Singular Non-circular Complex Elliptically Symmetric Distributions: New Results and Applications

Absolutely Continuous non-singular complex elliptically symmetric distributions (referred to as the non-singular CES distributions) have been extensively studied in various applications under the assumption of nonsingularity of the scatter matrix for which the probability density func-tions (p.d.f’s) exist. These p.d.f’s, however, can not be used to characterize the CES distributions with a singular scatter matrix (referred to as the singular CES distributions). This paper presents a generalization of the singular real elliptically symmetric (RES) distributions studied by D´ıaz-Garc´ıa et al to singular CES distributions. An explicit expression of the p.d.f of a multivariate non-circular complex random vector with singular CES distribution is derived. The stochastic representation of the singular non-circular CES (NC-CES) distributions and the quadratic forms in NC-CES random vector are proved. As special cases, explicit expressions for the p.d.f’s of multivariate complex random vectors with singular non-circular complex normal (NC-CN) and singular non-circular complex Compound-Gaussian (NC-CCG) distributions are also derived. Some useful properties of singular NC-CES distributions and their conditional distributions are derived. Based on these results, the p.d.f’s of non-circular complex t -distribution, K distribution, and generalized Gaussian distribution under sin-gularity are presented. These general results degenerate to those of singular circular CES (C-CES) distributions when the pseudo-scatter matrix is equal to the zero matrix. Finally, these results are applied to the problem of estimating the parameters of a complex-valued non-circular multivariate linear model in the presence either of singular NC-CES or C-CES distributed noise terms by proposing widely linear estimators.


Introduction
Non-singular CES distributions have recently been the focus of active research in engineering applications involving non-Gaussian data models [1][2][3][4][5][6][7][8]. Very comprehensive reviews of non-singular C-CES and NC-CES distributions are given in [5,9,10] and [11], respectively. The CES distribution includes various distributions, such as the circular complex normal (C-CN) distribution [12,13], NC-CN distribution [14,22], complex t-distribution [1,9], K-distribution [10,23,24], complex generalized Gaussian (CGG) distribution [5,15,25]. These distributions are characterized by the associated p.d.f.'s which exist for non-singular scatter or covariance matrix. However, a problem that has not been tackled completely is related to the p.d.f. of singular C-CES, NC-CES, C-CN, and NC-CN distributions, which are not unusual in theoretical and practical engineering problems. However, it was proved by [26] that the singular RES distributions have p.d.f's on a subspace of smaller dimension and equal to the rank of the scatter matrix. This same reference also gives an explicit expression for the p.d.f of singular RES distributions.
Complex-valued signals are widely used for modeling many systems in a wide range of fields (e.g., optics, communications, radar, and biomedicine). Linear solutions for complex-valued signals have been studied in detail in the literature for both circular and non-circular complex signals. Among these solutions, the linear and widely linear minimum mean-squared error (LMMSE and WLMMSE) estimators introduced in [27,28] work under the assumption that the covariance matrix of mea-Singular Non-circular Complex Elliptically Symmetric Distributions: New Results and Applications surement is non-singular. The performance of these estimators are compared in [29,30] and widely used in many practical applications [31][32][33][34]. However, these estimators can not be applied when the scatter or covariance matrix of the measurement data is singular. This paper presents the derivation of explicit expressions for the p.d.f.'s of multivariate singular C-CES, NC-CES, C-CN and NC-CN distributed random variables (r.v.'s) following the reasoning proposed in [26]. Stochastic representations of singular C-CES distributions, singular NC-CES distributions and quadratic forms in singular C-CES and NC-CES r.v.'s are also given. Some useful properties of singular NC-CES distributions and their conditional distributions are derived. The complex t-distribution, K-distribution, and generalized Gaussian distribution associated with singular multivariate CES r.v.'s are also derived. These results are applied to the problem of estimating the parameters of a complex-valued linear model in the presence of either singular NC-CES or C-CES distributed noise terms and followed by the derivation of widely linear estimators.
The remainder of this paper is organized as follows. Section 2 presents a brief overview of general characteristics of the continuous non-singular NC-CES distributions, followed by the derivation of the p.d.f. and the stochastic representation of singular NC-CES distributions. In the same section, some useful properties of singular NC-CES distributions and their conditional distributions are proved. Section 3 provides the p.d.f of singular NC-CN distribution as a special case of NC-CES distributions. Section 4 introduces practical singular circular complex compound Gaussian (C-CCG) and singular NC-CCG distributions, followed by the derivation of the stochastic representation of the quadratic forms in singular C-CCG and NC-CCG distributions. Section 5 presents the complex t−distribution, K−distribution and the CGG distribution associated to multivariate singular C-CES and NC-CES r.v.'s. Section 6 derives the singular widely linear mean square estimation of a signal from singular distributed measurement data vector. The problem of parametric estimation of complexvalued linear models with singular C-CES and NC-CES distributed error terms are examined in section 7 where the associated residuals of the widely linear estimators are shown to have singular CES distributions. Finally, conclusion is given in section 8.
The following notations are used throughout the paper. Matrices and vectors are represented by bold upper case and bold lower case characters, respectively. I is the identity matrix. Vectors are by default in column orientation, while T , H, * and # stand for transpose, conjugate transpose, conjugate and MoorePenrose inverse,respectively. E(.), Tr(.), rank(.), (.) and . are the expectation, trace, rank, real part, and norm operators, respectively. Cov(z) def = (z − E(z))(z − E(z)) H and pcov(z) def = (z−E(z))(z−E(z)) T are respectively the covariance matrix and the pseudo-covariance matrix of a complex r.v. z. Symbol = d means equal in distribution and U (CS m ) denotes the uniform distribution on the unit complex-sphere

