Signal Constellation Design Research Articles

Analysis and optimization of wavelet-coded communication systems

Since its advent, wavelet coding has been recognized as a promising technique for wireless transmission systems, especially due to its low decoding complexity and good performance over multipath fading. However, this technique requires unusual modulation schemes in order to be fully exploited. A BER analysis of wavelet-coded transmission systems over Rayleigh fading channels is presented in this work. This analysis is used in the design of novel signal constellations for wavelet-coded systems by means of numerical performance optimization. Numerical results obtained from analysis and simulations show that the wavelet-coded systems derived herein have very good performance characteristics over fading channels.

Pairwise optimization of modulation constellations for non-uniform sources

The design of two-dimensional signal constellations for the transmission of binary non-uniform memoryless sources over additive white Gaussian noise channels is investigated. The main application of this problem is the implementation of improved constellations where transmitted data is highly non-uniform. A simple algorithm, which optimizes a constellation by re-arranging its points in a pairwise fashion (i.e., two points are modified at a time, with all other points remaining fixed), is presented. In general, the optimized constellations depend on both the source statistics and the signal-to-noise ratio (SNR) in the channel. We show that constellations designed with source statistics considered can yield symbol error rate (SER) performance that is substantially better than rectangular quadrature amplitude modulation signal sets used with either Gray mapping or more recently developed maps. SER gains as high as 5 dB in E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> /N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o</sub> SNR are obtained for highly non-uniform sources. Symbol mappings are also developed for the new constellations using a similar pairwise optimization method whereby we assign and compare a weighted score for each pair. These maps, when compared to the mappings used in conjunction with the standard rectangular QAM constellation, again achieve considerable performance gains in terms of bit error rate (BER). Gains as high as 4 dB were achieved over rectangular QAM with Gray mapping, or more than 1 dB better than previously improved mappings. Finally, the uncoded pairwise optimized system is compared to a standard tandem (separate) source and channel coding system. Although neither system is universally better, the uncoded system with optimized constellations outperforms the tandem coding system for low-to-mid SNRs. Performance/complexity trade-off s between the two systems are also discussed.

Effect of error in channel state information on the channel capacity with QAM signalling and the design of robust signal constellations

The effects of channel estimation errors on the channel capacity of a discrete time, discrete input, infinite output Rayleigh fading channel are investigated. The case of conventional modulation methods such as phase shift keying (PSK) and quatrature amplitude modulation (QAM) is investigated at first where it was observed that the capacity degrades rapidly with increasing channel estimation errors. The effect of error in the channel estimation is similar to the effect of higher noise in the channel that depends on the transmitted signal. A genetic algorithm is used to optimise the signal constellation in order to maximise the capacity for a given finite number of signal points. The aim of trying to maximise the capacity is to estimate the remaining gap in performance between a traditional modulation scheme such as QAM and the best possible constellation that is optimised for the channel. The constellations obtained from the genetic algorithm are, in general, not directly implementable. A method to design practical robust signal constellations that overcome the effect of channel state information (CSI) error is presented. The robust signal constellations obtained show a performance that is very close to the otimal constellations. In this work, the probability distribution of the error in CSI is assumed to be known.

Signal Constellation Design for Symbol–Level Rate Adaptation in AWGN Channels

This paper considers rate adaptation for the Additive White Gaussian Noise (AWGN)-channel with unknown channel capacity. We show how multi-layer signal constellations can be optimized layer-wise for particular Signal-to-Noise Ratio (SNR)- points, to provide simple and efficient ARQ-based symbol-level rate adaptation with fine granularity. A turbo-coded system is used as an example showing that the proposed design provides actual communication rates that are close to the channel capacity.

Signal Design and Detection in Presence of Nonlinear Phase Noise

In optical fiber transmission systems using inline amplifiers, the interaction of a signal and amplifier noise through the Kerr effect leads to nonlinear phase noise that can impair the detection of phase-modulated signals. We present analytical expressions for the maximum-likelihood (ML) decision boundaries and symbol-error rate (SER) for phase-shift keying and differential phase-shift keying systems with coherent and differentially coherent detection, respectively. The ML decision boundaries are in the form thetas(r) = c2r2 + c1r + c0, where thetas and r are the phase and the amplitude of the received signal, respectively. Using the expressions for the SER, we show that the impact of phase error from carrier synchronization is small, particularly for transoceanic links. For modulation formats such as 16-quadrature amplitude modulation, we propose various transmitter and receiver phase rotation strategies such that the ML detection is well approximated by using straight-line decision boundaries. The problem of signal constellation design for optimal SER performance is also studied for a system with four signal points.

Space–Time Constellation Design for Partial CSI at the Receiver

The design of signal constellations for a communication system using multiple transmitter antennas over a Rayleigh-fading channel is considered under the assumption that no channel state information (CSI) is available at the transmitter and the receiver has acquired a CSI estimate with known error covariance. This setup encompasses the well-studied scenarios of perfect and no channel knowledge at the receiver and allows a smooth transition between these two cases. The data detection performance as a function of the CSI error covariance is analyzed and used to investigate the design of training blocks if such blocks are transmitted to provide CSI to the receiver. Moreover, two approaches to design constellations adapted to the error covariance of the receiver CSI are presented. Whereas the first approach is based on a generic gradient search method, the second approach uses an appropriate combination of constellations designed for perfect and no CSI at the receiver. Simulations confirm the benefits of the presented designs.

