Block Coded Modulation Using Two-Level Group Codes Over Dihedral and Dicyclic Groups

Jyoti Bali

Department of Electrical Engineering,
Indian Institute of Technology, Delhi
Hauz Khas, New Delhi 110 016

October 22, 1998
Advisor: Dr. B.Sundar Rajan
Indian Institute of Science, Bangalore, India

Keywords: Coded Modulation, Group Codes, Multilevel codes.

Abstract: Two-Level block coded phase modulation schemes with a linear binary code C$_{1}$ and a linear code C$_{2}$ over Z$_{M}$, (the residue class integer ring modulo M) as component codes and a PSK signal (symmetric and asymmetric) set matched to a dihedral group D$_{M}$ or a 4-dimensional signal set, called (PSK)$^2$ signal set, matched to a dicyclic group DC$_{M}$ as basic signal sets have been studied. A pair of codewords each from C$_{1}$ and C$_{2}$ specify an n-tuple over D$_{M}$ (or DC$_{M}$) is called a label code which, in general, is a subset, not necessarily a subgroup of D$_{M}^{n}$ (or DC$_{M}^{n}$). Extending the matched labelling component wise on all the elements of the label code results in a signal set in 2n dimensions (or in 4n dimensions) called the Two-Level signal space code where n is the length of the codes C$_{1}$ and C$_{2}$.

A set of necessary and sufficient conditions on the component codes for the label code to result in a group code over D$_{M}^{n}$ and DC$_{M}^{n}$ are proved. For D$_{M}^{n}$, the algebraic conditions obtained by Garello and Benedetto for semidirect product group code are shown to be equivalent to part of the conditions obtained.

The performance of the signal space codes depend both on set of component codes and the matched labellings used. Given a component code pair different matched labellings will give different performance. We call the problem of identifying the best labelling the Initial Labelling Problem and provide a solution for a rather restrictive class of component codes for codes over D$_{M}$ and show that Initial Labelling Problem does not arise for codes over DC$_{M}$, i.e., the performance of the signal space code depends only on the set of component codes and not on the matched labelling used. For codes over D$_{M}$, through a series of theorems it is shown that based on the ratio of the Hamming distances of the component codes several Euclidean distance properties can be obtained for both symmetric and asymmetric PSK signal sets. Based on these results superiority of certain labellings over others is established. Moreover, conditions under which introduction of asymmetry to the symmetric PSK signal set will improve performance of the resulting signal space code are discussed and it is shown that when the conditions are satisfied upto certain angle performance improvement is guaranteed. These results are discussed in detail for the special cases of 4,6,8,12 and 16-PSK signal sets and several classes of codes are identified and their coding gains tabulated.

It turns out that the conditions on the component codes to result in a group code over DC$_{M}$ include the conditions for the same component codes to result in a group code over D$_{M}$. This means that such pairs of component codes can be used to label both the signal sets matched to D$_{M}$ as well as DC$_{M}$ and obtain different signal space codes. For such component codes we identify conditions, specifically rates of the code, under which one will perform better than the other.

Rotational invariance properties of the label codes, for symmetric and asymmetric PSK signal sets as well as (PSK)$^2$ signal sets are discussed and conditions for several angles of phase rotational invariance are proved. Based on these a two-stage differential encoding and decoding scheme is proposed. It is pointed out that the codes investigated in this thesis admit minimal trellises and hence are amenable to efficient soft decision decoding.

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Jyoti Bali
IBM Solutions Research Centre
Block I, Indian Institute of Technology, Delhi
Hauz Khas, New Delhi 110 016, INDIA
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