Singular and non-singular noncircular complex elliptical distribution
This section firstly briefly reviews of non-singular NC-CES distributions (called also Generalized CES distributions) presented in [11], secondly the main characteristics of singular NC-CES distributions are proven, i.e., singular with respect to Lebesgue measure as the scatter matrix is rank-deficient, by presenting an explicit expression of the p.d.f. of singular NC-CES distributions that exist on a subspace. Finally we provide some useful properties of singular NC-CES distributions and their conditional distributions.
for some function φ : R + → R, as the characteristic generator. Positive semi-definite Hermitian matrix Σ ∈ C m×m is the scatter matrix, complex symmetric matrix Ω ∈ C m×m denotes the pseudo-scatter matrix, with symmetry center µ ∈ C m . Note that the c.f. in (1) exists even though Σ is singular (rank(Σ) < m). In addition, Σ, Ω and φ(.) do not uniquely define a particular NC-CES distribution, an additional scale constraint, either on Σ and Ω either on φ(.), needs to be imposed for identifiability purposes. Therefore, an r.v. z ∈ C m has singular or non-singular NC-CES distribution, depending on whether rank(Σ) = r < m or r = m, respectively. A singular NC-CES distributed r.v. will be denoted as z ∼ EC r m (µ, Σ, Ω, φ) and for non-singular CES distributed r.v. the superscript will be omitted in EC r m . For clarity, the singular and non-singular NC-CES distributions are presented separately in the following subsections.

