Showing papers on "Complex normal distribution published in 2003"

Analysis of transmit-receive diversity in Rayleigh fading

Abstract: We analyze the error performance of a wireless communication system employing transmit-receive diversity in Rayleigh fading. By focusing on the complex Gaussian statistics of the independent and identically distributed entries of the channel matrix, we derive a formula for the characteristic function (c.f.) of the maximum output signal-to-noise ratio. We use this c.f. to obtain a closed-form expression of the symbol error probability (SEP) for coherent binary keying. The method is easily extended to obtain the SEP for the coherent reception of M-ary modulation schemes.

On multivariate Rayleigh and exponential distributions

Abstract: In this paper, expressions for multivariate Rayleigh and exponential probability density functions (PDFs) generated from correlated Gaussian random variables are presented. We first obtain a general integral form of the PDFs, and then study the case when the complex Gaussian generating vector is circular. We consider two specific circular cases: the exchangeable case when the variates are evenly correlated, and the exponentially correlated case. Expressions for the multivariate PDF in these cases are obtained in integral form as well as in the form of a series of products of univariate PDFs. We also derive a general expression for the multivariate exponential characteristic function (CF) in terms of determinants. In the exchangeable and exponentially correlated cases, CF expressions are obtained in the form of a series of products of univariate gamma CFs. The CF of the sum of exponential variates in these cases is obtained in closed form. Finally, the bivariate case is presented mentioning its main features. While the integral forms of the multivariate PDFs provide a general analytical framework, the series and determinant expressions for the exponential CFs and the series expressions for the PDFs can serve as a useful tool in the performance analysis of digital modulation over correlated Rayleigh-fading channels using diversity combining.

A limit theorem at the edge of a non-Hermitian random matrix ensemble

Abstract: The study of the edge behaviour in the classical ensembles of Gaussian Hermitian matrices has led to the celebrated distributions of Tracy–Widom. Here we take up a similar line of inquiry in the non-Hermitian setting. We focus on the family of N × N random matrices with all entries independent and distributed as complex Gaussian of mean zero and variance 1/N. This is a fundamental non-Hermitian ensemble for which the eigenvalue density is known. Using this density, our main result is a limit law for the (scaled) spectral radius as N ↑ ∞. As a corollary, we get the analogous statement for the case where the complex Gaussians are replaced by quaternion Gaussians.

Capacity of a Gaussian MIMO channel with nonzero mean

Abstract: We characterize the input covariance that maximizes the ergodic capacity of a flat-fading, multiple-input-multiple-output (MIMO) channel with additive white Gaussian noise, when the entries of the channel matrix are independent, circularly symmetric, complex Gaussian random variables of nonzero (and possibly different) means and identical variances. We show that the optimal transmit covariance must have the same eigenvectors as the squared mean channel, thereby reducing the computation of the optimal covariance to a simple convex optimization. This generalizes existing results for multiple-input-single-output (MISO) channels and MIMO channels restricted to have a mean of unit rank.

The pseudo-Wishart distribution and its application to MIMO systems

Abstract: The pseudo-Wishart distribution arises when a Hermitian matrix generated from a complex Gaussian ensemble is not full-rank. It plays an important role in the analysis of communication systems using diversity in Rayleigh fading. However, it has not been extensively studied like the Wishart distribution. Here, we derive some key aspects of the complex pseudo-Wishart distribution. Pseudo-Wishart and Wishart distributions are treated as special forms of a Wishart-type distribution of a random Hermitian matrix generated from independent zero-mean complex Gaussian vectors with arbitrary covariance matrices. Using a linear algebraic technique, we derive an expression for the probability density function (p.d.f.) of a complex pseudo-Wishart distributed matrix, both for the independent and identically distributed (i.i.d.) and non-i.i.d. Gaussian ensembles. We then analyze the pseudo-Wishart distribution of a rank-one Hermitian matrix. The distribution of eigenvalues of Wishart and pseudo-Wishart matrices is next considered. For a matrix generated from an i.i.d. Gaussian ensemble, we obtain an expression for the characteristic function (cf) of eigenvalues in terms of a sum of determinants. We present applications of these results to the analysis of multiple-input multiple-output (MIMO) systems. The purpose of this work is to make researchers aware of the pseudo-Wishart distribution and its implication in the case of MIMO systems in Rayleigh fading, where the transmitted signals are independent but the received signals are correlated. The results obtained provide a powerful analytical tool for the study of MIMO systems with correlated received signals, like systems using diversity and optimum combining, space-time systems, and multiple-antenna systems.

