On the equivalence of TCM encoders

Abstract

Optimal trellis-coded modulation (TCM) schemes are obtained by jointly designing the convolutional encoder and the binary labeling of the constellation. Unfortunately this approach is infeasible for large encoder memories or constellation sizes. Traditional TCM designs circumvent this problem by using a labeling that follows the set-partitioning principle and by performing an exhaustive search over the encoders. Therefore, traditional TCM schemes are not necessarily optimal. In this paper, we study binary labelings for TCM and show how they can be grouped into classes, which considerably reduces the search space in a joint design. For the particular case of 8-ary modulation the search space for the labelings is reduced from 8! to 240. Using this classification, we formally prove that for any channel it is always possible to design a TCM system based on the binary-reflected Gray code with identical performance to the one proposed by Ungerboeck in 1982. Moreover, the classification is used to tabulate asymptotically optimal TCM schemes.

Polar-Coded Modulation

“…It has been shown [23] that for one-dimensional constellations, natural labeling and binary reflected Gray labeling can be transformed into each other by a (bijective) linear transform.…”

“…As shown in [23], a set-partitioning labeling of an M -ASK/PSK constellation can be transformed into a binary reflected Gray labeling via an (m, m) binary matrix…”

“…We point out benefits and similarities from an information-theoretic conceptual point of view. Building on the relation of Gray and natural labeling, recently shown in [5], we proof the equivalence of polar-coded BICM and MLC for 4-QAM under the constraint of equal structural delay.…”

Method of blind reconstruction for trellis‐coded modulation codes

“…The equivalence of TCM encoders was introduced in [15]. Two TCM encoders are equivalent if they give the same output symbol sequence for the same information bit sequence.…”

“…8a is the trellis of the four mapping modes (1)(2)(3)(4). The information weight polynomial of the free distance is (14) Then differentiating g(Y ) against Y gives g ′ (Y ) = 2Y + 6Y 2 + 4Y 3 (15) Letting Y = 1, the Hamming distance is H(d free ) = 12. Fig.…”