Non-singular NC-CES distributions
Suppose rank(Σ) = m, which is a necessary condition for z ∼ EC m (µ, Σ, Ω, φ) to be absolutely continues with respect to Lebesgue measure in R 2m , therefore the p.d.f. of z exist and can be expressed as [11], where c m,g is a normalizing constant ensuring that p(z) integrates to one and it is given by c m,g is the surface area of CS m , and g(.) is a non-negative function (density generator), which satisfies δ m,g = ∞ 0 u m−1 g(u)du < ∞. q(z) has the quadratic form q(z) Σ Ω Ω * Σ * . In the absolutely continuous case, we use the notation z ∼ EC m (µ, Σ, Ω, g) in place of z ∼ EC m (µ, Σ, Ω, φ). Note that the p.d.f. (2) depends on z only through the quadratic form q(z).
Since Σ and Ω are hermitian positive definite and complex symmetric matrices, respectively, it follows from [35,Corollary 4.6.12(b)], that there exist a non-singular matrix A ∈ C m×m such that Σ = AA H and Ω = A∆A T where ∆ = Diag(κ 1 , . . . , κ m ) is a real diagonal matrix with nonnegative diagonal entries κ k for k = 1 . . . m. Let v ∈ C m be a r.v. obtained via an R-linear transformation of u ∼ U (CS m ) as follows [5]: In the sequel, this vector will be written as v ∼ U κ (CS m ) where κ def = (κ 1 , . . . , κ m ) T . The stochastic representation provides the tool to generate r.v. deviating from the EC m (µ, Σ, Ω, g) distributions.
Result 1 z ∼ EC m (µ, Σ, Ω, g) with rank(Σ) = m if and only if it admits the stochastic representation where the non-negative real random variable R def = √ Q, called the modular variate, is independent of the complex r.v.
The uniform spherical distribution can be obtained from a C-CN distributed random vector, y ∼ CN m (0, I), when dividing it by its length, u = d y ||y||2 . Since δ m,g < ∞, the covariance matrix R def = Cov(z) and the pseudo covariance matrix R def = pcov(z) exist and respectively equal to the scatter matrix and pseudo-scatter matrix up to a positive real constant c 0 [11, Theorem 3], i.e., R = c 0 Σ and R = c 0 Ω. Nevertheless, the constant c 0 can be chosen to be equal to 1, that is, if E(R 2 ) = 2rank(Σ). Note that while Σ always exists, R does not exist for some CES distributions (e.g. Cauchy distribution).
For the special case when Ω = O (or equivalently ∆ = O), the non-singular NC-CES distributions degenerate to the nonsingular C-CES distributions [5,10], for which the c.f. (1) and p.d.f. (2) take the forms similar to the real case: where q(z) . It follows also from result 1 that the stochastic representation of non-singular C-CES distributed r.v. z ∼ EC m (µ, Σ, g) has the form where A ∈ C m×m is a non-singular matrix such that Σ = AA H .

