This article is focused on the performance evaluation of linear codes under optimal maximum-likelihood (ML) decoding. Though the ML decoding algorithm is prohibitively complex for most practical codes, their performance analysis under ML decoding allows to predict their performance without resorting to computer simulations. It also provides a benchmark for testing the sub-optimality of iterative (or other practical) decoding algorithms. This analysis also establishes the goodness of linear codes (or ensembles), determined by the gap between their achievable rates under optimal ML decoding and information theoretical limits. In this article, upper and lower bounds on the error probability of linear codes under ML decoding are surveyed and applied to codes and ensembles of codes on graphs. For upper bounds, we discuss various bounds where focus is put on Gallager bounding techniques and their relation to a variety of other reported bounds. Within the class of lower bounds, we address de Caen's based bounds and their improvements, and also consider sphere-packing bounds with their recent improvements targeting codes of moderate block lengths.

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Performance Analysis of Linear Codes Under Maximum-Likelihood Decoding: A Tutorial

Serially concatenated and interleaved continuous phase modulation (CPM) with iterative decoding is investigated. An a posteriori probability (APP) algorithm for CPM is developed based on the classic APP algorithm for channel codes. The system is analyzed through upper bounds on the average bit error probability. For coded and interleaved minimum shift keying, the weight spectrum is computed, resulting in a transfer function bound. This is cumbersome for a general CPM system; instead, only the most significant error events contributing to the weight spectrum are identified. Simulations show that, firstly, these events give a satisfactory view of system performance when equal outer codes are used, and secondly, that remarkably good performance can be obtained for some simple systems. Finally, power spectral densities and bandwidths are computed, allowing for a bandwidth/performanee comparison of different combinations.

Serially concatenated continuous phase modulation with iterative decoding

We develop a reduced-complexity approach for the detection of coded shaped-offset quadrature phase-shift keying (SOQPSK), a highly bandwidth-efficient and popular constant-envelope modulation. The complexity savings result from viewing the signal as a continuous-phase modulation (CPM). We give a simple and convenient closed-form expression for a recursive binary-to-ternary precoder for SOQPSK. The recursive nature of this formulation is necessary in serially concatenated systems where SOQPSK serves as the inner code. We show that the proposed detectors are optimal in the full-response case, and are near-optimal in the partial-response case due to some additional complexity reducing approximations. In all cases, the proposed detectors achieve large coding gains for serially concatenated coded SOQPSK. These gains are similar to those reported recently by Li and Simon, which were obtained using a more complicated cross-correlated trellis-coded quadrature modulation (XTCQM) interpretation.

Reduced-Complexity Approach to Iterative Detection of Coded SOQPSK

In this paper, serially concatenated continuous phase modulation (SCCPM) is considered for the uplink of satellite communications. A three-step design procedure is proposed to optimize the association of the outer code and the CPM for a wide range of spectral efficiencies, ranging from 0.75 to 2.25 bit/s/Hz. Firstly, EXIT chart analysis is applied to derive general guidelines for choosing SCCPM parameters. A significant result is that a high-rate outer code is required to achieve good convergence threshold, given a spectral efficiency. At a second stage, union bounds to the error probability are considered to choose the outer code under the constraints arising from the EXIT charts analysis. From this analysis, extended BCH codes and extended Hamming codes are proposed as outer code for broadband and narrowband transmission, respectively. For the latter, double-binary convolutional codes and symbol interleaving is also proposed as a valid alternative. Finally, combining both EXIT charts and union bounds we optimize the association of code and CPM to achieve both low error rates and good convergence. The proposed concatenated structure offers very low error floors and good performance in the waterfall region for all considered spectral efficiencies. A significant improvement with respect to previous concatenated CPM schemes is shown.

Serially concatenated continuous phase modulation for satellite communications

The performance of a serially concatenated system which includes a channel with memory preceded by a precoder as a rate-1 inner coder is presented. The effect of different precoders on the maximum-likelihood bit-error performance is analyzed. The precoder weight gain, which explains the good bit-error rate (BER) performance, is identified through a union bound analysis. Precoders are divided into two groups based on an analysis of the Euclidean distance and its multiplicity, and each precoder group shows a distinct BER curve behavior. It is shown that the BER curves for two precoder groups cross over each other. Convolutional codes are considered as outer codes in simulations on various intersymbol interference channels. Several important design considerations for the choice of precoders are derived based on the analysis and these are confirmed through simulations with an iterative decoding algorithm.

The effect of a precoder on serially concatenated coding systems with an ISI channel

Iterative methods for concatenated coding and modulation in digital communication systems are considered. It is assumed that the code and modulation can be described by finite-state machines (FSM). An iterative decoder for such a system typically consists of a posteriori probability (APP) algorithms for the constituent FSMs. Starting with a detailed examination of these algorithms, it is found that their initialization values can be formally justified. Then, possible iterative methods such as fix-point iteration, Jacobi over-relaxation, damped substitution, and Newton’s method are presented and evaluated. The result is that fix-point iteration seems to be the best choice in most situations. As an application of iterative decoding in serially concatenated systems, coded and interleaved continuous phase modulation (CPM) is investigated. An APP algorithm (or maximum APP (MAP) detector) for CPM is derived, based on the classic APP algorithm for channel codes. It is then shown that a good suboptimal algorithm can be obtained using only a small set of receiver filters, which are optimal with respect to the size of the set. The resulting system is analyzed with the aim to determine the main parameters affecting performance. The weight spectrum for coded and interleaved minimum shift keying (MSK) is computed, resulting in a transfer function bound. This is cumbersome for a general CPM system; instead only the most significant error events contributing to the weight spectrum are identified, resulting in a new performance measure. Extensive computer simulations show good coherence with theory, and also remarkably good performance for some simple systems. Finally, the power spectral density and bandwidth are computed, allowing for a bandwidth/performance comparison of different combinations.

Serially concatenated systems : an iterative coding approach with application to continuous phase modulation

Serially concatenated continuous phase modulation (SCCPM), or coded and interleaved CPM, is examined. Like turbo codes, SCCPM allows for iterative decoding based on a posteriori probability algorithms. The system is analyzed through an upper bound on the average bit error probability with maximum likelihood sequence detection, showing that SCCPM is capable of providing interleaver gain - and thus turbo-like performance - if a proper mapping of the CPM symbols is used. Simulations indicate that firstly, the analysis gives a satisfactory view of system performance when equal outer codes are used, and secondly, that remarkably good performance can be obtained with some simple systems. For unequal outer codes, the convergence of the iterative decoder will also have a significant effect on performance. Finally, a bandwidth/performance comparison of some selected systems shows that SCCPM can achieve large gains compared to the best known coded CPM systems without interleaving.


Keywords:

concatenated codes;
serial concatenation;
continuous phase modulation;
iterative decoding;
APP algorithm

Pär Moqvist

Papers

Serially concatenated continuous phase modulation with iterative decoding

Serially concatenated systems : an iterative coding approach with application to continuous phase modulation

Serially Concatenated Continuous-Phase Modulation with Iterative Decoding