Noncentral Wishart Matrices Research Articles

Noncentral Wishart matrices, asymptotic normality of vec and smooth statistics

Wishart matrices play an important role in normal multivariate statistical analysis. In this work, we present an approach that has been already used for normal vectors and is now applied to noncentral Wishart matrices. We show that, under general conditions, the vec of the Wishart matrix and a large class of its statistics have asymptotic normal distributions when the norm of the noncentrality parameter diverges ∞. These statistics are called smooth and are given by functions whose component functions have continuous second-order partial derivatives in a neighbourhood of a ‘pivot’ point. Moreover, we derive the application domain of the asymptotic normal distributions for the vec of the Wishart matrix and its smooth statistics. Thus we have an attraction to the normal model, for the increasing predominance of noncentrality and not for increasing sample sizes. A simulation study shows that the threshold for the use of asymptotic normal distributions is quite acceptable.

Lower bound on the capacity of distributed MIMO systems with spatially correlated Rician/Lognormal fading

This paper presents an analytic lower bound of ergodic capacity for distributed MIMO (D-MIMO) systems that experience not only Rician fading but also shadowing effects of Lognormal distributed. In particularly, we consider that the Rician fading channel is spatially correlated at both transmitter and receiver. In the communication environment corresponding to this setting, the angle spread at the transmitter and the angle spread at the receiver are both insufficient, and the non-fading components co-exists with the fading components. Such communication environment is very common. In the process of deriving the analytic lower bound, in order to avoid Hayakawa polynomials that cannot be analytically expressed, non-central quadratic forms are transformed to non-central Wishart matrices by use of inequality. The validity of the presented lower bound is verified by computer simulations. The simulation results show the influence of the number of radio ports at the transmitter, the number of antennas at the transmitter or receiver, correlation level (angle spreads) and Rician K-factor on the capacity of systems. In all cases, the presented bound remains tight across the entire signal-to-noise ratio (SNR) regime.

Tightness of Jensen’s Bounds and Applications to MIMO Communications

Due to the difficulty in manipulating the distribution of Wishart random matrices, the performance analysis of multiple-input-multiple-output (MIMO) channels has mainly focused on deriving capacity bounds via Jensen's inequality. However, to the best of our knowledge, the tightness of Jensen's bounds has not yet been rigorously quantified in the general MIMO context. This paper proposes a new methodology for measuring the tightness of Jensen's bounds via the sandwich theorem. In particular, we first compare the tightness of two different pairs of upper/lower bounds for a general class of MIMO channels based on the unordered eigenvalue of the instantaneous correlation matrix and for arbitrary numbers of antennas. The tightness of Jensen's bounds in different channel scenarios is investigated including multiuser MIMO with maximal ratio combining. Our analysis is facilitated by deriving some new results for finite-dimensional Wishart matrices, i.e., for the arbitrary moments of the unordered eigenvalue of central and non-central Wishart matrices. Our results provide very interesting insights into the implications of the system parameters, such as the number of antennas, and signal-to-noise ratio, on the tightness of Jensen's bounds, and showcase the suitability and limitations of Jensen's bounds.

Some new results on the eigenvalues of complex non-central Wishart matrices with a rank-1 mean

Let W be an n × n complex non-central Wishart matrix with m ( ≥ n ) degrees of freedom and a rank-1 mean. In this paper, we consider three problems related to the eigenvalues of W . To be specific, we derive a new expression for the cumulative distribution function (c.d.f.) of the minimum eigenvalue ( λ min ) of W . The c.d.f. is expressed as the determinant of a square matrix, the size of which depends only on the difference m − n . This further facilitates the analysis of the microscopic limit of the minimum eigenvalue. The microscopic limit takes the form of the determinant of a square matrix with its entries expressed in terms of the modified Bessel functions of the first kind. We also develop a moment generating function based approach to derive the probability density function of the random variable tr ( W ) / λ min , where tr ( ⋅ ) denotes the trace of a square matrix. Moreover, we establish that, as m , n → ∞ with m − n fixed, tr ( W ) / λ min scales like n 3 . Finally, we find the average of the reciprocal of the characteristic polynomial det [ z I n + W ] , | arg z | < π , where I n and det [ ⋅ ] denote the identity matrix of size n and the determinant, respectively.