Singular NC-CES distributions
The singular NC-CES distributions has still not been studied in the literature, in despite of the various studies that have been published. The results of this section are generalizations of [26] for the singular RES distribution to the singular NC-CES distributions, as the scatter matrix is singular with rank(Σ) = r < m. This section first proves an explicit expression of the p.d.f. for singular NC-CES distributions, and provides some useful properties of singular NC-CES distributions and their conditional distributions.
Note that the conditions of the scatter matrix Σ and the schur complement matrix Σ * − Ω H Σ −1 Ω begin hermitian positive definite are necessary to ensure that the matrixΓ is hermitian positive definite [14]. However, if the matrix Σ is singular with rank(Σ) = r < m,Γ is singular as well and the p.d.f in (2) has no meaning. Therefore the following question arises: Does the p.d.f. exist for singular NC-CES distributed r.v. z ∼ EC r m (µ, Σ, Ω, g)? The answer is given by the result 2 where the p.d.f. exists on a subspace. To derive the p.d.f. of singular NC-CSE distributions we need the following lemma (proved in the Appendix A) that provides a factorization of the pseudoscatter matrix Ω.
Lemma 1 Let rank(Σ) = r < m and rank(Ω) = p ≤ m, the pseudo-scatter matrix Ω can be factorized as: where Λ r is a diagonal matrix containing the r non-zero eigenvalues {λ k } r k=1 of Σ, and the columns of the complex matrix U r ∈ C m×r are the corresponding non-zero eigenvectors. V p ∈ C r×p is an (r×p) matrix with orthonormal columns and ∆ p def = Diag(κ 1 , . . . , κ p ) is a (p×p) complex-valued nonnegative diagonal matrix with κ l = 0, and |κ l | < 1 for l = 1, . . . , p.
The following result (proved in the Appendix B) provides the p.d.f. of the singular NC-CES distributions.
Result 2 Let z ∼ EC r m (µ, Σ, Ω, g) with parameters µ ∈ C m , Σ ∈ C m×m is the scatter complex hermitian matrix assumed singular with rank(Σ) = r < m and Ω ∈ C m×m is the pseudo-scatter complex symmetric matrix with rank(Ω) = p ≤ m . In such case, the p.d.f. of a singular NC-CES distributed r.v. z is given by and where c r,p Remark 1 It follows from lemma 1 that if rank(Σ) = rank(Ω) = r, there is a unitary matrix V r ∈ C r×r such that where ∆ r = Diag(κ), κ = (κ 1 , . . . , κ r ) T ∈ R r with κ k = 0 for k = 1 . . . r. Let W r be a (r×m) complex matrix defined as r U H r which satisfies the following equalities: The following result extends result 1 to give a stochastic representation of a r.v. distributed as a singular NC-CES distribution.
Result 3 A r.v. z follows a singular NC-CES distribution, i.e., z ∼ EC r m (µ, Σ, Ω, g) with rank(Σ) = rank(Ω) = r if and only if it admits the stochastic representation (3), U r and Λ r are defined in (52) and V r is defined in (10).
. Therefore, result 2 can be simplified to the following result. and where c r The following corollary proved in Appendix C gives the distribution of the quadratic forms.
• Let z ∼ EC r m (µ, Σ, g). Then The following result proved in Appendix D on the conditional distributions of singular NC-CES distributed r.v.'s will be used in the derivation of singular widely linear mean square estimation of a signal from singular distributed measurement data vector in section 6.
where z 1 and µ 1 are (d × 1) vectors (d < m), Σ 11 and where Note that the singular C-CES distribution is obtained if Ω = O and the result 5 degenerates to the following corollary.

Singular non-circular complex normal distribution
This section derives explicit expression of the p.d.f. for a singular NC-CN distribution. Let us first remind the reader that the non-singular NC-CN distribution was introduced in [5,14,22], which can be viewed as a class of non-singular NC-CES distributions [11]. The non-singular NC-CN distribution has been recently widely used in various statistical signal processing applications such as: DOA methods [16][17][18], blind source separation methods [19][20][21], signal detection methods [15,36,37], etc. Also Cramér-Rao performance bounds based on non-singular NC-CN distribution have been proposed for DOA estimation in [38] and source separation in [39]. Since non-singular NC-CN distribution is a member of non-singular NC-CES distributions, it follows that the p.d.f. of non-singular NC-CN distribution given below can be obtained from (2) by letting the density generator g(t) equal to g(t) = exp(−t), which gives c m,g = π −m and µ = E(z), the hermitian covariance matrix R def = Cov(z) = Σ and the complex pseudo- In the special case where the pseudo-covariance matrix R = O, (21) reduces to the following p.d.f. of non-singular C-CN distribution, where Q(z) . Thus, non-singular C-CN distribution can be seen as a special case of non-singular NC-CN distribution.
Recall that the matrixR is positive definite if and only if R and its schur complement R s = R − R R − * R * are definite positive [14]. However if these conditions are not met,R is a singular matrix and therefore the p.d.f. (21) does not exist. The following result gives the p.d.f. of singular NC-CN distribution which is obtained from result 2 by replacing g(t) in (8) by g(t) = exp(−t), which gives c m,g = π −m and Σ = R, Ω = R .
with R ∈ C m×m being a singular hermitian covariance matrix with rank(R) = r < m and R ∈ C m×m being a complex symmetric pseudocovariance matrix with rank(R ) = p . In such case, the p.d.f. of a singular NC-CN distributed r.v. z is given by and where Q(z) is a quadratic form Q(z) with µ ∈ C m , R ∈ C m×m and rank(R) = r < m. In such case, the p.d.f. of z is given by and Remark 2 Note that the c.f. of singular NC-CN distributed r.v. z ∼ CN r m (µ, R, R ) always exists and identical to the c.f. of non-singular NC-CN distribution given by [14] It follows from result 5 that the p.d.f. of conditional distribution of two singular NC-CN distritbuted r.v.'s are summarized as the following result.
For the circular case which is characterized by all the matrices R x being zero, result 8 reduces to the following corollary.
4 Singular circular and non-circular Compound-

