A Fast Recursive Algorithm and Architecture for Pruned Bit-reversal Interleavers

In this paper, pruned bit-reversal permutations employed in variable-length interleavers and their associated fast pruning algorithms and architectures are considered. Pruning permutations is mathematically formulated as a counting problem in a set of k integers and any subset of α consecutive integers under some permutation, where integers from this subset that map into indices less than some β < k are to be counted. A solution to this problem using sums involving integer floors and related functions is proposed. It is shown that these sums can be evaluated recursively using integer operations. Specifically, a mathematical treatment for bit-reversal permutations (BRPs) and their permutation statistics are presented. These permutations have been mainly addressed using numerical techniques in the literature to speed up in-place computations of fast Fourier and related transforms. Closed-form expressions for BRP statistics including inversions, serial correlations, and a new statistic called permutation inliers that characterizes the pruning gap of pruned interleavers, are derived. Using the inliers statistic, a recursive algorithm that computes the minimum number of inliers in a pruned BR interleaver (PBRI) in logarithmic time complexity is presented. This algorithm enables parallelizing a serial PBRI algorithm by any desired parallelism factor by computing the pruning gap in lookahead rather than a serial fashion, resulting in significant reduction in interleaving latency and memory overhead. Extensions to 2-D block and stream interleavers are also presented. Moreover, efficient hardware architectures for the proposed algorithms employing simple logic gates are Part of this work has been presented in [1]. M. M. Mansour ( ) ECE Department, American University of Beirut, Beirut, Lebanon e-mail: [email protected] presented. Simulation results of interleavers employed in modern communication standards demonstrate 3 to 4 orders of magnitude improvement in interleaving time compared to existing approaches.