A hybrid M-algorithm/sequential decoder for convolutional and trellis codes

The Viterbi Algorithm (VA) is optimum in the sense of being maximum likelihood for decoding codes with a trellis structure. However, since the VA is in fact an exhaustive search of the code trellis, the complexity of the VA grows exponentially with the constraint length upsilon. This limits its application to codes with small values of upsilon and relatively modest coding gains. The M-Algorithm (MA) is a limited search scheme which carries forward M paths in the trellis, all of the same length. All successors of the M paths are extended at the next trellis depth, and all but the best M of these are dropped. Since a limited search convolutional decoder will flounder indefinitely if one of the paths in storage is not the correct one, the data are usually transmitted in blocks. It has been shown that the performance of the MA approaches the VA at high signal to noise ratios (SNR's) with an M which is far less than the 2 sup upsilon states in the full trellis. Thus the MA can be used with larger values of upsilon, making larger coding gains possible at high SNR's. However, it still requires a relatively large fixed computational effort to achieve good performance

A bandwidth efficient coding scheme for the Hubble Space Telescope

As a demonstration of the performance capabilities of trellis codes using multidimensional signal sets, a Viterbi decoder was designed. The choice of code was based on two factors. The first factor was its application as a possible replacement for the coding scheme currently used on the Hubble Space Telescope (HST). The HST at present uses the rate 1/3 nu = 6 (with 2 (exp nu) = 64 states) convolutional code with Binary Phase Shift Keying (BPSK) modulation. With the modulator restricted to a 3 Msym/s, this implies a data rate of only 1 Mbit/s, since the bandwidth efficiency K = 1/3 bit/sym. This is a very bandwidth inefficient scheme, although the system has the advantage of simplicity and large coding gain. The basic requirement from NASA was for a scheme that has as large a K as possible. Since a satellite channel was being used, 8PSK modulation was selected. This allows a K of between 2 and 3 bit/sym. The next influencing factor was INTELSAT's intention of transmitting the SONET 155.52 Mbit/s standard data rate over the 72 MHz transponders on its satellites. This requires a bandwidth efficiency of around 2.5 bit/sym. A Reed-Solomon block code is used as an outer code to give very low bit error rates (BER). A 16 state rate 5/6, 2.5 bit/sym, 4D-8PSK trellis code was selected. This code has reasonable complexity and has a coding gain of 4.8 dB compared to uncoded 8PSK (2). This trellis code also has the advantage that it is 45 deg rotationally invariant. This means that the decoder needs only to synchronize to one of the two naturally mapped 8PSK signals in the signal set

Error control techniques for satellite and space communications

The performance of bandwidth efficient trellis inner codes using two-dimensional MPSK signal constellations in a NASA concatenated coding is summarized. Work was also continued on trellis coded modulation using multi-dimensional signal sets. Achievable lower bounds on free distance trellis codes were proved and the existence of good trellis coded modulation (TCM) schemes were established for a variety of signal constellations. The performance of TCM schemes on fading channels is being investigated. Preliminary results indicate that bandwidth efficient trellis coding is feasible on such channels, but that the important design parameter is no longer the minimum free Euclidean distance

Error Control Techniques for Satellite and Space Communications

In this report, we present the results of our recent work on turbo coding in two formats. Appendix A includes the overheads of a talk that has been given at four different locations over the last eight months. This presentation has received much favorable comment from the research community and has resulted in the full-length paper included as Appendix B, 'A Distance Spectrum Interpretation of Turbo Codes'. Turbo codes use a parallel concatenation of rate 1/2 convolutional encoders combined with iterative maximum a posteriori probability (MAP) decoding to achieve a bit error rate (BER) of 10(exp -5) at a signal-to-noise ratio (SNR) of only 0.7 dB. The channel capacity for a rate 1/2 code with binary phase shift-keyed modulation on the AWGN (additive white Gaussian noise) channel is 0 dB, and thus the Turbo coding scheme comes within 0.7 DB of capacity at a BER of 10(exp -5)

Error control techniques for satellite and space communications

This report focuses on the results obtained during the PI's recent sabbatical leave at the Swiss Federal Institute of Technology (ETH) in Zurich, Switzerland, from January 1, 1995 through June 30, 1995. Two projects investigated various properties of TURBO codes, a new form of concatenated coding that achieves near channel capacity performance at moderate bit error rates. The performance of TURBO codes is explained in terms of the code's distance spectrum. These results explain both the near capacity performance of the TURBO codes and the observed 'error floor' for moderate and high signal-to-noise ratios (SNR's). A semester project, entitled 'The Realization of the Turbo-Coding System,' involved a thorough simulation study of the performance of TURBO codes and verified the results claimed by previous authors. A copy of the final report for this project is included as Appendix A. A diploma project, entitled 'On the Free Distance of Turbo Codes and Related Product Codes,' includes an analysis of TURBO codes and an explanation for their remarkable performance. A copy of the final report for this project is included as Appendix B

Error control techniques for satellite and space communications

The results included in the Ph.D. dissertation of Dr. Fu Quan Wang, who was supported by the grant as a Research Assistant from January 1989 through December 1992 are discussed. The sections contain a brief summary of the important aspects of this dissertation, which include: (1) erasurefree sequential decoding of trellis codes; (2) probabilistic construction of trellis codes; (3) construction of robustly good trellis codes; and (4) the separability of shaping and coding

Error control techniques for satellite and space communications

Worked performed during the reporting period is summarized. Construction of robustly good trellis codes for use with sequential decoding was developed. The robustly good trellis codes provide a much better trade off between free distance and distance profile. The unequal error protection capabilities of convolutional codes was studied. The problem of finding good large constraint length, low rate convolutional codes for deep space applications is investigated. A formula for computing the free distance of 1/n convolutional codes was discovered. Double memory (DM) codes, codes with two memory units per unit bit position, were studied; a search for optimal DM codes is being conducted. An algorithm for constructing convolutional codes from a given quasi-cyclic code was developed. Papers based on the above work are included in the appendix