On improved bounds on the decoding error probability of block codes over interleaved fading channels, with applications to turbo-like codes

Abstract

We derive here improved upper bounds on the error probability of block codes which are transmitted over fully interleaved Rician fading channels, coherently detected and maximum likelihood decoded. We assume that the fading coefficients during each symbol are statistically independent (due to a perfect channel interleaver), and that perfect estimates of these fading coefficients are provided to the receiver. The improved upper bounds on the block and bit error probabilities are derived for fully interleaved fading channels with various orders of space diversity, and are found by generalizing some recently introduced upper bounds for the binary-input AWGN channel. The advantage of these bounds over the ubiquitous union bound is demonstrated for some ensembles of turbo codes and low-density parity check codes, and it is especially pronounced in a portion of the rate region exceeding the cutoff rate. Our generalization of the Duman and Salehi bound which is based on certain variations of Gallager's bounding technique, is demonstrated to be the tightest reported upper bound. We therefore apply it to calculate numerically upper bounds on the thresholds of some ensembles of turbo-like codes, referring to the optimal maximum likelihood decoding. For certain ensembles of uniformly interleaved turbo codes, the upper bounds derived here also indicate good match with computer simulation results of efficient iterative decoding algorithms.

An improved sphere-packing bound for finite-length codes over symmetric memoryless channels

“…Further details on BPSK modulated signals transmitted over fully interleaved fading channels, including expressions for the capacity, cutoff rate and various bounds on the decoding error probability, are provided in [23] and references therein. Figure 1 also depicts the capacity limit bound (CLB) for these two channels (calculated from [23, Eq.…”

Coding theorems for turbo code ensembles

“…(Examples of the thresholds obtainable using these techniques are given in Tables I and II.) Additionally, the recent techniques of Divsalar [9], Duman and Salehi [16], [15], and Sason and Shamai [28], [32], [29], [30], which build on Gallager's technique [17], are all potentially capable of producing far stronger results than possible using the union bound.…”

Tight exponential upper bounds on the ml decoding error probability of block codes over fully interleaved fading channels

“…The generalization of the second version of Duman & Salehi bounds (i.e., the generalization of the DS2 bound in [11]), which relies on the Gallager bounding technique yields the following upper bound on P e (d) In contrast to [10], where the optimal function ψ (which can be viewed as a tilting measure with respect to the two measurements y and a) was pursued, here for the sake of closed form expressions, only an exponential tilting measure is examined. Let ψ(·, ·) be the following pdf:…”

“…The parameters above lie in the regions: Following the arguments in [10], a similar upper bound on the bit error probability is derived for binary and linear block codes. To that end, one relies on the input-output weight distribution of the examined code or ensemble (the reader is referred to [4,5,6] which derive several techniques for calculating the input-output weight distribution of codes and ensembles).…”

“…By comparing the upper bounds (10) and (11), we want to determine the parameters in (10) so that the upper bound (11) results as a particular case of (10). To this end, we set α = 1 in (10), and then we try to choose the remaining parameters in (10) so that the following equalities hold:…”