An Upper Bound on the Capacity of Distributed MIMO Systems With Rician/Lognormal Fading

In many communication environments for distributed MIMO (D-MIMO) systems, insufficient angular spread and dominant nonfading paths exist simultaneously. In such situations, a spatially correlated Rician/Lognormal channel model is appropriate. In this letter, analytic upper bound of ergodic capacity is presented for D-MIMO systems operating on Rician/Lognormal fading channels. In the analysis, it is assumed that receive correlation and transmit correlation exist simultaneously, and the rank of channel mean matrix is arbitrary. To avoid the appearance of Hayakawa polynomials, which cannot be simplified, the determinant in the expression of ergodic capacity is expanded by use of the linear sum of determinants of noncentral wishart matrices. The results of computer simulation show that the bound remains tight across the entire signal-to-noise ratio (SNR) regime.

Hydrodynamical spectral evolution for random matrices

The eigenvalues of the matrix structure where X is a random Gaussian Hermitian matrix and is non-random or random independent of X, are closely related to Dyson Brownian motion. Previous works have shown how an infinite hierarchy of equations satisfied by the dynamical correlations become triangular in the infinite density limit, and give rise to the complex Burgers equation for the Green’s function of the corresponding one-point density function. We show how this and analogous partial differential equations, for chiral, circular and Jacobi versions of Dyson Brownian motion follow from a macroscopic hydrodynamical description involving the current density and continuity equation. The method of characteristics gives a systematic approach to solving the PDEs, and in the chiral case we show how this efficiently reclaims the characterization of the global eigenvalue density for non-central Wishart matrices due to Dozier and Silverstein. Collective variables provide another approach to deriving the complex Burgers equation in the Gaussian case, and we show that this approach applies equally as well to chiral matrices. We relate both the Gaussian and chiral cases to the asymptotics of matrix integrals.

Zero-Forcing Precoding Performance in Multiuser MIMO Systems With Heterogeneous Ricean Fading

An accurate approximation is developed for the distribution of the instantaneous per-terminal signal-to-noise-ratio (SNR) of a downlink multiuser multiple-input multiple-output system with zero-forcing (ZF) precoding. Our analysis assumes a Ricean fading environment, where we show that the SNR at a given terminal is well approximated by the gamma distribution and we derive its parameters. The analysis relies on densities of an arbitrary eigenvalue and a pair of arbitrary eigenvalues of the uncorrelated complex non-standard, noncentral Wishart matrices. Unlike previous studies, we consider microwave and millimeter-wave channel parameters with a unique Rice factor for each terminal. We demonstrate that stronger line-of-sight adversely impacts the ZF SNR, while increasing the Rice factor variability results in higher peak ZF SNR. Our approximations are insensitive to changes in the system dimension and operating SNRs.

POLARIMETRIC TARGET DETECTION USING STATISTIC OF THE DEGREE OF POLARIZATION

The degree of polarization (DoP) can be utilized as a detection statistic in the polarimetric radar to achieve target detection performance improvement. In this paper, a polarimetric radar model is established, which includes reflections from both target and clutter at first. Then, probability density functions (PDFs) of the estimated DoP are expressed in closed form, which is derived from joint eigenvalue distributions of complex noncentral Wishart matrices. The detector is developed and evaluated theoretically on the basis of the statistical properties of the DoP. Finally, a comparison between the new DoP detector and single-polarization detector is presented against real data. The performance improvement is demonstrated by the comparison results.

Connections Between the Generalized Marcum $Q$-Function and a Class of Hypergeometric Functions

This paper presents a new connection between the generalized Marcum-Q function and the confluent hypergeometric function of two variables, phi3. This result is then applied to the closed-form characterization of the bivariate Nakagami-m distribution and of the distribution of the minimum eigenvalue of correlated non-central Wishart matrices, both important in communication theory. New expressions for the corresponding cumulative distributions are obtained and a number of communication-theoretic problems involving them are pointed out.

