The performance and complexity of turbo decoding using rate k/n constituent codes are investigated. The conventional, minimal and sectionalized trellis modules of the constituent convolutional codes are utilized. The performance metric is the bit error rate (BER), while complexity is analyzed based on the number of multiplications, summations and comparisons required by the max-log-MAP decoding algorithm. Our results show that the performance depends on how the systematic bits are grouped in a trellis module. The best performance is achieved when the k systematic bits are grouped together in the same section of the module, so that the log-likelihood ratio (LLR) of the k-bit vector is calculated at once. This is a characteristic of the conventional trellis module and of some of the sectionalizations of the minimal trellis module. Moreover, we show that it is possible to considerably reduce the decoding complexity with respect to the conventional trellis if a particular sectionalization of the minimal trellis module is utilized. In some cases, this sectionalization is found within the best performing group, while in some other cases a small performance loss can be traded off for a large complexity reduction.

We consider high-rate systematic recursive convolutional encoders to be adopted as constituent encoders in turbo schemes. It has been shown by Douillard and Berrou that the construction of high-rate turbo codes by means of high-rate constituent encoders offers several advantages over the typical construction based on the puncturing of rate-1/2 constituent encoders. To reduce the decoding complexity associated with high-rate codes, we adopt the “minimal” trellis representation of convolutional codes introduced by McEliece and Lin. While in the literature this trellis has been obtained for nonrecursive nonsystematic generator matrices, we herein introduce the construction of the “minimal” trellis for a systematic recursive convolutional encoding matrix. We also derive expressions for the arithmetic decoding complexity when the max-log-MAP algorithm is applied over the conventional and the “minimal” trellises. Examples are provided, which show that significant savings in decoding complexity are obtained, while keeping the same error performance of conventional schemes, when the minimal trellis is used. Finally, a code search is conducted and examples are provided which indicate that a refinement in terms of decoding complexity-error performance trade-off is obtained.