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ComplexValued Signal Processing: The Proper Way to Deal With Impropriety
"... Abstract—Complexvalued signals occur in many areas of ..."
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Abstract—Complexvalued signals occur in many areas of
Circularity and Gaussianity detection using the complex generalized Gaussian distribution
 IEEE Signal Proc. Lett
, 2009
"... Abstract—Knowing the statistical properties of a complexvalued 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 generali ..."
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Abstract—Knowing the statistical properties of a complexvalued 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. Index Terms—Complexvalued signal processing, detection, noncircularity. I.
THE HANALYTIC SIGNAL
"... We consider the extension of the analytic signal concept known for real valued signals to the case of complex signals. This extension is based on the Quaternion Fourier Transform (QFT) and leads to the socalled Hanalytic signal. After defining the Hanalytic signal and giving some of its properti ..."
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We consider the extension of the analytic signal concept known for real valued signals to the case of complex signals. This extension is based on the Quaternion Fourier Transform (QFT) and leads to the socalled Hanalytic signal. After defining the Hanalytic signal and giving some of its properties, we present a new notation for quaternions, named the polar CayleyDickson form, which allows the extension of instantaneous phase and amplitude for the Hanalytic signal. Identification of the components of a complex signal are then performed through the analysis of its Hanalytic signal. We illustrate these new ideas on simulations. 1.
On ICA of improper and noncircular sources
 in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process. (IEEE ICASSP
, 2009
"... We provide a review of independent component analysis (ICA) for complexvalued improper and noncircular random sources. An improper random signal is correlated with its complex conjugate, and a noncircular random signal has a rotationally variant probability distribution. We present methods for ICA ..."
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We provide a review of independent component analysis (ICA) for complexvalued improper and noncircular random sources. An improper random signal is correlated with its complex conjugate, and a noncircular random signal has a rotationally variant probability distribution. We present methods for ICA using secondorder statistics, and higherorder statistics. For ICA based on secondorder statistics, we emphasize the key role played by the circularity coefficients, which are the canonical correlations between the source and the complex conjugate. For ICA based on higherorder statistics, we show how to extend algorithms for realvalued ICA to the complex domain using Wirtinger calculus. Index Terms — Independent component analysis, noncircular, improper, circularity coefficients, Wirtinger calculus
On Testing the Extent of Noncircularity
"... Abstract—In this correspondence, we provide a multiple hypothesis test to detect the number of latent noncircular signals in a complex Gaussian random vector. Our method sequentially tests the results of individual generalized likelihood ratio test (GLRT) statistics with known asymptotic distributio ..."
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Abstract—In this correspondence, we provide a multiple hypothesis test to detect the number of latent noncircular signals in a complex Gaussian random vector. Our method sequentially tests the results of individual generalized likelihood ratio test (GLRT) statistics with known asymptotic distributions to form the multiple hypothesis detector. Specifically, we are able to set a threshold yielding a precise probability of error. This test can be used to statistically determine if a given complex observation is circular Gaussian, and if not, how many latent signals in the observation are noncircular. Simulations are used to quantify the performance of the detector as compared to a detector based on the minimum description length (MDL) criterion. The utility of the detector is shown by applying it to a beamforming application using independent component analysis (ICA). Index Terms—Canonical coordinates, circularity, circularity coefficients, generalized likelihood ratio test. I.
Complexvalued signal processing – essential models, tools and statistics
 Information Theory and Applications Workshop (ITA), 2011, La Jolla
"... AbstractComplexvalued signals arise in many diverse fields such as communications, radar, biomedical sciences, physical sciences, and related fields. This paper briefly reviews some important tools, statistics, models and estimators that are useful for handling complexvalued random signals. Over ..."
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AbstractComplexvalued signals arise in many diverse fields such as communications, radar, biomedical sciences, physical sciences, and related fields. This paper briefly reviews some important tools, statistics, models and estimators that are useful for handling complexvalued random signals. Over the past four decades, circularity (i.e. invariance of the distribution under multiplication by a unit complex number) or secondorder circularity (i.e. uncorrelatedness of the random vector with its complex conjugate) has been a common implicit assumption. Hence in this paper a special emphasis is put on this circularity property, as optimal signal processing methods for circular and noncircular signals are often different and choosing the right type of processing can provide significant performance gains. Topics reviewed in this paper include different types of circularity measures and detectors of circularity, complex elliptical symmetry of random variables, CramérRao lower bounds on the estimation of complexvalued parameters, optimization of a realvalued cost function with respect to complexvalued parameters using CRcalculus, and complexvalued independent component analysis.
Contents lists available at ScienceDirect Signal Processing
"... journal homepage: www.elsevier.com/locate/sigpro Asymptotic distribution of circularity coefficients estimate of ..."
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journal homepage: www.elsevier.com/locate/sigpro Asymptotic distribution of circularity coefficients estimate of
Open Access
"... GLRTbased array receivers for the detection of a known signal with unknown parameters corrupted by noncircular interferences Pascal Chevalier 1,2 * , Abdelkader Oukaci 3 and JeanPierre Delmas 4 The detection of a known signal with unknown parameters in the presence of noise plus interferences (cal ..."
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GLRTbased array receivers for the detection of a known signal with unknown parameters corrupted by noncircular interferences Pascal Chevalier 1,2 * , Abdelkader Oukaci 3 and JeanPierre Delmas 4 The detection of a known signal with unknown parameters in the presence of noise plus interferences (called total noise) whose covariance matrix is unknown is an important problem which has received much attention these last decades for applications such as radar, satellite localization or time acquisition in radio communications. However, most of the available receivers assume a second order (SO) circular (or proper) total noise and become suboptimal in the presence of SO noncircular (or improper) interferences, potentially present in the previous applications. The scarce available receivers which take the potential SO noncircularity of the total noise into account have been developed under the restrictive condition of a known signal with known parameters or under the assumption of a random signal. For this reason, following a generalized likelihood ratio test (GLRT) approach, the purpose of this paper is to introduce and to analyze the performance of different array receivers for the detection of a known signal, with different sets of unknown parameters, corrupted by an unknown noncircular total noise. To simplify the study, we limit the analysis to rectilinear known useful signals for which the baseband signal is real, which concerns many applications.
2.2 Design of Complex Circular Random Variables................ 4
, 2007
"... This material is a part of the second year report for the EPSRC grant EP/D061709/1. Personal use of this material is permitted. ..."
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This material is a part of the second year report for the EPSRC grant EP/D061709/1. Personal use of this material is permitted.