Asymptotic regime for impropriety tests of complex random vectors

Impropriety testing for complex-valued vector has been considered lately due to potential applications ranging from digital communications to complex media imaging. This paper provides new results for such tests in the asymptotic regime, i.e. when the vector dimension and sample size grow commensurately to infinity. The studied tests are based on invariant statistics named impropriety coefficients. Limiting distributions for these statistics are derived, together with those of the Generalized Likelihood Ratio Test (GLRT) and Roy's test, in the Gaussian case. This characterization in the asymptotic regime allows also to identify a phase transition in Roy's test with potential application in detection of complex-valued low-rank subspace corrupted by proper noise in large datasets. Simulations illustrate the accuracy of the proposed asymptotic approximations.

[1]  Iain M Johnstone,et al.  APPROXIMATE NULL DISTRIBUTION OF THE LARGEST ROOT IN MULTIVARIATE ANALYSIS. , 2010, The annals of applied statistics.

[2]  Alfred Hanssen,et al.  A generalized likelihood ratio test for impropriety of complex signals , 2006, IEEE Signal Processing Letters.

[3]  Patrick Rubin-Delanchy,et al.  On Testing for Impropriety of Complex-Valued , 2009 .

[4]  Tülay Adali,et al.  On Testing the Extent of Noncircularity , 2011, IEEE Transactions on Signal Processing.

[5]  T. Adali,et al.  Complex ICA of Brain Imaging Data [Life Sciences] , 2007, IEEE Signal Processing Magazine.

[6]  Eric Moreau,et al.  Blind Identification and Separation of Complex-Valued Signals: Moreau/Blind Identification and Separation of Complex-Valued Signals , 2013 .

[7]  Michael D. Perlman,et al.  Two testing problems relating the real and complex multivariate normal distributions , 1984 .

[8]  Florent Chatelain,et al.  Exact Distribution and High-dimensional Asymptotics for Improperness Test of Complex Signals , 2019, ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[9]  Mahmood R. Azimi-Sadjadi,et al.  Detection of Spatially Correlated Time Series From a Network of Sensor Arrays , 2014, IEEE Transactions on Signal Processing.

[10]  Peter J. Schreier,et al.  Determining the Dimension of the Improper Signal Subspace in Complex-Valued Data , 2017, IEEE Signal Processing Letters.

[11]  Jean Pierre Delmas.  Asymptotically minimum variance second-order estimation for noncircular signals with application to DOA estimation , 2004, IEEE Transactions on Signal Processing.

[12]  Visa Koivunen,et al.  Complex random vectors and ICA models: identifiability, uniqueness, and separability , 2005, IEEE Transactions on Information Theory.

[13]  Marco Chiani,et al.  Distribution of the largest root of a matrix for Roy's test in multivariate analysis of variance , 2014, J. Multivar. Anal..

[14]  Andrew T. Walden,et al.  A Frequency Domain Test for Propriety of Complex-Valued Vector Time Series , 2016, IEEE Transactions on Signal Processing.

[15]  J. W. Silverstein.  The Limiting Eigenvalue Distribution of a Multivariate F Matrix , 1985 .

[16]  Erik G. Larsson,et al.  Spectrum Sensing for Cognitive Radio : State-of-the-Art and Recent Advances , 2012, IEEE Signal Processing Magazine.

[17]  Pierre Comon.  Circularité et signaux aléatoires à temps discret , 1994 .

[18]  Jean Pierre Delmas,et al.  On the asymptotic distribution of GLR for impropriety of complex signals , 2011, Signal Process..

[19]  Jiang Hu,et al.  Canonical correlation coefficients of high-dimensional Gaussian vectors: Finite rank case , 2014, The Annals of Statistics.

[20]  I. Johnstone,et al.  Testing in high-dimensional spiked models , 2015, The Annals of Statistics.

[21]  Bernard C. Picinbono,et al.  On circularity , 1994, IEEE Trans. Signal Process..

[22]  V. Koivunen,et al.  Generalized complex elliptical distributions , 2004, Processing Workshop Proceedings, 2004 Sensor Array and Multichannel Signal.

[23]  Wolfgang Utschick,et al.  Measuring impropriety in complex and real representations , 2019, Signal Process..

[24]  L. Scharf,et al.  Statistical Signal Processing of Complex-Valued Data: The Theory of Improper and Noncircular Signals , 2010 .

[25]  Nicolas Le Bihan,et al.  Spectral Analysis of Stationary Random Bivariate Signals , 2017, IEEE Transactions on Signal Processing.

[26]  Jonathan Dong,et al.  Spectral Method for Multiplexed Phase Retrieval and Application in Optical Imaging in Complex Media , 2018, ICASSP 2019 - 2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[27]  Robert Schober,et al.  I/Q Imbalance Aware Widely-Linear Receiver for Uplink Multi-Cell Massive MIMO Systems: Design and Sum Rate Analysis , 2015, IEEE Transactions on Wireless Communications.

[28]  N. L. Johnson,et al.  Multivariate Analysis , 1958, Nature.

[29]  Walid Hachem,et al.  Correlation test for high dimensional data with application to signal detection in sensor networks , 2014, 2014 22nd European Signal Processing Conference (EUSIPCO).

[30]  Samuel Kotz,et al.  The beta exponential distribution , 2006, Reliab. Eng. Syst. Saf..

[31]  E. Lehmann.  Testing Statistical Hypotheses , 1960 .

[32]  Tülay Adali,et al.  Complex-Valued Signal Processing: The Proper Way to Deal With Impropriety , 2011, IEEE Transactions on Signal Processing.

[33]  Yuan Wang,et al.  Passive Detection of Correlated Subspace Signals in Two MIMO Channels , 2017, IEEE Transactions on Signal Processing.

[34]  R. Muirhead.  Aspects of Multivariate Statistical Theory , 1982, Wiley Series in Probability and Statistics.