On interleaved, differentially encoded convolutional codes

We study a serially interleaved concatenated code construction, where the o uter code is a standard convolutional code, and the inner code is a recursi ve convolutional code of rate 1, We focus on the ubiquitous inner different ial encoder (used, in particular, to resolve phase ambiguities), double dif ferential encoder (used to resolve both phase and frequency ambiguities), a nd another rate 1 recursive convolutional code of memory 2, We substantiate analytically the rather surprising result, that the error probabilities co rresponding to a maximum-likelihood (ML) coherently detected antipodal modu lation over the additive white Gaussian noise (AWGN) channel for this const ruction are advantageous as compared to the stand-alone outer convolutional code. This is in spite of the fact that the inner code is of rate 1. The a nalysis is based on the tangential sphere upper bound of an ML decoder, inc orporating the ensemble weight distribution (WD) of the concatenated code, where the ensemble is generated by all random and uniform interleavers, Thi s surprising result is attributed to the WD thinning observed for the conca tenated scheme which shapes the WD of the outer convolutional code to resem ble more closely the binomial distribution (typical of a fully random code of the same length and rate). This gain is maitained regardless of a rather dramatic decrease, as demonstrated here, in the minimum distance of the co ncatenated scheme as compared to the minimum distance of the outer stand-al one convolutional code, The advantage of the examined serially interleaved concatenated code given In terms of bit and/or block error probability whic h is decoded by a practical suboptimal decoder over optimally decoded stand ard convolutional code is demonstrated by simulations, and some insights In to the performance of the iterative decoding algorithm are also discussed. Though we have investigated only specific constructions of constituent inne r (rate 1) and outer codes, Ive trust, hinging on the rational of the argum ents here, that these results extend to many other constituent convolutiona l outer codes and rate 1 inner recursive convolutional codes. Union bounds on the performance of serial and hybrid concatenated codes were addressed i n [8], where differential encoding was also examined, and shown efficient.