Sequence MAP decoding of trellis codes for Gaussian and Rayleighchannels

This paper considers the use of sequence maximum a posteriori (MAP) decoding of trellis codes. A MAP receiver can exploit any “residual redundancy” that may exist in the channel encoded signal in the form of memory and/or a nonuniform distribution, thereby providing enhanced performance over very noisy channels, relative to maximum likelihood (ML) decoding. The paper begins with a first-order two-state Markov model for the channel encoder input. A variety of different systems with different source parameters, different modulation schemes, and different encoder complexities are simulated. Sequence MAP decoding is shown to substantially improve performance under very noisy channel conditions for systems with low-to-moderate redundancy, with relative gain increasing as the rate increases. As a result, coding schemes with multidimensional constellations are shown to have higher MAP gains than comparable schemes with two-dimensional (2-D) constellations. The second part of the paper considers trellis encoding of the code-excited linear predictive (CELP) speech coder's line spectral parameters (LSPs) with four-dimensional (4-D) QPSK modulation. Two source LSP models are used. One assumes only intraframe correlation of LSPs while the second one models both intraframe and interframe correlation. MAP decoding gains (over ML decoding) as much as 4 dB are achieved. Also, a comparison between the conventionally designed codes and an I-Q QPSK scheme shows that the I-Q scheme achieves better performance even though the first (sampler) LSP model is used     

Source-channel optimized trellis codes for bitonal imagetransmission over AWGN channels

We consider the design of trellis codes for transmission of binary images over additive white Gaussian noise (AWGN) channels. We first model the image as a binary asymmetric Markov source (BAMS) and then design source-channel optimized (SCO) trellis codes for the BAMS and AWGN channel. The SCO codes are shown to be superior to Ungerboeck's codes by approximately 1.1 dB (64-state code, 10^-5 bit error probability), We also show that a simple “mapping conversion” method can be used to improve the performance of Ungerboeck's codes by approximately 0.4 dB (also 64-state code and 10 ^-5 bit error probability). We compare the proposed SCO system with a traditional tandem system consisting of a Huffman code, a convolutional code, an interleaver, and an Ungerboeck trellis code. The SCO system significantly outperforms the tandem system. Finally, using a facsimile image, we compare the image quality of an SCO code, an Ungerboeck code, and the tandem code, The SCO code yields the best reconstructed image quality at 4-5 dB channel SNR     

MAP algorithms for decoding linear block codes based onsectionalized trellis diagrams

The maximum a posterioriprobability (MAP) algorithm is a trellis-based MAP decoding algorithm. It is the heart of turbo (or iterative) decoding that achieves an error performance near the Shannon limit. Unfortunately, the implementation of this algorithm requires large computation and storage. Furthermore, its forward and backward recursions result in a long decoding delay. For practical applications, this decoding algorithm must be simplifled and its decoding complexity and delay must be reduced. In this paper, the MAP algorithm and its variation's, such as log-MAP and max-log-MAP algorithms, are first applied to sectionalized trellises for linear block codes and carried out as two-stage decodings. Using the structural properties of properly sectionalized trellises, the decoding complexity and delay of the MAP algorithms can be reduced. Computation-wise optimum sectionalizations of a trellis for MAP algorithms are investigated. Also presented in this paper are bidirectional and parallel MAP decodings     

Computer design of super-orthogonal space-time trellis codes

Super-orthogonal space-time trellis codes (SOSTTCs) designed by hand can significantly improve the performance of space-time trellis codes. This paper introduces a new representation of SOSTTCs based on a generator matrix that allows a systematic and exhaustive search of all possible codes. This will verify that some of the known codes are optimal, and provides a means to easily implement encoders and decoders with a large number of states without relying on a graphical representation. New codes with up to 256 states that outperform previously known codes are presented     

Design of ring convolutional trellis codes for MAP decoding of MPEG-4 images

We propose a trellis-coded modulation system using continuous-phase frequency-shift keying (CPFSK) and ring convolutional codes for transmitting the bits generated by an embedded zerotree wavelet encoder. Improved performance is achieved by using maximum a posteriori decoding of the zerotree symbols, and ring convolutional trellis codes are determined for this decoding method. The CPFSK transmitter is decomposed into a memoryless modulator and a continuous phase encoder over the ring of integers modulo 4; the latter is combined with a polynomial convolutional encoder over the same ring. In the code design process, a search is made of the combined trellis, where the branch metrics are modified to include the source transition matrix. Simulation results of image transmission are provided using the optimized system, including mismatched channel cases.     