Complex random matrices and Rayleigh channel capacity

Abstract: The eigenvalue densities of complex central Wishart matrices are investigated with the objective of studying an open problem in channel capacity. These densities are represented by complex hypergeometric functions of matrix arguments, which can be expressed in terms of complex zonal polynomials. The connection between the complex Wishart matrix theory and information theory is given. This facilitates the evaluation of the most important information-theoretic measure, the so-called channel capacity. In particular, the capacity of multiple input, multiple output (MIMO) Rayleigh distributed channels are fully investigated. We consider both correlated and uncorrelated channels and derive the corresponding channel capacity formulas. It is shown how the channel correlation degrades the capacity of the communication system. To appear in Communications in Information & Systems

Capacity analysis of correlated mimo channels

Abstract: This paper gives expressions for the capacity of ergodic multiple-input multiple-output channels with finite dimensions, in which the channel gains have a correlated complex normal distribution and receivers experience independent Gaussian noise. The particular correlated normal distribution considered corresponds to flat Rayleigh fading with arbitrary transmit and receive correlation. Knowledge of the correlation matrices is assumed at both the transmitter and receiver, while the receiver, but not the transmitter, has complete knowledge of the channel realization. The optimal input density is characterized via a necessary and sufficient condition for optimality, along with an iterative algorithm for its numerical computation. The resulting capacity is expressed in terms of hypergeometric functions of matrix argument, which depend on the channel correlation matrices only through their eigenvalues. Some closed-form expressions are also given in the case of single-sided correlation. Some consideration is given to high- and low-power asymptotics. Easily computable asymptotic expressions are also given for receive-side only correlation in the case that the number of transmitters is large. In that case, the capacity can be divided into two components: one arising from the dominant eigenvalues of the receiver-end correlation matrix, and the other from the remaining spherically distributed eigenvalues. Some numerical results are also presented.

Power and complex envelope correlation for modeling measured indoor MIMO channels: a beamforming evaluation

Abstract: The multivariate complex normal distribution is often employed as a tractable and convenient model for MIMO wireless systems. Several models may result, depending on how the covariance matrix is specified, i.e. power or complex envelope correlation and full or separable (Kronecker) correlation. This paper investigates the differences of the various models by applying a joint transmit/receive beamformer to recent wideband MIMO radio channel measurements at 5.2 GHz. It is found that the Kronecker model, especially for power correlation, significantly alters the joint beamformer spectrum. A multipath clustering model is applied whose parameters are estimated directly from the measured data. The clustering model is able to match capacity pdfs, and resulting simulated joint beamformer spectra look more realistic than those generated with conventional separable correlation functions.

Correlated MIMO Rayleigh Fading Channels: Capacity, Optimal Signaling, and Asymptotics 1

Abstract: The capacity of the MIMO channel is investigated under the assumption that the elements of the channel matrix are zero mean proper complex Gaussian random variables with a general correlation structure. It is assumed that the receiver knows the channel perfectly but that the transmitter knows only the channel statistics. The analysis is carried out using an equivalent virtual representation of the channel that is obtained via a spatial discrete Fourier transform. It is shown that in the virtual domain, the capacity achieving input vector consists of independent zero-mean proper complex entries, whose variances can be computed numerically. Furthermore, in the asymptotic regime of low signal-to-noise ratio (SNR), it is shown that beamforming along one virtual transmit angle is asymptotically optimal. Necessary and sucien t conditions for the optimality of beamforming are also derived. Finally, the capacity is investigated in the asymptotic regime where the number of receive and transmit antennas go to innit y, with their ratio kept constant. Using a result of Girko, an expression for the asymptotic capacity scaling with the number of antennas is obtained in terms of the two-dimensional spatial scattering function of the channel.

Topics in complex random matrices and information theory

Abstract: The eigenvalue distribution of both central and noncentral complex Wishart matrices are investigated with the objective of studying several open problems in information theory and numerical analysis, etc. Specifically, the largest, kth largest, and the smallest eigenvalue distributions of complex Wishart matrices and the condition number distribution of complex Gaussian random matrices are derived. These distributions are represented by complex hypergeometric functions of matrix arguments, which can be expressed in terms of complex zonal polynomials. We derive several results on complex hypergeometric functions and zonal polynomials that are used to evaluate these distributions. We also give a method to compute these complex hypergeometric functions. Then the connection between the complex Wishart matrix theory and information theory is formulated. This facilitates the evaluation of the most important information-theoretic measure, the so-called channel capacity. The capacity of the communication channel expresses the maximum rate at which information can be reliably conveyed by the channel. In particular, the capacities of the multiple input, multiple output Rayleigh and Rician distributed channels are fully investigated. We consider both correlated and uncorrelated channels and derive the corresponding channel capacity formulas. These studies show how the channel correlations degrade the capacity of the communication system.

