Multiple trellis coded frequency and phase modulation

Multiple trellis coded modulation of constant envelope frequency and phase modulated signal sets (MTCM/FPM) is investigated for performance on the additive white Gaussian noise (AWGN) channel and on the one-sided normal, Rayleigh and Rician fading channels. The Nakagami- m fading model is used as an alternative to the Rician fading model to calculate the error probability upper bound for trellis-coded schemes on the fading channel. The likeliness and the disparity between the upper bounds to the error probability for the two fading models are discussed. The design criteria for the one-sided normal fading channel, modeled by the Nakagami-m distribution, are observed to be the same as those for the Rayleigh-fading channel. For the MTCM/FPM schemes, it is demonstrated that the set partitioning designed to maximize symbol diversity (optimum for fading channels) is optimum for performance on the AWGN channel as well. The MTCM/FPM schemes demonstrate improved performance over MTCM/MPSK schemes and TCM/FPM schemes on the AWGN channel and the fading channel     

Performance bounds for optimum multiuser DS-CDMA systems

In this letter we estimate the bit error probability (BEP) of optimum multiuser detection for synchronous and asynchronous code division multiple access (CDMA) systems on Gaussian and fading channels. We first compute an upper bound and a lower bound on the bit error probability for a given spreading code, then average the bounds over a few thousand sets of spreading codes. These bounds are obtained from a partial distance spectrum. On Gaussian channels, the upper bound converges to the lower bound at moderate to large signal-to-noise ratios. However, on fading channels the upper bound does not converge, hence we present our results for the lower bound only. The numerical results show that: 1) the BEP of a 31-user CDMA system with binary random spreading codes of length 31 is only two to four times higher than the BEP of the single user system; 2) the number of users that can be accommodated in an asynchronous CDMA system is larger than the processing gain; and 3) optimum multiuser detection outperforms linear detection (e.g., the decorrelating detector) by about 2.8 to 5.7 dB     

Improved tangential sphere bound on the bit-error probability ofconcatenated codes

The tangential sphere bound (TSB) of Poltyrev (1994) is a tight upper bound on the word error probability P_w of linear codes with maximum likelihood decoding and is based on the code's distance spectrum. An extension of the TSB to a bound on the bit-error probability P_b is given by Sason/Shitz (see IEEE Trans. Inform. Theory, vol.46, p.24-47, 2000). We improve the tangential sphere bound on P_b and apply the new method to some examples. Our comparison to other bounds as well as to simulation results shows an improved tightness, particularly for signal-to-noise ratios below the value corresponding to the computational cutoff rate R_o     

The performance of reference-based M-ary PSK with trelliscoded modulation in Rayleigh fading

The performance of trellis-coded M-ary phase shift keying (MPSK) is analyzed for three types of phase detection system; differential detection, pilot-symbol-aided detection and pilot-tone-aided detection. A near-exact solution to the probability of event error in Rayleigh fading is derived, and two extremely tight upper bounds generated. They improve the Chernoff upper bound by at least a factor of four at medium and high signal-to-noise ratios (SNR)     

Performance of RCPC codes with soft-decision decoding overNakagami-m fading channels

The analytical upper bounds on the pairwise error probability of rate compatible punctured convolutional (RCPC) codes, using coherent BPSK signals over slow frequency nonselective Nakagami-m fading channels with AWGN, are evaluated. With perfect channel state information (CSI) assumption, we use a direct integral with the Nakagami-m probability density function to obtain a closed form upper bound. For the case without CSI, we find an approximated upper bound for the high SNR cases and the approximation can be justified for signal to noise ratio (SNR) E _s/N₀ ≫ 1.5 dB     

Performance bounds for coded free-space optical communications through atmospheric turbulence channels

Error-control codes can help to mitigate atmospheric turbulence-induced signal fading in free-space optical communication links using intensity modulation/direct detection (IM/DD). Error performance bound analysis can yield simple analytical upper bounds or approximations to the bit-error probability. We first derive an upper bound on the pairwise codeword-error probability for transmission through channels with correlated turbulence-induced fading, which involves complicated multidimensional integration. To simplify the computations, we derive an approximate upper bound under the assumption of weak turbulence. The accuracy of this approximation under weak turbulence is verified by numerical simulation. Its invalidity when applied to strong turbulence is also shown. This simple approximate upper bound to the pairwise codeword-error probability is then applied to derive an upper bound to the bit-error probability for block codes, convolutional codes, and turbo codes for free-space optical communication through weak atmospheric turbulence channels. We also discuss the choice of interleaver length in block codes and turbo codes based on numerical evaluation of our performance bounds.     

