formed by an outer 4-state convolutional code with rate 1/2 and an inner 4-state convolutional code with rate 213, joined by a uniform interleaver of length N = 200,400,600,800,1000and 2000. Both outer and inner encoders are systematic and recursive with the following generator matrices:

Example 2 Consider again the CCs of Example 1, linked by an interleaver of length N = 4m, and use equation (8) and (3). The so obtained upper bound t o the bit error probability is plotted in Fig. 2 for various values of N . The curves show the interleaver gain, defined as the factor by which the bit error probability is decreased with the interleaver length. Contrary t o parallel concatenated block codes [7], the curves do not exhibit the interleaver gain saturation. Rather, the bit error probability seems to decrease regularly with N as N - l .

Using the previously outlined analysis, we have obtained the bit error probability curves shown in Fig. 4. The performance show a very significant interleaver gain, i.e. lower values of the bit error probability for higher values of N . The interleaver gain seems t o behave as N - 3 . This behavior is explained in a forthcoming paper [9].

0

In this section, we will use the bit error probability bounds previously derived to compare the performance of parallel ("turbo codes", [7]) and serially concatenated block and convolutional codes.

111. SERIALLY CONCATENATED

CONVOLUTIONAL CODES

The structure of a serially concatenated convolutional code (SCCC) is shown in Fig. 3. It refers to the case of two convolutional CCs of rate 1/2 and 2/3 joined by an interleaver of length N generating a rate 1/3 SCCC. The exact analysis of this scheme requires, as illustrated in [7], the use of a hypertrellzs having as states pairs of states of the outer and inner code. The hyperstates Sij and S,, are joined by a hyperbranch which consists of all pairs of paths with length N/2 that join the states si,SI of the outer code and the states s i , s, of the inner code, respectively. Each hyperbranch is thus an equivalent SCBC labeled with an IOWEF that can be evaluated as explained before. From the hypertrellis, the

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IV. COMPARISON BETWEEN

PARALLEL AND SERIALLY

CONCATENATED C O D E S

A . Parallel and serially concatenated block codes To obtain a fair comparison, we have chosen the following PCBC and SCBC: the PCBC have parameters ( l l m , 3m, N ) and employs two equal (7,3) systematic cyclic codes with generator g ( D ) = (1 D)(1 D D 3 ) ;the SCBC, instead, is a (15m, 4m, N ) SCCC, obtained by the concatenation of the (7,4) Hamming code with a (15,7) BCH code. They have almost the same rate (Res = 0.266, RcP = 0.273), and have been compared choosing the interleaver length in such a way that the decoding delay due t o the interleaver, measured in terms of input information bits, is

+

+ +

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109

B. Seraally and parallel concatenated convolutaonal codes To obtain a fair comparison, we have chosen the following

10-

PCCC and SCCC: the PCCC is a rate 1/3 code obtained

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the San bits, W I PCBC, The error p various the inp other h length and a h mance

. As an example, to obtain a delay equal t o 12 input must choose an interleaver length N = 12 for the .nd N = 12/Rg = 21 for the SCBC. sults are reported in Fig. 5, where we plot the bit Nbability versus the signal-to-noise ratio Eb/No for iput delays. The results show that, for low values of delay, the performance is almost the same. On the nd, increasing the delay (and thus the interleaver ) yields a significant interleaver gain for the SCBC, bst no gain for the PCBC. The difference in perfor3 dB at Pb(e) = in favor of the SCBC.

concatenating two equal rate 1/2, 4-state systematic recursive convolutional codes with the generator matrix of the outer code of Example 3. The SCCC is the same used in the Example 3. Also in this case, the interleaver lengths have been chosen so as t o yield the same decoding delay, due to the interleaver, in terms of input bits. The results are reported in Fig. 6, where we plot the bit error probability versus the signal-tonoise ratio Eb/No for various input delays. The results show the great difference in the interleaver gain. In particular, the PCCC shows an interleaver gain going as N - l , whereas the interleaver gain of the SCCC goes as N - 3 . This means, for Pb(e) = a gain of more than 2 dB in favour of the SCCC. Previous comparisons have shown that serial concatenation is advantageous with respect to parallel concatenation, in terms of maximum-likelihood performance. For mediumlong interleaver lengths, however, this significant result remains a theoretical one, as maximum-likelihood decoding is an almost impossible achievement. For parallel concatenated codes ("turbo codes"), iterative decoding algorithms have been proposed, which yield performance close t o optimum, with limited complexity. In [9], we present a new iterative decoding scheme capable of decoding serially concatenated codes, and prove, with several examples of applications, that the performance gain with respect to parallel concatenation is maintained.

