Optimization of Two-Dimensional Signal Constellations in the Presence of Gaussian Noise
TL;DR: In this paper an asymptotic (large signal-to-noise ratio) expression is derived for the error rate and it is rigorously proved in the Appendix that the optimum constellations tend toward an equilateral structure, and become uniformly distributed in a circle.
Abstract: A considerable amount of literature exists on the problem of selecting an efficient set of N digital signals with in-phase and quadrature components for use in a suppressed carrier data transmission system. However, the signal constellation which minimizes the probability of error in Gaussian noise, under an average power constraint, has not been determined when the number of signals is greater than two. In this paper an asymptotic (large signal-to-noise ratio) expression, of the minimum distance type, is derived for the error rate. Using this expression, a gradient-search procedure, which is initiated from several randomly chosen N -point arrays, converges in each case to a locally optimum constellation. The algorithm incorporates a radial contraction technique to meet the average signal power constraint. The best solutions are described for several values of N and compared with well-known signal formats. As an example, the best locally optimum 16-point constellation shows an advantage of about 0.5 dB in signal-signal-to-noise ratio over quadrature amplitude modulation. The locally optimum constellations are the vertices of a trellis of (almost) equilateral triangles. As N \rightarrow \infty , it is rigorously proved in the Appendix that the optimum constellations tend toward an equilateral structure, and become uniformly distributed in a circle.
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TL;DR: In this article, the capacity limit of fiber-optic communication systems (or fiber channels?) is estimated based on information theory and the relationship between the commonly used signal to noise ratio and the optical signal-to-noise ratio is discussed.
Abstract: We describe a method to estimate the capacity limit of fiber-optic communication systems (or ?fiber channels?) based on information theory. This paper is divided into two parts. Part 1 reviews fundamental concepts of digital communications and information theory. We treat digitization and modulation followed by information theory for channels both without and with memory. We provide explicit relationships between the commonly used signal-to-noise ratio and the optical signal-to-noise ratio. We further evaluate the performance of modulation constellations such as quadrature-amplitude modulation, combinations of amplitude-shift keying and phase-shift keying, exotic constellations, and concentric rings for an additive white Gaussian noise channel using coherent detection. Part 2 is devoted specifically to the "fiber channel.'' We review the physical phenomena present in transmission over optical fiber networks, including sources of noise, the need for optical filtering in optically-routed networks, and, most critically, the presence of fiber Kerr nonlinearity. We describe various transmission scenarios and impairment mitigation techniques, and define a fiber channel deemed to be the most relevant for communication over optically-routed networks. We proceed to evaluate a capacity limit estimate for this fiber channel using ring constellations. Several scenarios are considered, including uniform and optimized ring constellations, different fiber dispersion maps, and varying transmission distances. We further present evidences that point to the physical origin of the fiber capacity limitations and provide a comparison of recent record experiments with our capacity limit estimation.
2,135 citations
Cites background from "Optimization of Two-Dimensional Sig..."
...A general geometric property of any optimum -ary modulation scheme is that its center of gravity is the origin, as this minimizes the average symbol energy [22], [210]....
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...The latter is the optimum 8-ary constellation in terms of minimum distance [210]....
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TL;DR: This paper attempts to present a comprehensive tutorial survey of the development of efficient modulation techniques for bandlimited channels, such as telephone channels, with principal emphasis on coded modulation techniques, in which there is an explosion of current interest.
Abstract: This paper attempts to present a comprehensive tutorial survey of the development of efficient modulation techniques for bandlimited channels, such as telephone channels. After a history of advances in commercial high-speed modems and a discussion of theoretical limits, it reviews efforts to optimize two-dimensional signal constellations and presents further elaborations of uncoded modulation. Its principal emphasis, however, is on coded modulation techniques, in which there is an explosion of current interest, both for research and for practical application. Both block-coded and trellis-coded modulation are covered, in a common framework. A few new techniques are presented.
770 citations
TL;DR: Simulation results show that the proposed STBCs from quasi-orthogonal design can outperform the codes from orthogonal designs at both low and high SNRs.
Abstract: Space-time block codes (STBCs) from orthogonal designs proposed by Alamouti, and Tarokh-Jafarkhani-Calderbank have attracted considerable attention lately due to their fast maximum-likelihood (ML) decoding and full diversity. However, the maximum symbol transmission rate of an STBC from complex orthogonal designs for complex signals is only 3/4 for three and four transmit antennas, and it is difficult to construct complex orthogonal designs with rate higher than 1/2 for more than four transmit antennas. Recently, Jafarkhani, Tirkkonen-Boariu-Hottinen, and Papadias-Foschini proposed STBCs from quasi-orthogonal designs, where the orthogonality is relaxed to provide higher symbol transmission rates. With the quasi-orthogonal structure, the quasi-orthogonal STBCs still have a fast ML decoding, but do not have the full diversity. The performance of these codes is better than that of the codes from orthogonal designs at low signal-to-noise ratio (SNR), but worse at high SNR. This is due to the fact that the slope of the performance curve depends on the diversity. It is desired to have the quasi-orthogonal STBCs with full diversity to ensure good performance at high SNR. In this paper, we achieve this goal by properly choosing the signal constellations. Specifically, we propose that half of the symbols in a quasi-orthogonal design are chosen from a signal constellation set A and the other half of them are chosen from a rotated constellation e/sup j/spl phi// A. The resulting STBCs can guarantee both full diversity and fast ML decoding. Moreover, we obtain the optimum selections of the rotation angles /spl phi/ for some commonly used signal constellations. Simulation results show that the proposed codes outperform the codes from orthogonal designs at both low and high SNRs.
