On the equivalence of rate R = k non-systematic feed-forwardconvolutional codes and recursive systematic convolutional codes

摘要

This paper presents a new algorithm toconvert non-systematic feed-forward convolutionalcodes (NSCC) of rate R = k to recursive systematiccodes (RSC), while preserving code equivalence i.e.the code distance spectrum. Traditional conversionalgorithms are limited to codes of rate R = k/(k+1),i.e. to n = k+1 when using encoder implementationsin observer canonical form (OCF), or are restrictedto codes of rate R = 1, i.e. to k = 1 when encodersin controller canonical from (CCF) are used. Theproposed algorithm is suitable for any codes of rateR = k, including also n > k+1. Here, after codeconversion RSCs with optimum distance spectrum(ODS) are available in OCF. However, theseencoders require a more sophisticated terminationthan conventional methods. To provide an appropriatetermination, the most general notation forlinear convolutional encoders in terms of a linearequation system is introduced. This equation systemis described via the ABCD-matrix notation andleads to a compact termination method, which isapplicable to linear convolutional codes in general.Finally, the turbo code as specified in 3GPP [1] ischosen as a reference to demonstrate the performanceof ODS component encoders in comparisonwith traditional component encoders. Two performancemeasures have been applied: 1st the effectivefree distance of the overall turbo code depending onthe interleaver length and 2nd bit-error-figures vs.Eb/No. This unveils the potential of ODS codes tooffer superior performance compared to conventionalschemes like in 3GPP and IEEE 802.16ab [2].