On the design and maximum-likelihood decoding of space-time trellis codes

In this letter, we present a simple generalization of the maximum ratio combining principle for space-time coded systems. This result leads to a maximum-likelihood decoder implementation that does not depend on the number of receive antennas and avoids the loss in performance incurred in the decoders proposed by Tarokh and Lo (1998) and Biglieri et al. The insights offered by this decoding rule allow for a simple and elegant proof for the space-time code design criterion in systems with large number of receive antennas. We further present an upper bound on probability of error that captures the dependence of space-time code design on the number of receive antennas. Finally, we present a computationally efficient approach for constructing space-time trellis codes that exhibit satisfactory performance in systems with variable number of receive antennas.     

Designing lexicographic codes with a given trellis complexity

We generalize constructions of lexicographic codes to produce locally optimal codes with a desired trellis decoding complexity. These constructions are efficient for high-rate codes and provide a means for automated code design. As a byproduct, we improve known bounds on the parameters of lexicodes     

8-PSK trellis codes for a Rayleigh channel

A suboptimal trellis coding approach based on the concept of combining a good convolutional code and bit interleavers is presented. The aim is to improve the reliability of digital radio communication over a fading channel. It is shown that over a Rayleigh channel and for a fixed code complexity the proposed system is superior to the baseline system. Its performance is analyzed using the generalized R_o and the upper bound on the bit error rate. The results suggest that on a Rayleigh channel, the standard trellis codes may not be the correct approach for improving the reliability of the communication channel. The discussion is restricted to a rate 2/3 coded system with 8-PSK modulation     

On trellis codes with a delay processor and a signal mapper

It is already known that a trellis code T, which is constructed by using the encoder of a convolutional code C with short constraint length followed by a delay processor and a signal mapper, is equivalent to a trellis code with large constraint length. In this paper, we derive a new lower bound on the free distance of T, which, in some cases, is better than the previously derived bound. Moreover, instead of the decoding used in earlier publications, we apply iterative decoding on both tailbiting and zero-tail representations of T to take advantage of the new lower bound and, in the meantime, to decrease the associated error coefficient caused by the decoding used in earlier publications. Comparisons among various designs of such a trellis code and some well-known coding methods are also provided.     

Concatenated coding with inner trellis codes for fading channels

It is known that concatenated coding can significantly enhance the performance of digital communication systems operating over fading channels when compared to uncoded or single-coded systems. The price to be paid for such an improvement, however, is a substantial increase in the required bandwidth. In this paper, we consider the use of a concatenation scheme in which the inner code is a trellis-coded modulation (TCM) system. The analysis is carried out for two different outer codes: a binary Golay code and a non-binary Reed-Solomon code. Results obtained for the Rayleigh and Rician fading channels through analytical bounds indicate that the use of this system does provide a significant reduction in the bit error probability, a fact that is also verified through computer simulation. Unlike a traditional concatenated system, the proposed method achieves the coding gain while maintaining acceptable bandwidth efficiency.     

The design of joint source and channel trellis waveform coders

The generalized Lloyd algorithm is applied to the design of joint source and channel trellis waveform coders to encode discrete-time continuous-amplitude stationary and ergodic sources operating over discrete memoryless noisy channels. Experimental results are provided for independent and autoregressive Gaussian sources, binary symmetric channels, and absolute error and squared error distortion measures. Performance of the joint codes is compared with the tandem combination of a trellis source code and a trellis channel code on the independent Gaussian source using the squared error distortion measure operating over an additive white Gaussian noise channel. It is observed that the jointly optimized codes achieve performance close to or better than that of separately optimized tandem codes of the same constraint length. Performance improvement via a predictive joint source and channel trellis code is demonstrated for the autoregressive Gaussian source using the squared error distortion measure.     

Performance of trellis codes for a class of equalized ISI channels

The performance of trellis codes is examined for a class of intersymbol interference (ISI) channels that occur in high-frequency radio systems. The channels considered are characterized by in-band spectral nulls and by a rapid time variation. The baseline modulation technique is 4QAM (four-point quadrature amplitude modulation). When spectral nulls are absent, performance of fractionally spaced linear equalizers and trellis decoders is found to be near ideal and to be better than using symbol-spacing in the equalizer. However, error propagation in the feedback path, resulting from equalizer-based decisions, ruins the performance of the combination of decision-feedback equalizers and trellis decoders when spectral nulls are present. Their performance can be improved by using fractionally spaced feedforward equalizer sections and by designing the decoder to compensate for ISI. Rate 2/3 codes are found to outperform rate 1/2 codes in error performance     

Improved design criteria and new trellis codes for space-time codedmodulation in slow flat fading channels

