Complex Random Variables Research Articles

On a bound of Hoeffding in the complex case

It was proved by Hoeffding in 1963 that a real random variable X confined to [a, b] satisfies E e^(X--E X) $\le$ e^((b--a)^2/8). We generalise this to complex random variables.

Dependence among complex random variables as a fuel cell condition indicator

As various faults alter the PEM fuel cell impedance characteristic over a broad frequency range, the electrochemical impedance spectroscopy is frequently employed for the purpose of condition monitoring. The proposed methodology treats the impedance components among different frequencies as dependent complex random variables. The information about fuel cell condition is incorporated into the dependence structure of these complex random variables. This dependence is described through the corresponding joint cumulative density function by employing copula functions. The benefits of such an approach are threefold: (i) the estimation of the joint cumulative density function requires only several measurements of a fuel cell in a fault-free condition, (ii) the procedure is computationally efficient, and (iii) the output of the copula function is directly used as an overall unit-free condition indicator. The approach was evaluated on a kW-range PEM fuel cell stack subjected to water management faults of various severities. The results show that the ci corresponds with the severity of the induced faults.

The exact and near-exact distributions of the main likelihood ratio test statistics used in the complex multivariate normal setting

In this paper the authors show how it is possible to establish a common structure for the exact distribution of the main likelihood ratio test (LRT) statistics used in the complex multivariate normal setting. In contrast to what happens when dealing with real random variables, for complex random variables it is shown that it is possible to obtain closed-form expressions for the exact distributions of the LRT statistics to test independence, equality of mean vectors and the equality of an expected value matrix to a given matrix. For the LRT statistics to test sphericity and the equality of covariance matrices, cases where the exact distribution has a non-manageable expression, easy to implement and very accurate near-exact distributions are developed. Numerical studies show how these near-exact distributions outperform by far any other available approximations. As an example of application of the results obtained, the authors develop a near-exact approximation for the distribution of the LRT statistic to test the equality of several complex normal distributions.

Universality for random tensors

Nous démontrons deux théorèmes d’universalité pour les tenseurs aléatoires de rang $D$ quelconque. Nous montrons d’abord qu’un tenseur aléatoire dont les entrées sont $N^{D}$ variables complexes indépendantes identiquement distribuées converge en distribution dans la limite $N$ grand vers la même limite que la limite en distribution d’un modèle de tenseurs Gaussien. Cela généralise l’universalité des matrices aléatoires aux tenseurs aléatoires. Nous démontrons ensuite un deuxième théorème d’universalité, plus fort. Sous l’hypothèse que la distribution de probabilité jointe des entrées du tenseur est invariante, et en supposant que les cumulants de cette distribution invariante sont uniformément bornés, nous prouvons que dans la limite $N$ grand le tenseur converge à nouveau en distribution vers la même limite que la limite en distribution d’un modèle de tenseurs Gaussien. La covariance de la distribution Gaussienne à $N$ grand n’est pas universelle, mais dépend des détails de la distribution jointe.

A Locally Deterministic, Detector-Based Model of Quantum Measurement

This paper describes a simple, causally deterministic model of quantum measurement based on an amplitude threshold detection scheme. Surprisingly, it is found to reproduce many phenomena normally thought to be uniquely quantum in nature. To model an $N$-dimensional pure state, the model uses $N$ complex random variables given by a scaled version of the wave vector with additive complex noise. Measurements are defined by threshold crossings of the individual components, conditioned on single-component threshold crossings. The resulting detection probabilities match or approximate those predicted by quantum mechanics according to the Born rule. Nevertheless, quantum phenomena such as entanglement, contextuality, and violations of Bell's inequality under local measurements are all shown to be exhibited by the model, thereby demonstrating that such phenomena are not without classical analogs.

Moments Tensors, Hilbert's Identity, and k-wise Uncorrelated Random Variables

In this paper, we introduce a notion to be called k-wise uncorrelated random variables, which is similar but not identical to the so-called k-wise independent random variables in the literature. We show how to construct k-wise uncorrelated random variables by a simple procedure. The constructed random variables can be applied, e.g., to express the quartic polynomial (xTQx)2, where Q is an n × n positive semidefinite matrix, by a sum of fourth powered polynomial terms, known as Hilbert's identity. By virtue of the proposed construction, the number of required terms is no more than 2n4 + n. This implies that it is possible to find a (2n4 + n)-point distribution whose fourth moments tensor is exactly the symmetrization of Q ⊗ Q. Moreover, we prove that the number of required fourth powered polynomial terms to express (xTQx)2 is at least n(n + 1)/2. The result is applied to prove that computing the matrix 2 ↦ 4 norm is NP-hard. Extensions of the results to complex random variables are discussed as well.

