Spectral distribution of the exponentially windowed sample covariance matrix (original) (raw)
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Limiting spectral distribution of the sample covariance matrix of the windowed array data
EURASIP Journal on Advances in Signal Processing, 2013
In this article, we investigate the limiting spectral distribution of the sample covariance matrix (SCM) of weighted/windowed complex data. We use recent advances in random matrix theory and describe the distribution of eigenvalues of the doubly correlated Wishart matrices. We obtain an approximation for the spectral distribution of the SCM obtained from windowed data. We also determine a condition on the coefficients of the window, under which the fragmentation of the support of noise eigenvalues can be avoided, in the noise-only data case. For the commonly used exponential window, we derive an explicit expression for the l.s.d of the noise-only data. In addition, we present a method to identify the support of eigenvalues in the general case of signal-plus-noise. Simulations are performed to support our theoretical claims. The results of this article can be directly employed in many applications working with windowed array data such as source enumeration and subspace tracking algorithms.
IEEE Access, 2019
This paper addresses the asymptotic behavior of a particular type of information-plusnoise-type matrices, where the column and row number of the matrices are large and of the same order, while signals are diverged and time delays of the channel are fixed. We prove that the empirical spectral distribution (ESD) of the large dimension sample covariance matrix and a well-studied spiked central Wishart matrix converge to the same distribution. As an application, an asymptotic power function is presented for the general likelihood ratio statistics for testing the presence of signal in large array signal processing.
Analysis of the Limiting Spectral Distribution of Large Dimensional Random Matrices
Journal of Multivariate Analysis, 1995
Results on the analytic behavior of the limiting spectral distribution of matrices of sample covariance type, studied in Marčenko and Pastur [2] and Yin [8], are derived. Through an equation defining its Stieltjes transform, it is shown that the limiting distribution has a continuous derivative away from zero, the derivative being analytic wherever it is positive, and resembles |x − x 0 | for most cases of x 0 in the boundary of its support. A complete analysis of a way to determine its support, originally outlined in Marčenko and Pastur [2], is also presented.
CLT for linear spectral statistics of large-dimensional sample covariance matrices
The Annals of Probability, 2004
The limiting behavior, as n → ∞ with n/N approaching a positive constant, of functionals of the eigenvalues of B n , where each is given equal weight, is studied. Due to the limiting behavior of the empirical spectral distribution of B n , it is known that these linear spectral statistics converges a.s. to a nonrandom quantity. This paper shows their rate of convergence to be 1/n by proving, after proper scaling, that they form a tight sequence. Moreover, if EX 2 11 = 0 and E|X 11 | 4 = 2, or if X 11 and T n are real and EX 4 11 = 3, they are shown to have Gaussian limits.
Journal of Multivariate Analysis, 2007
A derivation of results on the analytic behavior of the limiting spectral distribution of sample covariance matrices of the "information-plus-noise" type, as studied in Dozier and Silverstein [3], is presented. It is shown that, away from zero, the limiting distribution possesses a continuous density. The density is analytic where it is positive and, for the most relevant cases of a in the boundary of its support, exhibits behavior closely resembling that of |x − a| for x near a. A procedure to determine its support is also analyzed.
2021
Given a random matrix X = (x1, . . . , xn) ∈ Mp,n with independent columns and satisfying concentration of measure hypotheses and a parameter z whose distance to the spectrum of 1 n XXT should not depend on p, n, it was previously shown that the functionals Tr(AR(z)), for R(z) = ( 1 n XXT −zIp) and A ∈ Mp deterministic, have a standard deviation of order O(‖A‖∗/ √ n). Here, we show that ‖E[R(z)] − R̃(z)‖F ≤ O(1/ √ n), where R̃(z) is a deterministic matrix depending only on z and on the means and covariances of the column vectors x1, . . . , xn (that do not have to be identically distributed). This estimation is key to providing accurate fluctuation rates of functionals of X of interest (mostly related to its spectral properties) and is proved thanks to the introduction of a semi-metric ds defined on the set Dn(H) of diagonal matrices with complex entries and positive imaginary part and satisfying, for all D,D′ ∈ Dn(H): ds(D,D) = maxi∈[n] |Di −D′ i|/(I(Di)I(D i)). Possibly most impor...
arXiv (Cornell University), 2023
1/2 n Xn) * , where Xn is a p × n matrix with independent standardized random variables, Rn is a p × n nonrandom matrix and Tn is a p ×p non-random, nonnegative definite Hermitian matrix. The matrix Bn is referred to as the information-plus-noise type matrix, where Rn contains the information and T 1/2 n Xn is the noise matrix with the covariance matrix Tn. It is known that, as n → ∞, if p/n converges to a positive number, the empirical spectral distribution of Bn converges almost surely to a nonrandom limit, under some mild conditions. In this paper, we prove that, under certain conditions on the eigenvalues of Rn and Tn, for any closed interval outside the support of the limit spectral distribution, with probability one there will be no eigenvalues falling in this interval for all n sufficiently large. u 1+yg − (1 + yst)z + t(1 − y) .
Eigenvalue density of correlated complex random Wishart matrices
Physical Review E, 2004
Using a character expansion method, we calculate exactly the eigenvalue density of random matrices of the form M † M where M is a complex matrix drawn from a normalized distribution P͑M͒ϳexp͑−Tr͕AMBM † ͖͒ with A and B positive definite (square) matrices of arbitrary dimensions. Such so-called correlated Wishart matrices occur in many fields ranging from information theory to multivariate analysis.
arXiv (Cornell University), 2023
In this paper, we derive the analytical behavior of the limiting spectral distribution of non-central covariance matrices of the "general information-plus-noise" type, as studied in [1]. Through the equation defining its Stieltjes transform, it is shown that the limiting distribution has a continuous derivative away from zero, the derivative being analytic wherever it is positive, and we show the determination criterion for its support. We also extend the result in [1] to allow for all possible ratios of row to column of the underlying random matrix.
Limiting empirical spectral distribution for products of rectangular matrices
Journal of Mathematical Analysis and Applications, 2021
In this paper, we consider m independent random rectangular matrices whose entries are independent and identically distributed standard complex Gaussian random variables and assume the product of the m rectangular matrices is an n by n square matrix. We study the limiting empirical spectral distributions of the product where the dimension of the product matrix goes to infinity, and m may change with the dimension of the product matrix and diverge. We give a complete description for the limiting distribution of the empirical spectral distributions for the product matrix and illustrate some examples.