The design criteria for space-time codes under slow flat Rayleigh fading environments are well known to be the rank criterion and the determinant criterion. However, the derivation of these two criteria is based on the assumption that the signal-to-noise ratio (SNR) is sufficiently high. Hence these criteria are loose when the SNR does not satisfy this assumption. In this letter, we consider this practical issue and derive tighter design criteria for moderate and low SNRs. Some new space-time trellis codes are found based on the new criteria through computer search. Simulation results show that these codes outperform existing codes under their designed SNR conditions     

New simple encoder and trellis decoder for Golay codes

New simple encoding and trellis decoding techniques for Golay codes, based on generalised array codes (GAGs) are proposed. The techniques allow the design of both (23,12,7) Golay and (24,12,8) extended Golay codes with minimal trellises. It is shown that these trellises differ only in the last trellis depth with different labelling digits.<>     

Joint synchronization and demodulation of trellis codes

This paper examines the joint synchronization and detection of Ungerboeck coded modulation. Estimation theory is used to derive a synchronization structure that is efficient in estimating carrier phase and symbol timing. The maximum likelihood receiver generates estimates of carrier phase and timing that are free of data-dependent jitter for any continuous pulse shape. Various feedback schemes to be used with Ungerboeck codes are presented and simulated. Simulations show that efficient estimates of carrier and clock can be found when joint data and parameter estimation of an Ungerboeck coded signal is performed     

A comparison of trellis modules for binary convolutional codes

A convolutional code can be represented by the conventional trellis module or the “minimal” trellis module based on the BCJR trellis for block codes. For many convolutional codes, the trellis complexity (TC) of the minimal module is significantly less than the TC of the conventional module. An alternative representation, consisting of an extended BCJR trellis module and a pruned version of the conventional module, was previously introduced. We prove that the overall TC of the two new modules is less than the TC of the conventional module for infinitely many codes. Furthermore, we show that the overall TC of the new modules is smaller than the TC of the minimal module for many codes considered in the literature     

Multilevel construction of block and trellis group codes

The theory of group codes has been shown to be a useful starting point for the construction of good geometrically uniform codes. In this paper we study the problem of building multilevel group codes, i.e., codes obtained combining separate coding at different levels in such a way that the resulting code is a group code. A construction leading to multilevel group codes for semi-direct and direct products is illustrated. The codes that can be obtained in this way are identified. New geometrically uniform Euclidean-space codes obtained from multilevel codes over abelian and nonabelian groups are presented     

Joint design of fixed-rate source codes and multiresolution channelcodes

We propose three new design algorithms for jointly optimizing source and channel codes. Our optimality criterion is to minimize the average end-to-end distortion. For a given channel SNR and transmission rate, our joint source and channel code designs achieve an optimal allocation of bits between the source and channel coders. Our three techniques include a source-optimized channel code, a channel-optimized source code, and an iterative descent technique combining the design strategies of the other two codes. The joint designs use channel-optimized vector quantization (COVQ) for the source code and rate compatible punctured convolutional (RCPC) coding for the channel code. The optimal bit allocation reduces distortion by up to 6 dB over suboptimal allocations and by up to 4 dB relative to standard COVQ for the source data set considered. We find that all three code designs have roughly the same performance when their bit allocations are optimized. This result follows from the fact that at the optimal bit allocation the channel code removes most of the channel errors, in which case the three design techniques are roughly equivalent. We also compare the robustness of the three techniques to channel mismatch. We conclude the paper by relaxing the fixed transmission rate constraint and jointly optimizing the transmission rate, source code, and channel code     

Super-quasiorthogonal space-time trellis codes for four transmit antennas with rectangular signal constellations

In this paper, we present the first super-quasiorthogonal space-time trellis codes (SQOSTTCs) for systems with four transmit antennas using various types of rectangular signal constellations to increase the spectral efficiency up to 5 bits/s/Hz. In our wireless communications system, we define an eight-dimensional (8D) signal constellation as Cartesian product of four two-dimensional (2D) rectangular signal sets. The transmission of an 8D point from the first antenna is achieved by transmitting four concatenated 2D points in four consecutive channel uses. The 2D symbols transmitted from the other three antennas are not independent but so chosen as to form, together with the symbols transmitted from the first antenna, the entries of a 4×4 quasiorthogonal transmission matrix. The union of two sets of quasiorthogonal transmission matrices forms a so-called super-quasiorthogonal signal set. With the 4×4 quasiorthogonal transmission matrices, we then label the state transitions of a trellis diagram describing the operation of the encoder. The simulation results of the frame error rate and the bit error rate demonstrate the excellent performance of our proposed SQOSTTCs.