Bit-error probability for maximum-likelihood decoding of linearblock codes and related soft-decision decoding methods

In this correspondence, the bit-error probability P_b for maximum-likelihood decoding of binary linear block codes is investigated. The contribution P_b(j) of each information bit j to P_b is considered and an upper bound on P_b(j) is derived. For randomly generated codes, it is shown that the conventional approximation at high SNR P_b≈(d_H/N).P_s, where P_s represents the block error probability, holds for systematic encoding only. Also systematic encoding provides the minimum P_b when the inverse mapping corresponding to the generator matrix of the code is used to retrieve the information sequence. The bit-error performances corresponding to other generator matrix forms are also evaluated. Although derived for codes with a generator matrix randomly generated, these results are shown to provide good approximations for codes used in practice. Finally, for soft-decision decoding methods which require a generator matrix with a particular structure such as trellis decoding, multistage decoding, or algebraic-based soft-decision decoding, equivalent schemes that reduce the bit-error probability are discussed. Although the gains achieved at practical bit-error rates are only a fraction of a decibel, they remain meaningful as they are of the same orders as the error performance differences between optimum and suboptimum decodings. Most importantly, these gains are free as they are achieved with no or little additional circuitry which is transparent to the conventional implementation     

Closed form expression and improved bound on pairwise error probability for performance analysis of turbo codes over Rician fading channels

This letter provides derivations for an exact expression and a bound on pair wise error probability over fully interleaved Rician fading channels under the assumption of ideal channel state information. The derivation which is based on the probability distribution of the sum of squared Rician random variables leads to an improved upper bound in comparison with the only known bound in literature. Pairwise error probability plots together with average union upper bounds for turbo codes having (1,7/5,7/5) and (1,5/7,5/7) generator polynomials are presented to demonstrate the effectiveness of the new results.     

On the performance of peaky capacity-achieving signaling on multipath fading channels

We analyze the error probability of peaky signaling on bandlimited multipath fading channels, the signaling strategy that achieves the capacity of such channels in the limit of infinite bandwidth under an average power constraint. We first derive an upper bound for general fading, then specialize to the case of Rayleigh fading, where we obtain upper and lower bounds that are exponentially tight and, therefore, yield the reliability function. These bounds constitute a strong coding theorem for the channel, as they not only delimit the range of achievable rates, but also give us a relationship among the error probability, data rate, bandwidth, peakiness, and fading parameters, such as the coherence time. They can be used to compare peaky signaling systems to other large bandwidth systems over fading channels, such as ultra-wideband radio and wideband code-division multiple access. We find that the error probability decreases slowly with the bandwidth W; under Rayleigh fading, the error probability varies roughly as W/sup -/spl alpha//, where /spl alpha/>0. With parameters typical of indoor wireless situations, we study the behavior of the upper and lower bounds on the error probability and the reliability function numerically.     

Analysis and performance of bidirectional decoding of convolutionalcodes over fading channels

The performance of suboptimal convolutional decoding over fading channels is explored. The suboptimal decoding algorithm used is the bidirectional algorithm. By estimating a “decoder weight spectrum” for the decoder, an “equivalent free distance” may be observed. Furthermore, by using this “decoder weight spectrum”, useful estimations of the error probabilities are obtained and compared to computer-simulation results in the case of very slow and very fast fading. The resultant curves are shown to be very tightly related. Computer-simulation results are also shown for various signal-to-noise ratios, normalized Doppler spreads, and frame length on three typical fading channels: the Rayleigh fading channel with exponential and Bessel autocorrelation functions and the Rician fading channel with exponential autocorrelation function. We show that considerable gains (up to 4 dB) can be obtained with respect to a similar-complexity Viterbi decoder at a frame error probability P_e =10^-3 and a slightly smaller gain (up to 1.8 dB) at a bit error probability P_b=10^-5     

Rotated TCM systems with dual transmit and multiple receive antennas on Nakagami fading channels

This paper analyzes the bit-error rate performance of a dual transmission-multiple reception antenna system in conjunction with a rotated trellis-coded modulation scheme over Nakagami fading channels. A simple way to combine the exact calculation of the pairwise error event probability with the transfer function bounding techniques is described. This approach allows us to evaluate an extremely tight theoretical upper bound on the average bit-error probability. The parameters which dominate the performance of these transmission schemes on fading channels are extracted from this evaluation. Optimum adaptive and fixed-phase rotations have been calculated, showing that most of the power efficiency obtained by using the optimum adaptive-phase rotation can be retained by using the optimum fixed-phase rotation.     