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Fag. 5. 'omparison of SCBC and PCBC with various interleaver I( gths, chosen so as t o yield the same input decoding delay.

r two interleaver lengths, chosen so as t o yaeld the same put decoding delay.

V. CONCLUSIONS We have proposed a new coding scheme consisting in the serial concatenation of block and convolutional codes sepa-

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110 rated by an interleaver, and obtained upper bounds to their average performance in terms of maximum-likelihood bit error probability. The obtained results, compared to parallel concatenated block and convolutional codes of the same rate complexity show significant advantages.

[14] Sergio Benedetto and Guido Montorsi, “Performance evaluation of turbo codes”, Electronics Letters, vol. 31, no. 3, pp. 163-165, Feb. 1995.

REFERENCES Claude Berrou, Alain Glavieux, and Punya Thitimajshima, “Near Shannon Limit Error-Correcting Coding and Decoding: Turbo-Codes” , in Proceedings of ICC’93, Geneve, Switzerland, May 1993, pp. 1064-1070. D. Divsalar and F. Pollara, “Turbo Codes for PCS Applications” , in Proceedings of ICC’95, Seattle, Washington, June 1995. Sergio Benedetto, Dariush Divsalar, Guido Montorsi, and Fabrizio Pollara, “Soft-output decoding algorithms for continuous decoding of parallel concatenated convolutional codes”, in Proceedings of ICC’96, Dallas, Texas, June 1996. Sergio Benedetto, Dariush Divsalar, Guido Montorsi, and Fabrizio Pollara, “Bandwidth efficient parallel concatenated coding schemes”, Electronics Letters, vol. 31, no. 24, pp. 2067-2069, Nov. 1995. Sergio Benedetto, Dariush Divsalar, Guido Montorsi, and Fabrizio Pollara, “Parallel concatenated trellis coded modulation”, in Proceedings of ICC’96, Dallas, Texas, June 1996. Patrick Robertson and Thomas Worz, “Coded modulation scheme employing turbo codes”, Electronics letters, vol. 31, no. 18, pp. 1546-1547, Aug. 1995. Sergio Benedetto and Guido Montorsi, “Unveiling turbo-codes: some results on parallel concatenated coding schemes”, IEEE Transactions on Information Theory, vol. 42, no. 2, pp. 409-429, Mar. 1996. Sergio Benedetto and Guido Montorsi, “Design of Parallel Concatenated Convolutional Codes” , IEEE Transactions on Communications, vol. 44, no. 5, pp. 591-600, May 1996. Sergio Benedetto, Dariush Divsalar, Guido Montorsi, and Fabrizio Pollara, “Serial concatenation of interleaved codes: Performance analysis, design and iterative decoding”, IEEE Transactions on Information Theory, submitted, July 1996. G.D. Forney J r . , Concatenated Codes, M.I.T., Cambridge, MA, 1966. Joachim Hagenauer and Peter Hoeher, “Concatenated Viterbi Decoding” , in Proceedings of Fourth Joint Swedish-Soviet Int. Workshop on Information Theory, Gotland, Sweden, Studenlitteratur, Lund, Aug. 1989, pp. 29-33. Punya Thitimajshima, Systematic recursive convolutional codes and their application to parallel concatenation, PhD thesis, UniversitC de Bretagne Occidentale, Dec. 1993, (in french). Sergio Benedetto and Guido Montorsi, “Average Performance of Parallel Concatenated Block Codes”, Electronics Letters, vol. 31, no. 3, pp. 156-158, Feb. 1995.

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