488 citations
Cites background from "Optimization of Two-Dimensional Sig..."
...3(b) is the best known constellation from a minimum Euclidean distance point of view, which is well known as Voronoi code [38]–[40]....
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...Moreover, Wayner in [38] proved that for a large number of points, the optimum constellation tends toward a lattice of equilateral triangles....
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TL;DR: In this paper, a 24-level format and an 8-level QPSK format were proposed for coherent optical communication systems, respectively, an extension and a subset of the commonly used 16-level dual-polarization (QPSK) format.
Abstract: Coherent optical transmission systems have a four-dimensional (4-D) signal space (two quadratures in two polarizations). These four dimensions can be used to create modulation formats that have a better power efficiency (higher sensitivity) than the conventional binary phase shift keying/quadrature phase shift keying (BPSK/QPSK) signals. Several examples are given, with some emphasis on a 24-level format and an 8-level format, including descriptions of how they can be realized and expressions for their symbol and bit error probabilities. These formats are, respectively, an extension and a subset of the commonly used 16-level dual-polarization QPSK format. Sphere packing simulations in 2, 3, and 4 dimensions, up to 32 levels, are used to verify their optimality. The numerical results, as the number of levels increases, are shown to agree with lattice-theoretical results. Finally, we point out that the use of these constellations will lead to improved fundamental sensitivity limits for optical communication systems, and they may also be relevant as a way of reducing power demands and/or nonlinear influence.
379 citations
Additional excerpts
...[40], and they are typically hexagonal packings...
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Book•
07 Jun 2000
TL;DR: In this article, the authors provide an overview of most major single and multi-carrier Quadrature Amplitude Modulation (QAM) techniques commencing with simple QAM schemes for the uninitiated through to complex, rapidly-evolving areas, such as arrangements for wideband mobile channels.
Abstract: Single- and Multi-carrier Quadrature Amplitude Modulation Principles and Applications for Personal Communications, WLANs and Broadcasting L. Hanzo Department of Electronics and Computer Science, University of Southampton, UK W. Webb Motorola, Arlington Heights, USA formerly at Multiple Access Communications Ltd, Southampton, UK T. Keller Ubinetics, Cambridge Technology Centre, Melbourn, UK formerly at Department of Electronics and Computer Science, University of Southampton, UK Motivated by the rapid evolution of wireless communication systems, this expanded second edition provides an overview of most major single- and multi-carrier Quadrature Amplitude Modulation (QAM) techniques commencing with simple QAM schemes for the uninitiated through to complex, rapidly-evolving areas, such as arrangements for wide-band mobile channels. Targeted at the more advanced reader, the multi-carrier modulation based second half of the book presents a research-orientated outlook using a variety of novel QAM-based arrangements. * Features six new chapters dealing with the complexities of multi-carrier modulation which has found applications ranging from Wireless Local Area Networks (WLAN) to Digital Video Broadcasting (DVB) * Provides a rudimentary introduction for readers requiring a background in the field of modulation and radio wave propagation * Discusses classic QAM transmission issues relevant to Gaussian channels * Examines QAM-based transmissions over mobile radio channels * Incorporates QAM-related orthogonal techniques, considers the spectral efficiency of QAM in cellular frequency re-use structures and presents a QAM-based speech communications system design study * Introduces Orthogonal Frequency Division Multiplexing (OFDM) over both Gaussian and wideband fading channels By providing an all-encompassing self-contained treatment of single- and multi- carrier QAM based communications, a wide range of readers including senior undergraduate and postgraduate students, practising engineers and researchers alike will all find the coverage of this book attractive.
354 citations
References
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TL;DR: The results indicate that in Gaussian noise alone an improvement in signal-to-noise ratio of as much as 2 dB may be realized by using quadrature amplitude modulation instead of conventional amplitude and phase modulation.