Gaussian Distributions
Non-singular C-CCG distributions presented in [10] under the assumption of non-singular scatter matrix Σ ∈ C m×m (i.e., rank(Σ) = m), represent an important subclass of nonsingular C-CES distributions. The non-singular C-CCG distributions are widely employed in radar signal processing to describe the heavy-tailed clutter process as a product of two independent random processes 'texture' and 'speckle'. More 1024 Singular Non-circular Complex Elliptically Symmetric Distributions: New Results and Applications precisely, a r.v. z has a non-singular C-CCG distribution if it admits a C-CCG-representation where τ is a positive real r.v. with p.d.f. f τ , called as texture, independent of n ∼ CN m (0, Σ), called as speckle. The p.d.f.'s of non-singular C-CCG-distributions are given by where q(z) = (z − µ) H Σ −1 (z − µ) and f τ (τ ) = ∂F τ (τ )/∂τ . Note that the p.d.f (27) can always be written in the form (5) with a density generator g(t) ∝ ∞ 0 τ −m exp(−t/τ )f τ (τ )dτ . A non-singular C-CCG distributed r.v. will be denoted as z ∼ CN m (µ, Σ). Different choices of distribution for f τ (.) lead to some well-known examples of CCG-distributions such as t-distribution and K-distribution presented in the next section.
Note that the p.d.f. of singular NC-CN distributions presented in section 3 exist on a subspace. Given the above definition of non-singular C-CCG distributions, the p.d.f. of singular NC-CCG distributions also exist on a subspace and can be defined as follows.
Definition 2 A r.v. z ∈ C m is said to have a singular NC-CCG distribution if it admits a NC-CCG representation where τ is a r.v. defined above and independent of n follows the singular NC-CN distribution CN r m (0, Σ, Ω). Also, the p.d.f.'s of singular NC-CCG distributions are given by and where c r,p λ,κ and the quadratic form q(z) are defined in result 2. A singular NC-CCG distributed r.v. will be denoted as z ∼ CN r m (µ, Σ, Ω). In the special case when Ω = O, (29) reduces to the following expression of the p.d.f. of singular C-CCG distributions (denoted as z ∼ CN r m (µ, Σ)) where q(z) and c r λ are defined in result 4. If τ has a finite second-order moments (i.e., E(τ ) < ∞), the mean and the second-order moments of z exist. It also follows that if E(τ ) = 1, the scatter matrix Σ and pseudo-scatter matrix Ω are, respectively, exactly equal to the covariance matrix Cov(z) = E(τ )Cov(n) = Σ and pseudo covariance matrix pcov(z) = E(τ )pcov(n) = Ω. The following result proved in Appendix E gives the stochastic representation of the quadratic form in singular NC-CES distributions.
Result 9 Let z = z r + jz i ∼ CN m (µ, Σ, Ω) and Q ∈ C 2m×2m be a hermitian matrix partitioned asQ def = Q Q Q * Q * where Q ∈ C m×m is a hermitian matrix and Q ∈ C m×m is a symmetric complex matrix. Then the stochastic representation of the quadratic formz H 0Qz0 wherẽ z 0 def =z −μ is given bỹ where the χ 2 1 (l) are independent central Chi-square random variables with one degree of freedom. The λ l are nonzero eigenvalues of the matrixΓQ of rank q whereΓ is defined in (2).
For the special case of non-singular C-CCG distributions where Ω = O, assume that Q = O, result (9) reduces to the following corollary: Corollary 4 Let z ∼ CN m (µ, Σ) and Q ∈ C m×m be a hermitian matrix. Then the stochastic representation of the quadratic form z H 0 Qz 0 where z 0 def = z − µ is given by where the λ l are nonzero eigenvalues of the matrix ΣQ of rank q c .

