Improved Upper Bounds on the Performance of Parallel and Serial Concatenated Turbo Codes via their Ensemble Distance Spectrum Igal Sason and Shlomo Shamai (Shitz)
Dept. of Electrical Engineering, Technion - Haifa 32000, Israel Email:
[email protected] and
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Abstract | The ensemble performance of parallel and serial concatenated turbo codes (TC) is considered for a binary-input AWGN channel and maximum likelihood (ML) decoding. Following the derivation of the ensemble distance spectrum of parallel and serial concatenated codes, improved upper bounds on the bit and block error probabilities of these ensembles of codes are derived and the in uence of the interleaver length N and the memory length of the component codes m are investigated. The improved bounding technique, based on the tangential sphere bound, is compared to the conventional union bound and to a recent alternative bounding technique by Duman and Salehi. The advantage of the bounds is demonstrated for a variety of parallel and serial concatenated coding schemes with either xed or random component codes, and it is especially pronounced in the region above the cuto rate, where the performance of TC is most appealing. I. Introduction
The discovery of TC in 1993, demonstrating near Shannon limit performance, has been widely lauded as one of the most signi cant recent advances in coding. This motivates the introduction of analytical methods to assess the performance of these ecient codes, widely investigated by simulations. Upper bounds on the block and bit error probabilities of parallel and serial concatenated coding schemes averaged over all the possible interleavers of a given length were proposed [1],[2]. These union bounds cannot be used for Eb =No values below that corresponding to the cuto rate [1]-[3], excluding thus the most interesting operation region of TC. An upper bound on the block and bit error probabilities of TC with ML decoding is derived, using a modi ed version of Gallager's bound [4]. These bounds are useful for some range below the channel cuto rate. However they do not cover the full range of usefulness of the TC. Our improved upper bounds on the bit and block error probabilities are based on the tangential sphere bound by Poltyrev [5]. It is demonstrated that this bounding technique is advantageous and it extends further than the region of Eb =No for which the upper bounds ad hoc are useful.
The comparison between serial and parallel concatenated codes [6] shows that the improved upper bounds on the bit error probability of the serially concatenated codes are advantageous for low and moderate values of Eb =No (for rates below the cuto rate). The outstanding performance of either parallel and serial concatenated TC with large interleavers is attributed to the distance spectrum thinning observed for the concatenated schemes, which shape its distance spectrum to resemble more closely to the binomial distribution (advocated as a measure for good capacity approaching codes). This phenomena has been shown, by our bounding method [6, ref. 31], to account for the surprising advantage of a serial concatenated system which comprises a rate 1 standard di erential encoder (acting as an inner recursive convolutional code) combined with a standard convolutional outer code. References
[1] S. Benedetto and G. Montorsi, \Unveiling turbo codes: some results on parallel concatenated coding schemes," IEEE Trans. on Inform. Theory , vol. 42, no. 2, pp. 409{428, March 1996. [2] A.J. Viterbi, A.M. Viterbi, J. Nicolas and N.T. Sindushyana, \Perspective on interleaved concatenated codes with iterative soft-output decoding," Proceedings of the Int. Symp. on Turbo Codes and Related Topics , pp. 47{54, Brest, France, 3-5 September, 1997. [3] E. Telatar and R. Urbanke, \On the ensemble performance of turbo codes," Proceedings 1997 IEEE Int. Symp. Information Theory (ISIT'97), pp. 105, Ulm Germany, June 29-July 4, 1997. [4] T.M. Duman and M. Salehi, \New performance bounds for turbo codes," Proceedings of 1997 Global Communications Conference (GLOBECOM'97), pp. 634{638, USA, Phoenix, Arizona, November 4-8, 1997. [5] G. Poltyrev, \Bounds on the decoding error probability of binary linear codes via their spectra," IEEE Trans. on Inform. Theory , vol. 40, no. 10, pp. 1261{1271, October 1996. [6] I. Sason and S. Shamai (Shitz), \Improved upper bounds on the decoding error probability of parallel and serial concatenated codes via their ensemble distance spectrum." Submitted in extended version to IEEE Trans. on Inform. Theory . -2
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II. Results
BIT/ BLOCK error probability
Some of the results of [6] are shown in the gure, comparing the upper bounds on the bit and block error probabilities of multiple TC in a binary-input AWGN channel with ML decoding. The three component codes of the hTC are xed i and each of them has the generator G(D) = 1; 1+1+DD+D2 2 . The two uniform interleavers are of length N = 800. The new bounds on bit and block error probabilities, demonstrate about 1.5 dB improvement over the union bound at error probability of 10;3 . Our bound has been applied for the speci c case of parallel concatenated TC with xed component codes of rate 21 and a uniform interleaver of length N = 500, as in [4]. The advantage in bit error probability as compared to [4] for Pb = 10;3 is about 0.4 dB. Comparison to simulated iterative decoding performance [4] demonstrates the suboptimality of iterative decoding for some Eb =No region.
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