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Statistics for complex variables and signals - Part I: VariablesAbstractThis paper is devoted to the study of higher-order statistics for complex random variables. We introduce a general framework allowing the direct manipulation of complex quantities: the separation between the real and the imaginary parts of a variable is avoided. We give the rules to integrate and derive probability density functions and characteristic functions, so that calculations may be carried out. In the case of multidimensional variables, we use the natural framework of tensors. The study of complex variables leads to the extension of the notion of complex circular random variables already known in the Gaussian case.1. IntroductionHigher-order statistics (HOS) are now an intensive field of research in Signal and Image Processing. This avenue of research is based on the use of a new characterization of variables and signals. Up to now this characterization was essentially based on second-order (energetic) measures: variance and covariance for variables, correlation and cross correlation for signals in the time domain, pectral power density and cross-spectral power density for signals in the frequency domain.After the pioneering papers of the potentialities of HOS are now used intensively. It would be very long to give a complete view of this domain in which new models are emerging that support the development of a large number of applications. A synthesis can be found in Furthermore, several special issues of journals have been devoted to this topic and a series of specialized workshops began in 1989.The essential features in research on HOS are found in modeling and in applications.In modeling, for random variables, HOS are essentially based on cumulants of order greater than 2. The higher-order description of signals is made through multicorrelations in the time domain, and multispectra in the frequency domain.Applications are being developed in a great number of fields.In nearly all theclassical domains of research in Signal and Image processing. HOS are introducing new methodologies. We can cite the blind source separation and blind deconvolution problems in a wide variety of situations: vibrations diagnostic, underwater acoustics, radar, satellite communications, seismic sounding, astronomy, etc. In nonlinear systems identification, HOS are a basic tool. Moreover, a close connexion exists between HOS and neuromimetic systems.This very active and fruitful field of research needs solid theoretical foundations. They were built a long time ago by mathematicians and statisticians who developed the theory of random variables and signals. The higher-order statistical properties of random variables are described in many classical textbooks . We found the development of atensorial approach particularly well fitted to the higher-order properties of multidimensional variables. The multicorrelations and multispectra are described.However, few authors have been concerned with the domain of complex random variables and signals, even if this situation appears in practical applications: in frequency domain processing after Fourier transformation, particularly in array processing, in single band systems of communications where analytic signals are commonly used, in time-frequency analysis by the Wigner-Ville distribution, etc.The Gaussian complex model, which is sufficient in the classical second-order approach. These authors have shown the algebraic simplifications brought by the use of a complex modeling. They have shown that new properties, like Gaussian complex circularity, are introduced by this complex modeling.More recently the lack of a general complex modeling was put in evidence: the authors noted that “paradoxically, one finds in the literature very few treatments of com plex random variables and processes”. They introduce the notion of “proper complex random processes” which is their denomination for circular processes. However, this approach is essentially limited to the second-order properties.This particular character of complex signals, when one is concerned with the bispectrum,has been exemplified.With the increasing use of higher-order statistics, it is now necessary to develop a general modeling for complex random variables and signals. It is the aim of this exposition, which is divided into two parts.In the first part, we are concerned with complex random variables. We begin by the definition of the probability laws using complex notations. We extend the results,which are already known for the Gaussian case to the general situations of monodimensional and multidimensional complex random variables, whether Gaussian or non-GausSian. Then, we extend the tensorial formalism developed in the real case to the multidimensional complex random variables. We show that, for a given order, different kinds of cumulants can be defined. This result is an extension of the pseudo-covariance introduced. With this modeling we can give a general definition of circularity, and we show that, in this specific case, many higher-order cumulants are null. We show the direct relation between the Fourier transform and circularity. Algorithms for the generation of complex circular non-Gaussian random variables are given and illustrated on simulations. An illustration of the new rules of calculation is given in Appendix B in the circula Gaussian case.Part II is devoted to the modeling and representation of complex random signals.For stationary signals, using the results given for the multidimensional random variables, we define the multicorrelations and multispectra for complex random stationary signals. We show that the complete characterization of complex signals at an order pdemands the introduction of different multicorrelations and multispectra. In the usual case of real valued signals these multicorrelations