Analysis and Design of Multiple Turbo Codes

A J-dimensional multiple turbo code consists of the parallel concatenation of J (J>2) convolutional encoders connected by J-1 permutors. This dissertation is devoted to the ananlysis and design of multiple turbo codes. The convergence behavior of multiple turbo codes, which is the most important factor that determines the bit error rate (BER) and frame error rate (FER) performance in the low signal-to-noise ratio (SNR) region, is studied using two approaches. First, an analytical approach to determine the transfer characteristics of 2-state convolutional codes is derived. The extrinsic information transfer (EXIT) chart method is then generalized to multiple turbo codes and used to search for low complexity multiple turbo codes with low convergence thresholds. The performance of a multiple turbo code in the high SNR region is largely affected by its minimum distance. To simplify the minimum distance analysis, the concept of summary distance is extended to J-dimensional (a set of J-1) permutors and used as a general design metric. A sphere packing upper bound on the minimum length-2 summary distance (spread) of J-dimensional permutors is derived. We derive a lower bound on the minimum summary distance in the ensemble of random permutors. Furthermore, we show that this lower bound can be improved by expurgating the 'bad' symbols that are associated with small summary distances. Two joint permutor construction algorithms are then presented, one for random permutors and another for linear permutors. The success probability of the random construction algorithm is estimated and a modified version of the algorithm is presented that takes the constituent encoder period into account. A constructive proof is given showing the existence of J-dimensional linear permutors with optimal spread. It is also shown that the minimum length-4 summary distance of a single linear permutor is greatly improved in the case of multiple linear permutors. A computer search for short linear permutors that result in large minimum distances for multiple turbo codes is performed. Compared to conventional turbo codes using the best DRP permutors, significant improvement has been achieved by using multiple turbo codes with linear permutors.