On the condition number distribution of complex wishart matrices

This paper investigates the distribution of the condition number of complex Wishart matrices. Two closely related measures are considered: the standard condition number (SCN) and the Demmel condition number (DCN), both of which have important applications in the context of multiple-input multiple-output (MIMO) communication systems, as well as in various branches of mathematics. We first present a novel generic framework for the SCN distribution which accounts for both central and non-central Wishart matrices of arbitrary dimension. This result is a simple unified expression which involves only a single scalar integral, and therefore allows for fast and efficient computation. For the case of dual Wishart matrices, we derive new exact polynomial expressions for both the SCN and DCN distributions. We also formulate a new closed-form expression for the tail SCN distribution which applies for correlated central Wishart matrices of arbitrary dimension and demonstrates an interesting connection to the maximum eigenvalue moments of Wishart matrices of smaller dimension. Based on our analytical results, we gain valuable insights into the statistical behavior of the channel conditioning for various MIMO fading scenarios, such as uncorrelated/semi-correlated Rayleigh fading and Ricean fading.

On analytical derivations of the condition number distributions of dual non-central Wishart matrices

In this paper, we explore the statistical characterization of Multiple-Input Multiple-Output (MIMO) channel correlation matrices with the main focus being on their condition number statistics. More specifically, novel expressions are derived for the probability density function (PDF) and cumulative distribution function (CDF) of the MIMO condition number. Contrary to the majority of related studies, where only the common case of Rayleigh fading was considered, our investigation is extended to account for the generalized case of Ricean fading where a deterministic Line-Of-Sight (LoS) component exists in the communication link. The overall analysis is based on the principles of random matrix theory and particularly of dual complex non-central Wishart matrices; the latter represent a practical class of MIMO systems, namely dual-branch systems which are equipped with two transmit and receive antenna elements. All the derived formulae are validated through extensive simulations with the attained accuracy being remarkably good.

Analytical Performance of MIMO-SVD Systems in Ricean Fading Channels with Channel Estimation Error and Feedback Delay

This paper analyzes bit error rate (BER) and outage probability of singular value decomposition-based multiple-input multiple-output systems with channel estimation error and feedback delay over uncorrelated Ricean fading channels. By utilizing marginal unordered and ordered eigenvalue distributions of complex noncentral Wishart matrices, we derive exact closed-form expressions on the average system performance and high signal- to-interference-plus-noise ratio (SINR) approximations on the individual eigen-subchannels, respectively, under the assumption of equal power allocation. Our expressions apply for various modulation formats and arbitrary numbers of transmit and receive antennas. Our results show that in low-to-moderate SINR regimes, both the BER and the outage probability increase with channel estimation error, feedback delay and the Ricean K-factor at a polynomial rate that is inversely proportional to the difference between the numbers of transmit and receive antennas. We also show that, with channel estimation error and feedback delay, the diversity orders of the BER and outage probability are zero and an irreducible error floor exists at high SINR.

MIMO multichannel beamforming: SER and outage using new eigenvalue distributions of complex noncentral Wishart matrices

This paper analyzes MIMO systems with multichannel beamforming in Ricean fading. Our results apply to a wide class of multichannel systems which transmit on the eigenmodes of the MIMO channel. We first present new closed-form expressions for the marginal ordered eigenvalue distributions of complex noncentral Wishart matrices. These are used to characterize the statistics of the signal to noise ratio (SNR) on each eigenmode. Based on this, we present exact symbol error rate (SER) expressions. We also derive closed-form expressions for the diversity order, array gain, and outage probability. We show that the global SER performance is dominated by the subchannel corresponding to the minimum channel singular value. We also show that, at low outage levels, the outage probability varies inversely with the Ricean A*-factor for cases where transmission is only on the most dominant subchannel (i.e. a singlechannel beamforming system). Numerical results are presented to validate the theoretical analysis.

Simple Approximation of Largest Eigenvalue Distribution in MIMO Channels Under Nakagami-Rice Fading

This paper presents an approximation method of statistical distribution of the largest eigenvalue of i.i.d. (independent and identically distributed) MIMO (multiple input multiple output) channel correlation matrices under Nakagami-Rice fading environment. The equation is actually derived for MIMO Nakagami m fading channel which is known as a good approximation of Nakagami-Rice fading, hence it well approximates the curves of the largest eigenvalue distribution of noncentral Wishart matrices. In the proposed approximation method, MIMO MRC (maximal ratio combining) system is ascribed to SIMO space diversity theory with the same number of branches, and the statistical distribution becomes a monomial gamma distribution. As a result, the derived marginal density function does not contain any special functions, and has a simple monomial gamma form which is suitable for various calculations of performance indices. Through computer simulations, it is shown that the proposed approximation formula is effective and has a better precision than conventional method.