and spectra are identical. The situation is different for analytic signals for which some multicorrelations and spectra are null. Extending the concept of circularity to the signals, we can show that, for this kind of signal, the only nonnull multicorrelations and spectra possess the same number of conjugated and nonconjugated terms.Furthermore, we show that band limited signals are circular up to a certain order. We come back to the choice between moments and cumulants and show that, except for the classical interest presented by cumulants due to their additivity and to their characterization of the Gaussian property, they make it possible to distinguish clearly between the properties at each order and to eliminate singularities in the multispectrum. This modeling is then extended to digital signals and to digital time-limited signals used in the Discrete Fourier Transform.2. Starting pointThe purpose of this paper is to introduce a general model of complex random variables. The usefulness of this modeling will be illustrated with examples. We will show that it leads to new characteristics in the description of signals and allows a new insight into the developing field of higher-order statistics.Complex random variables (CRV) appear as the output of a great number of processingr such as:- Fourier transforms,- Array processing,-- Hilbert transforms.When dealing with CRV two approaches can be used:-to consider a CRV as a two-dimensional real random variable (RRV),-to develop algebraic tools directly with CRV.The second approach has two advantages:-it makes all the derivations simpler,-it preserves the physical sense related to the complex nature of the data.This approach has been developed in the Gaussian case leading to the theory ofcomplex Gaussian random variables (CGRV). In this situation the consideration of the CGRV has given rise to the important notion of complex random Gaussian circular variables.The principal aim of this paper is to generalize these notions to the general case of Gaussian and non-Gaussian random variables. The primary motivation is that new algorithms using higher-order statistics (HOS) are being developed, and it is clear that in this field, it is absolutely necessary to deal with non-Gaussian data. Furthermore, a theory will be developed using tensors which constitute he natural framework of higher-order statistics. Hence, the second main issue of this paper is the extension of the framework introduced by MacCullagh to the complex case.After a definition of the basic principles on which our modeling is built, we will present the technical realization of the principal tools. We will give a generaldefinition of CRV and illustrate the usefulness of this new formalism in the context of complex circular random variables.2.1. Complex random variablesThe definition of CRV is well-known. From two real random variables (RRV) x and y , we define the complex random variable z by z x jy =+.(1)where 21j =.The turning point is to associate a probability density function (pdf) with this CRV.In the Gaussian circular case, this is done by onsidering both z and its complex conjugate z * defined as z x jy *=-.The …formal‟ pdf of the Gaussian circular variable is then()2,1,zz z z P z z e σπσ**-*=Thus, it appears from the Gaussian example that we must consider both z and z*in the definitions in order to extract all the statistical information. The preceding definition for the complex Gaussian variable shows that E[''z] equals zero. Hence, the only nonnull second-order moment is E[z z*]. This means that both z and z*give statistical (and perhaps different) information. Therefore, a theory of higher-order statistics in a general case must consider the variable and its complex conjugate. The information is in the statistics of the two variables, but also in their cross-statistics. We now introduce our formalism to handle the complex random variables in a more general way.The main problem which arises from the preceding discussion is an algebraic one, since the variables z and z*are algebraically linked. In order to overcome this, we propose to include the real world of z and z*in a larger space in which z and z*are not algebraically dependent.One way to do this is to consider x and y(real and imaginary parts of z)as complex random variables.In this context we will continue to*=-,but despite the notations,z and z*are no longer write z x jy=+and z x jycomplex conjugates. In order to introduce a continuity between the classical notations and the new ones, using tensors, that will be presented shortly, we have chosen to use these ambiguous notations of z and z*.We will introduce in the following an alternative presentation that avoids this problem.This …trick‟ will allow us to treat z and z*as algebraically independent variables. We will see that this greatly facilitates all the calculations. However, these purely conceptual CRV are only used as means for easier calculations. When we want to come back to the real physical world, we have to restrict z and z*to belong to the subset generated by the real numbers x and y.2.2The rulesFor this, we must establish some rules in order to obtain definitions which make sense. We will propose two laws:1. All the functions used must be well-defined mathematically, and all the operators, like integrals, must converge.2. We want to be able to recover the classical formulae when we consider the particular case of RRV.This new point of view applies to both one-dimensional random variables as multidimensional random variables.We will now see how it works.3. PDF and characteristic functionsWe will consider successively the one- and the multi-dimensional cases of random complex variables.3. 1One-dimensional complex random variablesWe have seen in Section 2.1 that the pdf is a function (),,z z P z z **of z and z *.Let us try to define the first characteristic function. Let ωand ω*错误！未找到引用源。