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L M Tavares - One of the best experts on this subject based on the ideXlab platform.
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on the statistics of the sum of squared Complex gaussian Random Variables
IEEE Transactions on Communications, 2007Co-Authors: Goncalo Tavares, L M TavaresAbstract:In this letter, we derive new results for the statistics of the Complex Random Variable z=DeltaSigman=1 N x n 2=z 1+jz Q = re jphi where {x n} is a set of mutually independent Complex-valued Gaussian Random Variables with arbitrary means and equal variances. Each Random Variable x n is assumed to have independent real and imaginary components with equal variance for all n. Expressions are derived for the joint probability density function (pdf) of (z 1,z Q), for the joint pdf of (r,phi) and also for the marginal pdf of the modulus r. An useful Fourier series expansion for the pdf of the phase phi is also derived. As an application of the results, a theoretical performance analysis of the well-known nondata-aided Viterbi and Viterbi feedforward carrier phase estimator operating with BPSK signals is presented. In particular, the expressions for the exact pdf, variance, and equivocation probability of the carrier phase estimates are derived.
Jain Vishesh. - One of the best experts on this subject based on the ideXlab platform.
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Quantitative invertibility of Random matrices : a combinatorial perspective
Massachusetts Institute of Technology, 2020Co-Authors: Jain Vishesh.Abstract:Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mathematics, May, 2020Cataloged from the official PDF of thesis.Includes bibliographical references (pages 101-106).In this thesis, we develop a novel framework for investigating the lower tail behavior of the least singular value of Random matrices - a subject which has been intensely studied in the past two decades. Our focus is on obtaining high probability bounds, rather than on estimating the least singular value of a 'typical' realisation of the Random matrix. In our main application, we consider Random matrices of the form Mn := M + Nn, where M is a fixed Complex matrix with operator norm at most exp(Nc), and Nn is a Random matrix, each of whose entries is an independent copy of a Complex Random Variable with mean 0 and variance 1. This setting, with some additional restrictions, has been previously considered in a series of influential works by Tao and Vu, most notably in connection with the strong circular law, and the smoothed analysis of the condition number, and our results extend and improve upon theirs in a couple of ways. As opposed to all previous works obtaining such bounds with error rate better than n-1, our proof makes no use either of the inverse Littlewood-Offord theorems, or of any sophisticated net constructions. Instead, we show how to reduce the optimization problem characterizing the smallest singular value from the (Complex) sphere to (Gaussian) integer vectors, where it is solved using direct combinatorial arguments.by Vishesh Jain.Ph. D.Ph.D. Massachusetts Institute of Technology, Department of Mathematic
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The strong circular law: a combinatorial view
2020Co-Authors: Jain Vishesh.Abstract:Let $N_n$ be an $n\times n$ Complex Random matrix, each of whose entries is an independent copy of a centered Complex Random Variable $z$ with finite non-zero variance $\sigma^{2}$. The strong circular law, proved by Tao and Vu, states that almost surely, as $n\to \infty$, the empirical spectral distribution of $N_n/(\sigma\sqrt{n})$ converges to the uniform distribution on the unit disc in $\mathbb{C}$. A crucial ingredient in the proof of Tao and Vu, which uses deep ideas from additive combinatorics, is controlling the lower tail of the least singular value of the Random matrix $xI - N_{n}/(\sigma\sqrt{n})$ (where $x\in \mathbb{C}$ is fixed) with failure probability that is inverse polynomial. In this paper, using a simple and novel approach (in particular, not using tools from additive combinatorics or any net arguments), we show that for any fixed matrix $M$ with operator norm at most $n^{0.51}$ and for all $\eta \geq 0$, $$\Pr\left(s_n(M+N_n) \leq \eta \right) \lesssim n^{C}\eta + \exp(-n^{c}),$$ where $s_n(M+N_n)$ is the least singular value of $M+N_n$ and $C,c$ are absolute constants. Our result is optimal up to the constants $C,c$ and the inverse exponential-type error rate improves upon the inverse polynomial error rate due to Tao and Vu. During the course of our proof, we extend the solution of the counting problem in inverse Littlewood-Offord theory, recently isolated by the author along with Ferber, Luh, and Samotij, from Rademacher Variables to general Complex Random Variables. This significantly improves on estimates for this problem obtained using the optimal inverse Littlewood-Offord theorem of Nguyen and Vu, and may be of independent interest.Comment: This version: extension to Complex case, application to the strong circular law. Comments welcome
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Quantitative invertibility of Random matrices: a combinatorial perspective
2020Co-Authors: Jain Vishesh.Abstract:We study the lower tail behavior of the least singular value of an $n\times n$ Random matrix $M_n := M+N_n$, where $M$ is a fixed Complex matrix with operator norm at most $\exp(n^{c})$ and $N_n$ is a Random matrix, each of whose entries is an independent copy of a Complex Random Variable with mean $0$ and variance $1$. Motivated by applications, our focus is on obtaining bounds which hold with extremely high probability, rather than on the least singular value of a typical such matrix. This setting has previously been considered in a series of influential works by Tao and Vu, most notably in connection with the strong circular law, and the smoothed analysis of the condition number, and our results improve upon theirs in two ways: (i) We are able to handle $\|M\| = O(\exp(n^{c}))$, whereas the results of Tao and Vu are applicable only for $M = O(\text{poly(n)})$. (ii) Even for $M = O(\text{poly(n)})$, we are able to extract more refined information -- for instance, our results show that for such $M$, the probability that $M_n$ is singular is $O(\exp(-n^{c}))$, whereas even in the case when $\xi$ is a Bernoulli Random Variable, the results of Tao and Vu only give a bound of the form $O_{C}(n^{-C})$ for any constant $C>0$. As opposed to all previous works obtaining such bounds with error rate better than $n^{-1}$, our proof makes no use either of the inverse Littlewood--Offord theorems, or of any sophisticated net constructions. Instead, we show how to reduce the problem from the (Complex) sphere to (Gaussian) integer vectors, where it is solved directly by utilizing and extending a combinatorial approach to the singularity problem for Random discrete matrices, recently developed by Ferber, Luh, Samotij, and the author.Comment: 37 pages; comments welcome. arXiv admin note: text overlap with arXiv:1904.1110
Goncalo Tavares - One of the best experts on this subject based on the ideXlab platform.
