[Paper Review] Punctured Trellis-Coded Modulation

Fabian Schuh, Andreas Schenk+1 more|arXiv (Cornell University)|Jan 17, 2013

Advanced Wireless Communication Techniques8 references4 citations

TL;DR

This paper proposes punctured trellis-coded modulation (P-TCM) as a flexible method to achieve arbitrary spectral efficiencies by puncturing the convolutional code of a TCM system, rather than using multi-dimensional constellations. It introduces a modified Viterbi algorithm for ML decoding on time-variant trellises and demonstrates through simulations that P-TCM enables smooth transitions between classical TCM rates, closely approximating the constellation capacity limit with improved spectral efficiency and power efficiency trade-offs.

ABSTRACT

In classic trellis-coded modulation (TCM) signal constellations of twice the cardinality are applied when compared to an uncoded transmission enabling transmission of one bit of redundancy per PAM-symbol, i.e., rates of $\frac{K}{K+1}$ when $2^{K+1}$ denotes the cardinality of the signal constellation. In order to support different rates, multi-dimensional (i.e., $\mathcal{D}$-dimensional) constellations had been proposed by means of combining subsequent one- or two-dimensional modulation steps, resulting in TCM-schemes with $\frac{1}{\mathcal{D}}$ bit redundancy per real dimension. In contrast, in this paper we propose to perform rate adjustment for TCM by means of puncturing the convolutional code (CC) on which a TCM-scheme is based on. It is shown, that due to the nontrivial mapping of the output symbols of the CC to signal points in the case of puncturing, a modification of the corresponding Viterbi-decoder algorithm and an optimization of the CC and the puncturing scheme are necessary.

Motivation & Objective

• To enable flexible transmission rates in TCM beyond integer rates by puncturing the convolutional code instead of relying on multi-dimensional constellations.
• To address the challenge of non-uniform metric computation and time-variant trellis structure introduced by puncturing in TCM.
• To develop a modified Viterbi algorithm that supports maximum-likelihood decoding for punctured TCM systems.
• To optimize the convolutional code, puncturing pattern, and labeling for ASK and QAM modulation under practical constraints.
• To demonstrate that punctured TCM can closely approximate the spectral efficiency of the constellation capacity limit.

Proposed method

• Puncturing is applied to the output of a rate-1/2 mother convolutional code, with puncturing patterns designed to achieve desired code rates.
• A modified Viterbi algorithm is proposed to handle time-variant trellises caused by cyclic puncturing, ensuring maximum-likelihood decoding.
• The system uses a combination of coded bits (from punctured CC) and uncoded bits to label signal points in a larger constellation, following the set partitioning principle.
• The trellis structure becomes time-variant due to cyclic puncturing, with varying numbers of branches per state depending on the puncturing pattern.
• The method supports both PAM and QAM by mapping two consecutive ASK symbols into a QAM symbol, increasing spectral efficiency without bandwidth expansion.
• An exhaustive computer search is performed over constraint length 5 convolutional codes, puncturing patterns, and labeling schemes to find optimal configurations.

Experimental results

Research questions

• RQ1 Can puncturing the convolutional code in TCM provide a flexible and efficient alternative to multi-dimensional TCM for achieving non-integer transmission rates?
• RQ2 How can the Viterbi algorithm be modified to support maximum-likelihood decoding on time-variant trellises arising from puncturing in TCM?
• RQ3 What is the optimal combination of convolutional code, puncturing pattern, and labeling for achieving near-capacity spectral efficiency in ASK and QAM systems?
• RQ4 To what extent can punctured TCM approximate the theoretical constellation capacity limit in terms of spectral and power efficiency?
• RQ5 Can the proposed P-TCM scheme support smooth transitions between standard TCM rates (e.g., 1, 2, 3 bits/symbol) while maintaining low error rates?

Key findings

• The proposed P-TCM scheme achieves spectral efficiencies of up to 8/5 = 1.6 bits/symbol and 12/7 ≈ 1.71 bits/symbol, with corresponding coding gains of 4.5 dB and 5.5 dB at BER = 10⁻³.
• For a 16-state code with rate 4/3, the system achieves a spectral efficiency of 1.33 bits/symbol with a coding gain of approximately 5.5 dB at BER = 10⁻³.
• The simulation results show that P-TCM closely approximates the constellation capacity limit, especially when combined with QAM modulation using two ASK symbols per QAM symbol.
• The modified Viterbi algorithm successfully enables maximum-likelihood decoding on time-variant trellises, outperforming simpler metric-based approaches in punctured systems.
• The method allows for soft transitions between classical TCM rates (e.g., 1, 2, 3 bits/symbol), enabling a continuous trade-off between spectral and power efficiency.
• The best-performing code for 8/5 rate is the constraint length 5 convolutional code with generator polynomials (34,31)₈ and a puncturing pattern with period Ω=6.