Examples of singular NC-CES distributions
Based on the results of section 2, and similar to the nonsingular case (Σ > 0) [5,10,11], we provide explicit expressions for the p.d.f.'s of three subclasses of CES distributions,i.e., complex K-distribution, complex t-distribution and complex generalized Gaussian (CGG) distribution, under the assumption of singular scatter matrix (rank(Σ) = r). These subclasses of distributions can be distinguished from each other only by their functional form of the density generator g(.) as shown below. Example 1: Singular non-circular complex K-distribution It follows from result 2 that the singular non-circular complex K-distribution can exist on subspace and its p.d.f. is given by the following definition.
Definition 3 A r.v. z ∈ C m is said to have a singular noncircular complex K-distribution with parameters µ ∈ C m ; Ω ∈ C m×m and rank(Ω) = p; Σ ∈ C m×m , Σ ≥ 0 and rank(Σ) = r if its p.d.f. is of the form f K (z) = c r,g K c r,p λ,κ g K (q(z)) , where the quadratic form q(z) defined in result 2, c r,g K = 2ν (r+ν)/2 /[Γ(ν)π r ] is a normalizing constant, g K (.) is the density generator given by g is the shape parameter which controls the shape of complex K-distribution, K (.) denotes the modified Bessel function of the second kind of order . The singular non-circular complex K-distribution is a class of singular NC-CCG distribution and it has the singular NC-CCG representation (28) where the unit mean texture variable τ follows a gamma distribution with shape parameter ν > 0 and scale parameter 1/ν, denoted τ ∼ Gamma(τ, 1/τ ). A singular non-circular complex Kdistribution will be denoted by CK r m,ν (µ, Σ, Ω).

Example 2: Singular non-circular complex t-distribution
It follows also from result 2 that the singular non-circular complex t-distribution can exist on subspace and its p.d.f. is given by the following definition.
Definition 4 A r.v. z ∈ C m is said to have a singular noncircular complex t-distribution with parameters µ ∈ C m ; Ω ∈ C m×m and rank(Ω) = p; Σ ∈ C m×m , Σ ≥ 0 and rank(Σ) = r if its p.d.f. is of the form where the quadratic form q(z) defined in result 2, is a normalizing constant and g T (.) is the density generator given by g T (t) = 1 + 2t ν −(2r+ν)/2 with ν degrees of freedom (2 < ν < ∞).

Example 3: Singular NC-CGG distribution
Similarly, the following definition provides the p.d.f. of singular NC-CGG distribution: Definition 5 A r.v. z ∈ C m is said to have a singular noncircular complex GG (NC-CGG) distribution with exponent s > 0 and scale b > 0 and parameters µ ∈ C m ; Ω ∈ C m×m and rank(Ω) = p; Σ ∈ C m×m , Σ ≥ 0 and rank(Σ) = r if its p.d.f. is of the form f G (z) = c r,g G c r,p λ,κ g G (q c s (z)) , where the quadratic form q(z) is defined in result 2 and g G (.) is the density generator given by g G (t) = exp(−t s /b), which gives c r,g G = sΓ(r)b −r/s / [π r Γ(r/s)] as the value of the normalizing constant. Note that for this singular NC-CGG distribution, the 2nd-order modular variate Q = R 2 is distributed as