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on the statistics of the sum of squared Complex gaussian Random Variables
IEEE Transactions on Communications, 2007Co-Authors: Goncalo Tavares, L M TavaresAbstract:In this letter, we derive new results for the statistics of the Complex Random Variable z=DeltaSigman=1 N x n 2=z 1+jz Q = re jphi where {x n} is a set of mutually independent Complex-valued Gaussian Random Variables with arbitrary means and equal variances. Each Random Variable x n is assumed to have independent real and imaginary components with equal variance for all n. Expressions are derived for the joint probability density function (pdf) of (z 1,z Q), for the joint pdf of (r,phi) and also for the marginal pdf of the modulus r. An useful Fourier series expansion for the pdf of the phase phi is also derived. As an application of the results, a theoretical performance analysis of the well-known nondata-aided Viterbi and Viterbi feedforward carrier phase estimator operating with BPSK signals is presented. In particular, the expressions for the exact pdf, variance, and equivocation probability of the carrier phase estimates are derived.
Tulay Adali - One of the best experts on this subject based on the ideXlab platform.
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Complex valued signal processing the proper way to deal with impropriety
IEEE Transactions on Signal Processing, 2011Co-Authors: Tulay Adali, Peter J Schreier, L L ScharfAbstract:Complex-valued signals occur in many areas of science and engineering and are thus of fundamental interest. In the past, it has often been assumed, usually implicitly, that Complex Random signals are proper or circular. A proper Complex Random Variable is uncorrelated with its Complex conjugate, and a circular Complex Random Variable has a probability distribution that is invariant under rotation in the Complex plane. While these assumptions are convenient because they simplify computations, there are many cases where proper and circular Random signals are very poor models of the underlying physics. When taking impropriety and noncircularity into account, the right type of processing can provide significant performance gains. There are two key ingredients in the statistical signal processing of Complex-valued data: 1) utilizing the complete statistical characterization of Complex-valued Random signals; and 2) the optimization of real-valued cost functions with respect to Complex parameters. In this overview article, we review the necessary tools, among which are widely linear transformations, augmented statistical descriptions, and Wirtinger calculus. We also present some selected recent developments in the field of Complex-valued signal processing, addressing the topics of model selection, filtering, and source separation.
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circularity and gaussianity detection using the Complex generalized gaussian distribution
IEEE Signal Processing Letters, 2009Co-Authors: M Novey, Tulay Adali, Anindya RoyAbstract:Knowing the statistical properties of a Complex-valued signal is important in many signal processing applications by providing the necessary information for choosing the appropriate algorithm. In this paper, we provide generalized likelihood ratio tests (GLRT), based on the Complex generalized Gaussian distribution (CGGD), for detecting two important signal properties: 1) the circularity of a Complex Random Variable, not constrained to the Gaussian case and 2) whether a Complex Random Variable is Complex Gaussian. These tests can be combined to statistically determine if a Complex Random Variable is, the often assumed, circular Gaussian. Simulations are used to quantify the performance of the detectors followed by application to communication signals and actual radar data.
Esa Ollila - One of the best experts on this subject based on the ideXlab platform.
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adjusting the generalized likelihood ratio test of circularity robust to non normality
International Workshop on Signal Processing Advances in Wireless Communications, 2009Co-Authors: Esa Ollila, Visa KoivunenAbstract:Recent research have elucidated that significant performance gains can be achieved by exploiting the circularity/non-circularity property of the Complex-valued signals. The generalized likelihood ratio test (GLRT) of circularity [1, 2] assuming Complex normal (Gaussian) sample has an asymptotic chi-squared distribution under the null hypothesis, but suffers from its sensitivity to Gaussianity assumption. With a slight adjustment, by diving the test statistic with an estimated scaled standardized 4th-order moment, the GLRT can be made asymptotically robust with respect to departures from Gaussianity within the wide-class of Complex elliptically symmetric (CES) distributions while adhering to the same asymptotic chi-squared distribution. Our simulations demonstrate the validity of the χ2 approximation even at small sample lengths. A practical communications example is provided to illustrate its applicability. In passing, we derive the connection with the kurtosis of a Complex Random Variable with a CES distribution with the kurtosis of its real and imaginary part.
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on the circularity of a Complex Random Variable
IEEE Signal Processing Letters, 2008Co-Authors: Esa OllilaAbstract:An important characteristic of a Complex Random Variable z is the so-called circularity property or lack of it. We study the properties of the degree of circularity based on second-order moments, called circularity quotient, that is shown to possess an intuitive geometrical interpretation: the modulus and phase of its principal square-root are equal to the eccentricity and angle of orientation of the ellipse defined by the covariance matrix of the real and imaginary part of z. Hence, when the eccentricity approaches the minimum zero (ellipse is a circle), the circularity quotient vanishes; when the eccentricity approaches the maximum one, the circularity quotient lies on the unit Complex circle. Connection with the correlation coefficient rho is established and bounds on rho given the circularity quotient (and vice versa) are derived. A generalized likelihood ratio test (GLRT) of circularity assuming Complex normal sample is shown to be a function of the modulus of the circularity quotient with asymptotic chi2 2 distribution.