Digital design and optimization of ultimate-shannon-limit-approaching channel codes

Abstract

This thesis focuses on digital communication systems in which the received signal­to-noise ratio is extremely low. Application examples include space communications and multiple-user environments using code division multiple access and interleaver division multiple access. Hence, only channel codes with performance very close to the ultimate Shannon limit, i.e., bit-energy-to-noise-power-spectral-density ratio Eb/N0 = -1.59 dB, are considered. In this thesis, we propose digital system designs of two ultimate-Shannon-limit-approaching codes, namely turbo Hadamard codes and concatenated zigzag Hadamard codes. Moreover, we propose ways to design punctured turbo Hadamard codes and to lower the error floor of turbo Hadamard codes. We also figure that efforts are needed to evaluate cycles in the design of a turbo Hadamard code. To estimate the computation effort required, we generalize the problem and propose a new method to evaluate the number of closed paths in an all-one base matrix. Firstly, we propose a pipelined digital design of a turbo Hadamard encoder/decoder system. To accomplish a high throughput, we make use of a multiple sub-decoder architecture to process multiple codes at the same time. Each sub-decoder is processing the data of one codeword at anytime; and data from the same codeword will be processed by different sub-decoders at different times. One iteration is completed when the data from the same codeword is processed by all the sub-decoders once. Moreover, tens of iterations are required to complete the decoding. Hence, the design challenges include control of data flow within a sub-decoder; control of data flow among sub-decoders; proper data storage to avoid data access conflicts; conversion of data formats to facilitate computations; effective interleavers that cause minimum latency. In order to achieve performance close to the ultimate Shannon limit, code lengths of 216450, 287235 and 358020 are used. Since the code lengths are relatively long, effective use of data storage is crucial. Also, the transmitter is required to send code bits in a way that facilitates storing and processing of data at the receiving end. Note that new codeword data are being received continuously and need to be stored as the decoder is processing the existing codewords. The theoretical throughput of our digital design is derived based on the given parameters such as hardware operating frequency, code length, number of iterations, and length of the turbo trellis. Results indicate that at an operating frequency of 100 MHz, the proposed digital turbo Hadamard encoder/decoder system achieves throughputs of 1.92 Gbps, 2.56 Gbps and 3.2 Gbps at code lengths of 216450, 287235 and 358020, respectively. The encoder/decoder system also realizes a bit error rate of 10-5 at Eb/N0 = -0.45 dB, i.e., 1.14 dB from the ultimate Shannon limit. Secondly we realize that in the design of the turbo Hadamard encoder/decoder system, relatively complex Bahl-Cocke-Jelinek-Raviv (BCJR) decoding is required and it limits the operating frequency to 100 MHz. Thus we propose a pipelined digital design of a concatenated zigzag Hadamard encoder/decoder system, in which BCJR decoding is not needed. Again we propose a multiple sub-decoder architecture to process multiple codes at the same time. However, different from the sub-decoders in the turbo Hadamard decoder system, each sub-decoder in the concatenated zigzag Hadamard decoder system processes a set of multiple codes at the same time; and the same set of multiple codes is processed by different sub-decoders at different times. Such an arrangement aims to improve the throughput and also the hardware utilization rate. We overcome challenges similar to those occurring in the design of turbo Hadamard encoder/decoder systems. The final concatenated zigzag Hadamard encoder/decoder system is found to work with 50% increase in operating frequency compared with the turbo Hadamard encoder/decoder system and with similar error performance. The drawbacks of the concatenated zigzag Hadamard encoder/decoder system, however, are higher decoding latency and higher memory requirement. Thirdly, we propose ways to optimize the turbo Hadamard codes. Punctured Hadamard codes and punctured turbo Hadamard encoder/decoder systems are first investigated. By puncturing some of the code bits and not sending those bits through the channel, the rate of a code is improved and sometimes the bit error rate performance can be improved too. Here, two methods to select the punctured Hadamard code bits, or equivalently the puncturing patterns, are proposed. The first scheme aims to maximize the minimum Hamming distance of the punctured Hadamard codes. The upper-bound of the minimum Hamming distance of punctured Hadamard codes is derived and then an algorithm is proposed to find punctured Hadamard codes achieving close to this bound. The second scheme aims to minimize the cross-correlations among the punctured Hadamard codes. For punctured code sets having the same minimum cross-correlation, a new metric has been proposed to identify sets that can further enhance the reliability of decoding. For Hadamard codes, the two proposed puncturing schemes have shown error improvements over the use of random puncturing. Moreover, the scheme that minimizing the cross-correlations outperform that maximizing the minimum Hamming distance. By applying the more superior scheme to puncture Hadamard codes in a turbo Hadamard encoder/decoder system, the code rate is improved with little change in error performance. Another way to optimize the turbo Hadamard codes is to lower its error floor. At the high Eb/N0 region, we observe that the error rate of the turbo Hadamard code may become flat. To tackle this issue, we investigate the overall turbo Hadamard code structure by looking into an associated parity-check matrix. By re-designing the interleavers and hence removing short cycles in the parity-check matrix, we observe that the error floor can be lowered. Finally, while studying the cycles of the turbo Hadamard code structure, we come up with a new method to evaluate the number of cycles for a given parity-check matrix. We further generalize the method and use it to evaluate the number of closed paths in an all-one base matrix. Theoretical results up to closed paths of length 10 have been derived and are verified by the exhaustive search method. Based on the theoretical work, results for closed paths of length larger than 10 can be further derived. The results are particularly useful for estimating computational resources required in designing parity-check matrices of low-density parity-check codes.