Outage and BER Performance Analysis of Cascade Channel in Relay Networks

In this paper, the outage and bit error rate (BER) performance of a relay networks is studied based on the cascade channel. Firstly, the distribution of the cascade channel, which is the integration of L double-Rayleigh complex random variables (RV), is derived in close form. To overcome the computational difficulty of the arising special function, the probability density function (pdf) of the channel is approximated by the concise Gamma distribution. Through matching the moments of both pdfs, the corresponding shape and scale parameters of the Gamma pdf are obtained. Based on the pdf approximation, the closed-form and accurate expressions of the metrics: outage probability and BER of the cascade channel, are deduced. Numerical results illustrate the accuracy of the proposed performance evaluation.

Widely linear general Kalman filter for stereophonic acoustic echo cancellation

The stereophonic acoustic echo cancellation (SAEC) problem is usually modelled as a two-input/two-output system with real random variables. Recently, the SAEC scheme was recast as a single-input/single-output system with complex random variables, thanks to the widely linear (WL) model. In this paper, we motivate the use of a more general form of the Kalman filter with the WL model for SAEC. Simulation results indicate that this algorithm outperforms the recursive least-squares (RLS) algorithm, which is usually considered as the benchmark for SAEC.

Local Marchenko-Pastur law at the hard edge of sample covariance matrices

Let XN be a N × N matrix whose entries are independent identically distributed complex random variables with mean zero and variance \documentclass[12pt]{minimal}\begin{document}$\frac{1}{N}$\end{document}1N. We study the asymptotic spectral distribution of the eigenvalues of the covariance matrix \documentclass[12pt]{minimal}\begin{document}$X_N^*X_N$\end{document}XN*XN for N → ∞. We prove that the empirical density of eigenvalues in an interval [E, E + η] converges to the Marchenko-Pastur law locally on the optimal scale, \documentclass[12pt]{minimal}\begin{document}$N \eta /\sqrt{E} \gg (\log N)^b$\end{document}Nη/E≫(logN)b, and in any interval up to the hard edge, \documentclass[12pt]{minimal}\begin{document}$\frac{(\log N)^b}{N^2}\lesssim E \le 4-\kappa$\end{document}(logN)bN2≲E≤4−κ, for any κ > 0. As a consequence, we show the complete delocalization of the eigenvectors.

The distribution of eigenvalues of randomized permutation matrices

In this article we study in detail a family of random matrix ensembles which are obtained from random permutations matrices (chosen at random according to the Ewens measure of parameter � > 0) by replacing the entries equal to one by more general non- vanishing complex random variables. For these ensembles, in contrast with more classical models as the Gaussian Unitary Ensemble, or the Circular Unitary Ensemble, the eigen- values can be very explicitly computed by using the cycle structure of the permutations. Moreover, by using the so-called virtual permutations, first introduced by Kerov, Olshanski and Vershik, and studied with a probabilistic point of view by Tsilevich, we are able to de- fine, on the same probability space, a model for each dimension greater than or equal to one, which gives a meaning to the notion of almost sure convergence when the dimension tends to infinity. In the present paper, depending on the precise model which is considered, we obtain a number of different results of convergence for the point measure of the eigenvalues, some of these results giving a strong convergence, which is not common in random matrix theory.

A widely linear model for stereophonic acoustic echo cancellation

The stereophonic acoustic echo, due to the coupling between two loudspeakers and two microphones, can be modelled by a two-input/two-output system with real random variables. In this paper, we recast the problem as a single-input/single-output system with complex random variables, by using the widely linear (WL) model, and propose a new distortion method that fits well in this context. In order to illustrate the behavior of this scheme, the recursive least-squares (RLS)-dichotomous coordinate descent (DCD) algorithm is used. Experimental results indicate that the RLS-DCD algorithm represents an attractive choice for this application since it has good numerical features in terms of stability and complexity.

On Complex Random Variables

In this paper, it is shown that a complex multivariate random variable is a complex multivariate normal random variable of dimensionality if and only if all nondegenerate complex linear combinations of have a complex univariate normal distribution. The characteristic function of has been derived, and simpler forms of some theorems have been given using this characterization theorem without assuming that the variance-covariance matrix of the vector is Hermitian positive definite. Marginal distributions of have been given. In addition, a complex multivariate t-distribution has been defined and the density derived. A characterization of the complex multivariate t-distribution is given. A few possible uses of this distribution have been suggested.

The distribution of the amplitude and phase of the mean of a sample of complex random variables

Edgeworth-type expansions are given for the distribution of (normalized versions of) the amplitude and phase of the mean of a sample of complex random variables. These expansions are transformed to polar forms with applications to modeling signals from a cell-phone. Limiting distributions of (normalized versions of) the amplitude and phase of the mean are given for the cases: (1) population mean is zero, and (2) population mean is non-zero. The results apply for distributions with finite cumulants.

On Approximate Diagonalization of Correlation Matrices in Widely Linear Signal Processing

The so called “augmented” statistics of complex random variables has established that both the covariance and pseudocovariance are necessary to fully describe second order properties of noncircular complex signals. To jointly decorrelate the covariance and pseudocovariance matrix, the recently proposed strong uncorrelating transform (SUT) requires two singular value decompositions (SVDs). In this correspondence, we further illuminate the structure of these matrices and demonstrate that for univariate noncircular data it is sufficient to diagonalize the pseudocovariance matrix-this ensures that the covariance matrix is also approximately diagonal. The proposed approach is shown to result in lower computational complexity and enhanced numerical stability, and to enable elegant new formulations of performance bounds in widely linear signal processing. The analysis is supported by illustrative case studies and simulation examples.