Efficient simulation of space-time correlated MIMO mobile fading channels

Abstract: Simulation of multiple-input multiple-output (MIMO) fading channels, with crosscorrelated subchannels, is of paramount importance in performance evaluation of space-time techniques in multiantenna systems. This paper focuses on four methods to simulate several spatio-temporally crosscorrelated stationary complex Gaussian processes: the spectral representation method, the sampling theorem method, the random polynomial method, and the circulant embedding method. The first three methods are based on parametric random representations, which consist of the superposition of deterministic functions with random coefficients and parameters, whereas the fourth one is built upon circulant embedding of the covariance matrix and the use of fast Fourier transform (FFT), to diagonalize a block circulant matrix. In this paper, we provide a comprehensive theoretical analysis of the computational complexity of all the four methods. The performance of these techniques are also assessed, via extensive simulations, in terms of the quality of the generated samples. Our theoretical analysis and simulation results show that for MIMO channel simulations, the spectral method has much less computational complexity, with the same simulation accuracy as other methods. Matlab/sup /spl copy// files for all the four methods are available at http://web.njit.edt//spl sim/abdi.

Stochastic Schrödinger equations with general complex Gaussian noises

Abstract: Within the framework of non-Markovian stochastic Schr\"odinger equations, we generalize the results of [W. T. Strunz, Phys. Lett. A 224, 25 (1996)] to the case of general complex Gaussian noises; we analyze the two important cases of purely real and purely imaginary stochastic processes.

Influence functions for array covariance matrix estimators

Abstract: An influence function (IF) measures the effects of infinitesimal perturbations on the estimator. In this paper, we study the influence functions of sensor array covariance matrix estimators. We derive general results concerning the IF of any affine equivariant (pseudo-)covariance matrix estimator and its eigenvectors and eigenvalues under complex elliptically symmetric model distributions. The complex Gaussian distribution, for example, is a prominent member in this class of distributions. We also derive the IF of the regular covariance matrix estimator and that of the M-functional of covariance. The knowledge of the IF of the covariance matrix estimator allows us to obtain directly the IF of the associated eigenvector and eigenvalue functionals. Consequently, the robustness and sensitivity properties of signal processing algorithms using the eigenvalue decomposition may be established.

Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process

Abstract: Consider the zero set of the random power series f(z)=sum a_n z^n with i.i.d. complex Gaussian coefficients a_n. We show that these zeros form a determinantal process: more precisely, their joint intensity can be written as a minor of the Bergman kernel. We show that the number of zeros of f in a disk of radius r about the origin has the same distribution as the sum of independent {0,1}-valued random variables X_k, where P(X_k=1)=r^{2k}. Moreover, the set of absolute values of the zeros of f has the same distribution as the set {U_k^{1/2k}} where the U_k are i.i.d. random variables uniform in [0,1]. The repulsion between zeros can be studied via a dynamic version where the coefficients perform Brownian motion; we show that this dynamics is conformally invariant.

Correlated MIMO Rayleigh fading channels: capacity and optimal signaling

Abstract: The capacity of the MIMO channel is investigated under the assumption that the elements of the channel matrix are zero mean proper complex Gaussian random variables with a general correlation structure. It is assumed that the receiver knows the channel perfectly but that the transmitter knows only the channel statistics. The analysis is carried out using an equivalent virtual representation of the channel that is obtained via a spatial discrete Fourier transform. The non-vanishing virtual channel coefficients are approximately independent and their variances correspond to samples of the underlying two-dimensional spatial scattering function that reflects the channel gain in different receive and transmit angular directions. It is shown that, in the virtual domain, the capacity achieving input vector is a zero mean proper complex Gaussian vector, with a diagonal covariance matrix /spl Lambda//spl deg/, whose components can be computed numerically. Furthermore, in the asymptotic regime of low signal-to-noise ratio (SNR), it is shown that only one element of /spl Lambda//spl deg/ is non-zero, i.e., that beamforming along one virtual transmit angle is asymptotically optimum. In the more general cast of arbitrary SNR, a necessary and sufficient condition for the optimality of beamforming is also derived.

A "Rayleigh-ness" test for DS/SS code acquisition

Abstract: Testing for statistical distribution has received increasing attention in signal processing for communications. Noncoherent receivers, based on correlation magnitude, are employed when the actual carrier phase of the received samples is unknown. This paper presents a new test, based on high-order statistics, to decide whether a real positive white series is a realization of a Rayleigh-distributed random process. Such a "Rayleigh-ness" test is based on a testing variable that measures the "Ricianity" of the series under investigation. That is, it estimates the possible presence of the mean of the complex Gaussian model generating both Rayleigh and Rice distributions. The asymptotic testing statistics have been derived as explicit functions of the higher order moments of the noise-plus-interference distribution. The performance of the Rayleigh-ness test has been analyzed in comparison with a conventional power detector. The devised test has application to noncoherent initial synchronization of the chip offset (code acquisition) in a symbol-length spreading sequence of direct-sequence code-division multiple-access systems. In such a case, the noise-plus-interference variance under the out-of-sync condition is much larger than the effective variance in the in-sync case. The obtained results evidence the robustness of the Rayleighness test, avoiding the severe performance degradation in the presence of large Gaussian and non-Gaussian noise, as well as multiuser interference in a downlink scenario.

