Trellis Coding with Asymmetric Modulations

Abstract

Traditionally symmetric signal constellations, i.e., those with uniformly spaced signal points, have been used for both uncoded and coded systems. Although symmetric signal constellations are optimum with no coding, the same is not necessarily true for coded systems. This paper shows that by designing the signal constellations to be asymmetric, one can, in many instances, obtain a performance gain over the traditional symmetric constellations combined With trellis coding. In particular, we consider the joint design of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n/(n + 1)</tex> trellis codes and asymmetric <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^{n+1}</tex> -point signal constellations, which has no bandwidth expansion relative to an uncoded 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> -point symmetric signal set. The asymptotic performance gains due to coding and asymmetry are evaluated in terms of the minimum free Euclidean distance d <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">free</inf> of the trellis. A comparison of the maximum value of this performance measure to the minimum distance d <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</inf> , of the uncoded system is an indication of the maxiamm reduction in required <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E_{b}/N_{0}</tex> that can be achieved for arbitrarily small system bit error rates. Bit error probability analysis is carried out for general cases. A few examples are given to show the performance gain due to the asymmetry of the signal set. It is to be emphasized that the introduction of asymmetry into the signal set does not affect the bandwidth or power requirements of the system; hence, the abovementioned improvements in performance come at little or no cost. Asymmetric signal sets in coded systems first appear in the work of Divsalar and Yuen [1], [2]. Here we expand upon these results by considering various types of asymmetric signal sets combined with the optimum (in the sense of maximum d <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">free</inf> ) trellis code having 2, 4, 8, and 16 states. The numerical results obtained will clearly demonstrate the tradeoff between the additional savings in required <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">E_{b}/N_{0}</tex> and the additional complexity (more trellis states) needed to achieve it.