Error probability for digital transmission over nonlinear channelswith application to TCM

The computation of upper and lower bounds to error probability in digital transmission over nonlinear channels with a finite memory is considered. By using orthogonal Volterra series, the authors derive a canonical representation for discrete nonlinear systems, based on a linear convolutional code and a memoryless mapper. This representation shows that finite-memory, discrete nonlinear systems can be analyzed in much the same way as TCM (trellis-coded modulation) schemes. In particular, TCM over nonlinear channels can be analyzed. A technique is derived that expresses an upper bound to error probability based on the computation of the transfer of a state diagram with N branches, and whose branch labels are matrices rather than scalars. Some examples of its application are given. In particular, error bounds are derived for nonlinear TCM schemes and for TCM schemes operating on nonlinear channels     

Finite-Dimensional Bounds on Zm and Binary LDPC Codes With Belief Propagation Decoders

This paper focuses on finite-dimensional upper and lower bounds on decodable thresholds of Zopf_m and binary low-density parity-check (LDPC) codes, assuming belief propagation decoding on memoryless channels. A concrete framework is presented, admitting systematic searches for new bounds. Two noise measures are considered: the Bhattacharyya noise parameter and the soft bit value for a maximum a posteriori probability (MAP) decoder on the uncoded channel. For Zopf _m LDPC codes, an iterative m-dimensional bound is derived for m-ary-input/symmetric-output channels, which gives a sufficient stability condition for Zopf_m LDPC codes and is complemented by a matched necessary stability condition introduced herein. Applications to coded modulation and to codes with nonequiprobably distributed codewords are also discussed. For binary codes, two new lower bounds are provided for symmetric channels, including a two-dimensional iterative bound and a one-dimensional noniterative bound, the latter of which is the best known bound that is tight for binary-symmetric channels (BSCs), and is a strict improvement over the existing bound derived by the channel degradation argument. By adopting the reverse channel perspective, upper and lower bounds on the decodable Bhattacharyya noise parameter are derived for nonsymmetric channels, which coincides with the existing bound for symmetric channels     

Pairwise error probability evaluation of multiple symbol trellis-coded continuous phase frequency shift keying in a slow fading channel

This article evaluates the pairwise error probability (PEP) of multiple symbol trellis-coded modulation applied to continuous phase frequency shift keying (MTCM/CPFSK) in a slow fading environment with and without channel state information (CSI). The fading amplitude is assumed to be constant during an error event and distributed as Rician. For the case with CSI, the PEP is approximated using the Gauss-Chebysev formula and a tight upper bound is presented. For the case without CSI, a simplified upper bound is derived by using the improved Chernoff bound technique. Simulation results are also presented.     

Error probability of coherent detection for trellis-coded partialresponse CPM on Rician fading channels

Trellis-coded continuous phase modulation (TC-CPM) schemes are attractive for bandwidth and power limited communication systems such as mobile satellite communications and land mobile radio communications. A coherent receiver for interleaved partial response TC-CPM is derived. A true upper bound on the bit error probability for flat fading channels is derived by showing that TC-CPM is equivalent to a trellis-coded modulation scheme. The upper bound is evaluated by defining an error-state diagram along with a set of characteristic distances and then applying transfer function bounding techniques. Comparison with simulation results shows the upper bound tight to within 1.5-2 dB     

Bit-error bounds for trellis-coded MPSK in mixed fading channels

Bit-error probability (BEP) bounds of trellis-coded MPSK systems over two classes of mixed fading channels are studied. These two classes of channels have been proposed as candidate models for mobile satellite communications. The first class consists of slow and frequency-nonselective fading channels whose output field strengths follow a probability law characterized by a convex combination of Rician and Rayleigh/lognormal distributions. For the other class of fading channels, the received signal amplitude has a convex combination of Rician and Rician/lognormal distributions. We analyze performance bounds for trellis codes that belong to the class of either geometrically uniform codes (GUCs) or quasi-regular codes (QRCs). Receivers with either ideal channel state information (CSI) or no CSI at all are considered. We examine asymptotic behaviors of these codes and identify key design parameters. Numerical results are provided to illustrate and compare the BEP performances of various codes and to validate the usefulness of the asymptotic analysis     