Accurate error-rate calculations through the inversion of mixed characteristic functions

Abstract: This article presents a new computational tool for use in general fading channel analysis when the detection scheme can be expressed as a quadratic form in zero-mean complex Gaussian random variables. We develop a simple numerical algorithm which is capable of inverting a characteristic function consisting of both simple and multiple poles. The approach benefits from the inherent symmetry in the residue calculations and uses the well-known Vandermonde matrix in order to take advantage of this symmetry. It is numerically stable, eliminates singularities, and circumvents the need for differentiation.

On the distribution of peak-to-average power ratio for non-circularly modulated OFDM signals

Abstract: In this paper we investigate the distribution of the peak-to-average power ratio for non-circularly modulated OFDM system. In contrast to circularly modulated OFDM systems, we find the following characteristics for non-circularly modulated OFDM symbols: i) they do not converge to complex Gaussian sequence asymptotically; and ii) the correlation among OFDM symbol powers does not diminish as the number of subcarriers goes to infinity. We further derive, based on the above two observations, an expression for the distribution of peak-to-average power ratio for non-circular constellation. While not in closed form, the obtained expression allows easy numerical evaluation. For the special case of BPSK, the expression can be simplified to a form that admits closed-form approximation. We show that the analytical prediction matches well with empirical result obtained through simulation.

Blind symbol timing, frequency offset and carrier phase estimation in OFDM systems

Abstract: This paper deals with the problem of joint symbol timing, frequency offset and carrier phase estimation in orthogonal frequency division multiplexing (OFDM) systems with noncircular transmissions. In this case, for a sufficiently large number of subcarriers, the OFDM signal results to be an improper complex Gaussian process since its pseudoautocorrelation function is different from zero. By exploiting the joint probability density function for improper complex Gaussian processes, maximum-likelihood (ML) estimators for the parameters of interest are derived, and moreover, their performance is compared with that of previously proposed ML estimators for proper complex Gaussian processes.

Error analysis of space-time codes for slow Rayleigh fading channels

Abstract: A communication system which employs L/sub T/ transmit and L/sub R/ receive antennas is considered. The channel is assumed to be quasistatic Rayleigh flat fading. It is assumed that only the receiver has knowledge of the path gains. The additive noise at receiver j at symbol interval t, N/sup j//sub t/, is assumed to be complex Gaussian with i.i.d. real and imaginary parts. We consider the pairwise error probability (PEP) of space-time orthogonal block (STOB) codes, and then generalize the solution to space-time trellis (STT) codes, linear dispersion (LD) codes, and BLAST with ML decoding.

Cramer–Rao lower bounds on the angle‐of‐arrival estimates for a wave propagating in a random medium: Geometric acoustics regime

Abstract: In the geometric acoustics regime, a propagating wave is weakly diffracted and weakly scattered by the medium. The variance of the real component of the signal is much less than the variance of the imaginary component, thus the signal may not be modeled as a complex Gaussian random variable (whose real and imaginary components have equal variance), as is often done in the Rytov extension region, where both scattering and diffraction are strong. A statistical model for a signal in the geometric acoustics regime has been previously developed [S. L. Collier and D. K. Wilson, J. Acoust. Soc. Am. 111, 2379 (2002)] and its properties have been further investigated [S. L. Collier and D. K. Wilson, ASA April 2003 Meeting on Signal Processing (submitted)]. This statistical model is applied here to an acoustic wave propagating in a random medium with fluctuations described by von Karman’s spectrum. Additive white Gaussian noise is also considered. The correlation functions of the phase and log‐amplitude fluctuations for a von Karman spectrum are derived in the geometric acoustics limit. The Cramer–Rao lower bounds (CRLBs) on the angle‐of‐arrival estimates are calculated assuming multiple unknown parameters. The range dependence of the CRLBs is studied in detail.

Studies on the analytical propagation equations of Gaussian beams through a mult i-apertured imaging system of B=0

Abstract: By using the method of matrix decomposition and expanding the aperture function into a finite sum of complex Gaussian functions, the closed-form propagation equ ations of Gaussian beams through a multi-apertured imaging system of B=0 are der ived and are illustrated with a few typical numerical examples. The advantages o f our treatment are pointed out.