Abstract: A long-standing communications problem is the efficient coding of a block of binary data into a pair of in-phase and quadrature components. This modulation technique may be regarded as the placing of a discrete number of signal points in two dimensions. Quadrature amplitude modulation (QAM) and combined amplitude and phase modulation (AM-PM) are two familiar examples of this signaling format. Subject to a peak or average power constraint, the selection of the signal coordinates is done so as to minimize the probability of error. In the design of high-speed data communication systems this problem becomes one of great practical significance since the dense packing of signal points reduces the margin against Gaussian noise. Phase jitter, which tends to perturb the angular location of the transmitted signal point, further degrades the error rate. Previous investigations have considered the signal evaluation and design problem in the presence of Gaussian noise alone and within the framework of a particular structure, such as conventional amplitude and phase modulation. We present techniques to evaluate and optimize the choice of a signal constellation in the presence of both Gaussian noise and carrier phase jitter. The performance of a number of currently used or proposed signal constellations are compared. The evaluation and the optimization are based upon a perturbation analysis of the probability density of the received signal given the transmitted signal. Laplace's asymptotic formula is used for the evaluation. Discretizing the signal space reduces the optimal signal design problem under a peak power constraint to a tractable mathematical programming problem. Our results indicate that in Gaussian noise alone an improvement in signal-to-noise ratio of as much as 2 dB may be realized by using quadrature amplitude modulation instead of conventional amplitude and phase modulation. New modulation formats are proposed which perform very well in Gaussian noise and additionally are quite insensitive to moderate amounts of phase jitter.
112 citations
TL;DR: A simple technique is presented for generating and optimally detecting the honeycomb (hexagonal.) signal set, i.e., the signal set that has the tightest sphere-packing properties and is shown to be slightly superior from an average power standpoint.
Abstract: Selection of a particular signal set array for a bandwidthConstrained multiple phase-and-amplitude-shift-keyed (MPASK) communication system for a linear additive Gaussian noise channel requires consideration of factors such as average and/or peak power versus symbol error probability, signal amplitude dynamic range, simplicity of generation and detection, and number of bit errors per symbol error (Gray code properties). A simple technique is presented for generating and optimally detecting the honeycomb (hexagonal.) signal set, i.e., the signal set that has the tightest sphere-packing properties. The symbol and bit error probability performance of this set is compared to other two-dimensional signal sets that have been investigated in the literature, and is shown to be slightly superior from an average power standpoint. The paper concludes with a comparison of all of these signal sets from the standpoint of the factors listed above.
93 citations
TL;DR: The theoretical properties of a class of digital systems where the signal space is two-dimensional, both amplitude-and phase-modulated, are considered, and the performance of one-dimensional systems, AM-only and PM-only, is compared with a complete set of curves for both peak and average power.
Abstract: A digital transmission system with n possible transmitted symbols is considered. If the time between transmitted symbols is T 0 seconds and the bandwidth is W , the n possible symbols correspond to n vectors in a 2WT_{0} dimensional signal space. This paper considers the theoretical properties of a class of digital systems where the signal space is two-dimensional. Such systems are both amplitude-and phase-modulated. Approximate expressions are derived for the average probability of error for these systems as a function of the placement of the n symbol vectors in the twodimensional signal space. Optimum placements are then given which minimize this probability of error for a given average or peak power SNR constraint. It is shown that the optimum channel structure is a function of the alphabet size n , and the type of power constraint, as well as the SNR. In general the optimum system is a phase-modulated system for low SNR's and for alphabet sizes n \leq 16 in the high SNR region. The performance of this optimum system in terms of channel capacity and probability of error is then compared with the performance of one-dimensional systems, AM-only and PM-only, in a complete set of curves for both peak and average power.
56 citations
TL;DR: In this paper, a special case of the result conjectured by H. Zassenhaus [6] and proved by N. Oler [l] is shown. But this result is not applicable to the case where the set of numbers {card(S): S is a packing in X}.
Abstract: Let X be a compact metric space. By a packing in X we mean a subset S ⊆ X such that, for x, y ∈ S with x ≠ y, the distance d(x, y) ≥ 1. Since X is compact, any packing of X is finite. In fact, the set of numbers {card(S): S is a packing in X} is bounded. The cardinality of the largest packing in X will be called the packing number of X and will be denoted by ρ(X). If A(X) and P(X) denote the area and perimeter, respectively, of a compact convex subset X of the plane, then a special case of a result conjectured by H. Zassenhaus [6] and proved by N. Oler [l] is the following.
55 citations
TL;DR: Experimental results indicate excellent agreement with the theory of combined amplitude and phase modulation (AM–PM), and the theory predicts a performance from 1 to 4 dB poorer than what can be realized from single sideband (SSB) modulation.
Abstract: This paper presents theoretical analysis and experimental verification of the performance of a digital data modem which uses combined amplitude and phase modulation (AM–PM). The theoretical model assumes operation over the bandlimited additive Gaussian channel. The receiver used in the experiment, and for which theoretical results are presented, uses an envelope detector in parallel with a phase detector to recover the data. The criteria of error rate and communication efficiency (measured in bits per cycle of bandwidth) as functions of average signal-to-noise ratio (S/N) are used to make comparisons with other modulation schemes. The theory predicts a performance from 1 to 4 dB poorer than what can be realized from single sideband (SSB) modulation. We present experimental results which indicate excellent agreement with the theory.
52 citations