Error exponents for channel coding with application to signal constellation design

This paper concerns error exponents and the structure of input distributions maximizing the random coding exponent for a stochastic channel model. The following conclusions are obtained under general assumptions on the channel statistics. 1) The optimal distribution has a finite number of mass points, or in the case of a complex channel, the amplitude has finite support. 2) A new class of algorithms is introduced based on the cutting-plane method to construct an optimal input distribution. The algorithm constructs a sequence of discrete distributions, along with upper and lower bounds on the random coding exponent at each iteration. 3) In some numerical example considered, the resulting code significantly outperforms traditional signal constellation schemes such as quadrature amplitude modulation and phase-shift keying for all rates below the capacity.

Design of multidimensional signal constellations

An algorithmic technique intended for signal constellation design for an N-dimensional Euclidean signal space is presented. Such signals are used for reliable and efficient digital communications on an additive white Gaussian noise channel. The minimum Euclidean distance between signals and the constellation-constrained capacity are used as performance measures. The basic idea is to spread out the signal vectors using an iterative method modelling the behaviour of equally charged particles in free space. Initially, the vectors are randomly distributed. The vectors reach equilibrium after a few iterations. Numerical calculations for N=2, 3, 4 and 5 dimensions and M=N+1, N+2, …,20 signals show that the generated codes are nearly as good as previously known codes, and in some cases even better.

Unitary signal constellations for differential space-time modulation with two transmit antennas: parametric codes, optimal designs, and bounds

We focus on the design of unitary signal constellations for differential space-time modulation with double transmit antennas. By using the parametric form of a two-by-two unitary matrix, we present a class of unitary space-time codes called parametric codes and show that this class of unitary space-time codes leads to a five-signal constellation with the largest possible diversity product and a 16-signal constellation with the largest known diversity product. Although the parametric code of size 16 is not a group by itself, we show that it is a subset of a group of order 32. Furthermore, the unitary signal constellations of sizes 32, 64, 128, and 256 obtained by taking the subsets of the parametric codes of sizes 37, 75, 135, and 273, respectively, have the largest known diversity products. We also use large diversity sum of unitary space-time signal constellations as another significant property for the signal constellations to have good performance in low-SNR scenarios. The newly introduced unitary space-time codes can lead to signal constellations with sizes of 5 and 9 through 16 that have the largest possible diversity sums. Subsequently, we construct a few sporadic unitary signal constellations with the largest possible diversity product or diversity sum. A four-signal constellation which has both the largest possible diversity product and the largest possible diversity sum and three unitary signal constellations with the largest possible diversity sums for sizes of 6, 7, and 8 are constructed, respectively. Furthermore, by making use of the existing results in sphere packing and spherical codes, we provide several upper and lower bounds on the largest possible diversity product and the largest possible diversity sum that unitary signal constellations of any size can achieve.

Asymptotic error probability analysis of quadratic receivers in Rayleigh-fading channels with applications to a unified analysis of coherent and noncoherent space-time receivers

A general, asymptotic (high signal-to-noise (SNR)) error analysis is introduced for quadratic receivers in frequency-flat and multipath Rayleigh-fading channels with multiple transmit and receive antennas. Asymptotically tight expressions for the pairwise error probabilities are obtained for coherent, noncoherent, and differentially coherent space-time receivers. Not only is our unified analysis applicable to more general modulation schemes and/or channel models than previously considered, but it also reveals a hitherto unrecognized eigenvalue structure that is common to all of these problems. In addition to providing an easy recipe for computing the asymptotic pairwise error rates, we make some conclusions regarding criteria for the design of signal constellations and codes such as (a) the same design criteria apply for both correlated and independent and identically distributed (i.i.d.) fading processes and (b) for noncoherent communications, unitary signals are optimal in the sense that they minimize the asymptotic union bound.

QAM transmission through a companding channel-signal constellations and detection

The effect of the companding process on QAM signals has been under investigation for the past several years. The compander, included in the PCM telephone network to improve voice performance, has an unusual affect on digital QAM data signals which are transmitted over the same channel. The quantization noise, generated by the companding process which is multiplicative (and asymmetric), degrades the detectability performance of the outermost points of the QAM constellation more than that of the inner points. The combined effect of the companding noise and the inherent white Gaussian noise of the system, leads to a re-examination of signal constellation design. The authors investigate the detectability performance of a number of candidates for signal constellations including, a typical rectangular QAM constellation, the same constellation with the addition of a smear-desmear operation, and two new improved QAM constellation designs with two-dimensional warping. >

Precoding technique for partial-response channels with applications to HDTV transmission

An equivalent partial-response (PR) channel 1-Z/sup -k/ arises in the envisioned terrestrial over-the-air broadcasting of digital high-definition television (HDTV) signals when a comb filter is used by an HDTV receiver to reduce the NTSC cochannel interference. The design of signal constellations and their associated precoders for this PR channel is considered. Besides PAM and square QAM, it is shown that generalized square and hexagonal constellations can also be used. Coded modulation and graceful degradation in the received signal quality are discussed. The results extended to a more general PR channel.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Nonlinear quantization and the design of coded and uncoded signal constellations

The author analyzes the effects of nonlinear quantizers, used in T1 carrier systems, on the performance of QAM (quadrature amplitude modulation) and TCM (trellis coded modulation) voiceband data communication modems, and introduces a method to counteract the effects of these nonlinearities. The effect of nonlinear quantizers is modeled as a source of multiplicative noise whose variance is proportional to the amplitude of the quantized signal. The relation between variance of the multiplicative noise, characteristics of the nonlinear quantizer, and impulse response of the pulse shaping filters used in the modems is derived. Effects of multiplicative noise on the performance of QAM and TCM modems are analyzed. A method for the design of QAM and TCM signal constellations is introduced which counteracts the harmful effects of nonlinear quantizers. The new constellations provide 2-3 dB performance improvements in multiplicative noise channels while losing less than 0.5 dB in additive noise channels.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>