On the Ergodic Capacity of Rank-$1$ Ricean-Fading MIMO Channels

This paper investigates the ergodic capacity of Ricean-fading multiple-input-multiple-output (MIMO) channels with rank-1 mean matrices under the assumption that the channel is unknown at the transmitter and perfectly known at the receiver. After introducing the system model and the concept of ergodic capacity of MIMO channels, we derive the explicit expressions for the expected values of the determinant and log-determinant of complex noncentral Wishart matrices. Subsequently, we obtain new upper and lower bounds on the ergodic capacity of rank-1 Ricean-fading MIMO channels at any signal-to-noise ratio (SNR). We show that our bounds are tighter than previously reported analytical bounds, and discuss the impact of spatial fading correlation and Ricean K-factor with the help of these bounds. Furthermore, we extend the analysis of ergodic capacity to frequency selective spatially correlated Ricean-fading MIMO channels. We demonstrate that the calculation of ergodic capacity of frequency selective fading MIMO channels can be converted to the calculation of the one of equivalent frequency flat-fading MIMO channels. Finally, we present numerical results that confirm the theoretical analysis

An upper bound on the ergodic mutual information in Rician fading MIMO channels

We consider Rician fading multiple-input multiple-output (MIMO) channels where the transmitted signal has complex Gaussian distribution, iid across the transmit antennas. Based on expected values of elementary symmetric functions of complex noncentral Wishart matrices, we derive an upper bound on the average (ergodic) mutual information for arbitrary SNR, arbitrary rank of the deterministic line-of-sight matrix, and arbitrary number of transmit/receive antennas. The Rayleigh fading signal component is allowed to have spatial correlation at one end of the link. Upper bounds for the cases of rank-1 line-of-sight component and pure Rayleigh fading emerge as special instances of the general result

Complex random matrices and Rician channel capacity

Eigenvalue densities of complex noncentral Wishart matrices are investigated to study an open problem in information theory. Specifically, the largest, smallest, and joint eigenvalue densities of complex noncentral Wishart matrices are derived. These densities are expressed in terms of complex zonal polynomials and invariant polynomials. A connection between the complex Wishart matrix theory and information theory is given. This facilitates evaluation of the most important information-theoretic measure, the so-called ergodic channel capacity. In particular, the capacity of multiple-input multiple-output (MIMO) Rician distributed channels is investigated. We consider both spatially correlated and uncorrelated MIMO Rician channels and derive exact and easily computable tight upper bound formulas for ergodic capacities. Numerical results are also given, which show how the channel correlation degrades the capacity of the communication system.

Some properties of invariant polynomials with matrix arguments and their applications in econometrics

Further properties are derived for a class of invariant polynomials with several matrix arguments which extend the zonal polynomials. Generalized Laguerre polynomials are defined, and used to obtain expansions of the sum of independent noncentral Wishart matrices and an associated generalized regression coefficient matrix. The latter includes thek-class estimator in econometrics.

On approximating the non-central wishart distribution by central wishart distribution a monte carlo study

This paper provides a Monte Carlo study of approximating the non-central Wishart distribution by Central Wishart distribution by mean of 1. Multivariate Gram-Chalier expansion and 2. Laguerre polynomial expansion. For assessing the closeness of these approximations, 1,000 independent 2×2 non-central Wishart matrices are generated by computer. The numerical results indicate that the multivariate Gram-Chalier expansion provides a close approximation to the non-central Wishart distribution as long as the correlation coefficient is less than 0.8. Also, it appears that the Gram-Chalier expansion approximation is better than the Laguerre polynomial expansion approximation when the probability values are large.

Asymptotic expansions for the distributions of latent roots ofS h S e −1 and of certain test statistics in MANOVA

S e andS n are independent central and noncentral Wishart matrices having Wishart distributionsW p (n e , Σ) andW p (n h , Σ; Ω) respectively. Asymptotic expansions are given for the distributions of latent roots ofS h S −1 and of certain test statistics in MANOVA under the assumption thatn=n e +n h becomes large with a fixed ration e ∶n h =e∶h(e+h=1,e>0,h>0) andΩ=O(n).