Spectrum of Non-Hermitian Heavy Tailed Random Matrices

Let (X jk ) j,k ≥ 1 be i.i.d. complex random variables such that |X jk | is in the domain of attraction of an α-stable law, with 0 < α < 2. Our main result is a heavy tailed counterpart of Girko’s circular law. Namely, under some additional smoothness assumptions on the law of X jk , we prove that there exist a deterministic sequence a n ~ n 1/α and a probability measure μ α on $${\mathbb{C}}$$ depending only on α such that with probability one, the empirical distribution of the eigenvalues of the rescaled matrix $${(a_n^{-1}X_{jk})_{1\leq j,k\leq n}}$$ converges weakly to μ α as n → ∞. Our approach combines Aldous & Steele’s objective method with Girko’s Hermitization using logarithmic potentials. The underlying limiting object is defined on a bipartized version of Aldous’ Poisson Weighted Infinite Tree. Recursive relations on the tree provide some properties of μ α . In contrast with the Hermitian case, we find that μ α is not heavy tailed.

On the convergence of random polynomials and multilinear forms

We consider different kinds of convergence of homogeneous polynomials and multilinear forms in random variables. We show that for a variety of complex random variables, the almost sure convergence of the polynomial is equivalent to that of the multilinear form, and to the square summability of the coefficients. Also, we present polynomial Khintchine inequalities for complex gaussian and Steinhaus variables. All these results have no analogues in the real case. Moreover, we study the L p -convergence of random polynomials and derive certain decoupling inequalities without the usual tetrahedral hypothesis. We also consider convergence on “full subspaces” in the sense of Sjögren, both for real and complex random variables, and relate it to domination properties of the polynomial or the multilinear form, establishing a link with the theory of homogeneous polynomials on Banach spaces.

Essential Statistics and Tools for Complex Random Variables

Complex random signals play an increasingly important role in array, communications, and biomedical signal processing and related fields. However, the fundamental properties of complex-valued signals and mathematical tools needed to process them are scattered in literature. We provide a concise, unified, and rigorous treatment of essential properties and tools of complex random variables, and apply these fundamentals to derive complex extensions of Leibniz rule, Faa di Bruno's formula, and Taylor's series. The extensions allow establishing relationships among complex moments and cumulants, and characterizing the circularity property. We propose measures for testing and quantifying circularity, and observe that non-circularity may be more common in practical applications than previously thought. All results are rigorously proved and supplemented with clarifying examples.

Complex Independent Component Analysis by Entropy Bound Minimization

We first present a new (differential) entropy estimator for complex random variables by approximating the entropy estimate using a numerically computed maximum entropy bound. The associated maximum entropy distributions belong to the class of weighted linear combinations and elliptical distributions, and together, they provide a rich array of bivariate distributions for density matching. Next, we introduce a new complex independent component analysis (ICA) algorithm, complex ICA by entropy-bound minimization (complex ICA-EBM), using this new entropy estimator and a line search optimization procedure. We present simulation results to demonstrate the superior separation performance and computational efficiency of complex ICA-EBM in separation of complex sources that come from a wide range of bivariate distributions.

Including Spatial Correlations in the Statistical MIMO Radar Target Model

Previous studies of statistical MIMO radar detection performance have used a target model that consists of a large number of point scatterers located within a rectangular target area. These point scatterers have scattering amplitudes that are complex random variables and are spatially uncorrelated, so that the target is a white noise process in space. Spatial correlations are introduced into the target model in this paper, and the impact of these correlations on MIMO radar system detection performance is analyzed.

The limiting spectral distribution of the product of the Wigner matrix and a nonnegative definite matrix

Let W n be n × n Hermitian whose entries on and above the diagonal are independent complex random variables satisfying the Lindeberg type condition. Let T n be n × n nonnegative definitive and be independent of W n . Assume that almost surely, as n → ∞ , the empirical distribution of the eigenvalues of T n converges weakly to a non-random probability distribution. Let A n = n − 1 / 2 T n 1 / 2 W n T n 1 / 2 . Then with the aid of the Stieltjes transforms, we show that almost surely, as n → ∞ , the empirical distribution of the eigenvalues of A n also converges weakly to a non-random probability distribution, a system of two equations determining the Stieltjes transform of the limiting distribution. Important analytic properties of this limiting spectral distribution are then derived by means of those equations. It is shown that the limiting spectral distribution is continuously differentiable everywhere on the real line except only at the origin and that a necessary and sufficient condition is available for determining its support. At the end, the density function of the limiting spectral distribution is calculated for two important cases of T n , when T n is a sample covariance matrix and when T n is the inverse of a sample covariance matrix.