Bounds on the pairwise error probability of coded DS/SSMAcommunication systems in Rayleigh fading channels

Arbitrarily tight upper and lower bounds on the pairwise error probability (PEP) of a trellis-coded or convolutional-coded direct-sequence spread-spectrum multiple-access (DS/SSMA) communication system over a Rayleigh fading channel are derived. A new set of probability density functions (PDFs) and cumulative distribution functions (CDFs) of the multiple-access interference (MAI) statistic is derived, and a modified bounding technique is proposed to obtain the bounds. The upper bounds and lower bounds together specify the accuracy of the resulting estimation of the PEP, and give an indication of the system error performance. Several suboptimum decoding schemes are proposed and their performances are compared to that of the optimum decoding scheme by the average pairwise error probability (APEP) values. The approach can be used to accurately study the multiple-access capability of the coded DS/SSMA system without numerical integrations     

Analysis of switched diversity TCM-MPSK systems on Nakagami fadingchannels

Space diversity reception and forward-error correction coding are powerful techniques to combat multipath fading encountered in mobile radio communications. In this paper, we analyze the performance of a discrete-time switched diversity system using trellis-coded modulation multiple phase-shift keying (TCM-MPSK) on slow, nonselective correlated Nakagami (1960) fading channels. Analytical upper bounds using the transfer function bounding technique are obtained and illustrated by several numerical examples. A simple integral expression for calculating the exact pairwise error probability is presented. The use of optimum adaptive and fixed switching thresholds is considered. Monte Carlo simulation results, which are more indicative of the exact system performance, are also given     

On improved bounds on the decoding error probability of block codesover interleaved fading channels, with applications to turbo-like codes

We derive here improved upper bounds on the decoding error probability of block codes which are transmitted over fully interleaved Rician fading channels, coherently detected and maximum-likelihood (ML) decoded. We assume that the fading coefficients during each symbol are statistically independent (due to a perfect channel interleaver), and that perfect estimates of these fading coefficients are provided to the receiver. The improved upper bounds on the block and bit error probabilities are derived for fully interleaved fading channels with various orders of space diversity, and are found by generalizing some previously introduced upper bounds for the binary-input additive white Gaussian nose (AWGN) channel. The advantage of these bounds over the ubiquitous union bound is demonstrated for some ensembles of turbo codes and low-density parity-check (LDPC) codes, and it is especially pronounced in a portion of the rate region exceeding the cutoff rate. Our generalization of the Duman and Salehi bound (Duman and Salehi 1998, Duman 1998) which is based on certain variations of Gallager's (1965) bounding technique, is demonstrated to be the tightest reported upper bound. We therefore apply it to calculate numerically upper bounds on the thresholds of some ensembles of turbo-like codes, referring to the optimal ML decoding. For certain ensembles of uniformly interleaved turbo codes, the upper bounds derived here also indicate good match with computer simulation results of efficient iterative decoding algorithms     

Bandwidth efficient coding for fading channels: code constructionand performance analysis

The authors apply a general method of bounding the event error probability of TCM (trellis-coded modulation) schemes to fading channels and use the effective length and the minimum-squared-product distance to replace the minimum-free-squared-Euclidean distance as code design parameters for Rayleigh and Rician fading channels with a substantial multipath component. They present 8-PSK (phase-shift-keying) trellis codes specifically constructed for fading channels that outperform equivalent codes designed for the AWGN (additive white Gaussian noise) channel when v⩾5. For quasiregular trellis codes there exists an efficient algorithm for evaluating event error probability, and numerical results which demonstrate the importance of the effective length as a code design parameter for fading channels with or without side information have been obtained. This is consistent with the case for binary signaling, where the Hamming distance remains the best code design parameter for fading channels. The authors show that the use of Reed-Solomon block codes with expanded signal sets becomes interesting only for large value of E_s/N₀, where they begin to outperform trellis codes