Singular widely linear mean square estimation
This section extends the results on linear or widely linear minimum mean-square error (LMMSE or WLMMSE) estimation of a signal from non-singular distributed measurement data vector [28] to the case of estimating a signal from singular distributed measurement data vector.
Let z 1 ∼ EC r1 d (0, Σ 11 , Ω 11 , g) be a singular NC-CES distributed r.v. that need to be estimated from a singular NC-CES distributed r.v. z 2 ∼ EC r2 n (0, Σ 22 , Ω 22 , g), as introduced in result 5. As usual z 1 is considered as signal or source and z 2 as the measurement or observation. We remind the reader here that the scatter matrix Σ 22 is singular and that δ n,g < ∞ such that the covariance matrix Cov(z 2 ) = Σ 22 and the pseudo-covariance matrix pcov(z) = Ω 22 exist. Let Σ 12 and Ω 12 be two matrices defined as Σ 12 and . It follows from result 5 that the conditional mean m(z 2 ) def = E(z 1 |z 2 ) can be expressed as a function of z 2 and z * 2 as follows: where E and F are two matrices defined in result 5, both of which depend on the pseudo-inverse operator. It is clear that m(z 2 ) is singular widely linear (SWL) in z 2 . Note that the estimator m(z 2 ) is called here the singular widely linear minimum mean-squared error (SWLMMSE) estimator of z 1 from z 2 . The error covariance matrix of the SWLMMSE estimator is the covariance matrix of the conditional distribution of z 1 given z 2 , and it follows from result 5 that it is given by The following subsections consider two cases when z 2 is a circular r.v. and z 1 is a real r.v.
Similarly, the error covariance matrix of the SLMMSE estimator is the covariance matrix of the conditional distribution of z 1 given z 2 , and it follows from corollary 2 that it is given by 1026 Singular Non-circular Complex Elliptically Symmetric Distributions: New Results and Applications

Real case
If z 1 is a real-valued parameter vector with singular CES distribution, it is singular NC-CES ditsributed r.v., and the application of SWLMMSE estimator is obvious. In this case, Ω 12 = Σ * 12 , and consequently from (34), the SWLMMSE estimator is of the form m r (z 2 ) = 2 (Ez 2 ).
Therefore, in this case the SWLMMSE estimator produces real-valued estimates, while the SLMMSE estimate is generally complex. Similarly, it follows from (35) that the error covariance matrix takes the form

Application
This section studies the problem of estimating the deterministic but unknown parameter vector of complex-valued linear model in the presence either of singular NC-CES or C-CES distributed error terms. After deriving the compact expressions of the maximum likelihood (ML) estimates of the parameters and their associated covariance matrices, we show that the associated residuals have singular NC-CES distributions. Consider the complex-valued non-circular multivariate linear model where ε, z ∈ C m , X ∈ C m×n is a known matrix of full column rank n and α ∈ C m is an unknown deterministic vector parameter to be estimated. Assume ε ∼ EC r m (0, Σ, Ω, g) such that z ∼ EC r m (Xα, Σ, Ω, g), Σ = σ 2 ε Σ is singular hermitian matrix with rank(Σ) = r and Ω = σ 2 ε Ω is complex symmetric matrix where σ 2 ε is assumed unknown but Σ and Ω are known. Since z is a non-circular complex r.v., the complexvalued linear model (38) in augmented form is

Singular C-CES distributed error term
For singular C-CES distributions, the matrix Ω = O and it follows that z ∼ EC r m (Xα, Σ, g). For fixed σ 2 ε , the ML estimator of α, denotedα, is values of α that maximizes the p.d.f. (12). Since the function g(.) is monotonically decreasing in [0 ∞), it follows that maximizing the p.d.f. (12) with respect to α is equivalent to maximizing the quadratic cost function Since X has full column rank and X H Σ # X is non-singular matrix, the ML estimatorα is given bŷ It is easy to verify that E(α) = α andα is unbiased with covariance matrix using Cov(z) = c 0 Σ and Σ # ΣΣ # = Σ # [40] where c 0 is a positive real scalar. Given that ML estimatorα is a linear transformation of multivariate non-singular C-CES distributed vector z, the ML estimator is non-singular C-CES distributed The residuals vector can be defined as Hence by (44) we have where c 0 is a constant defined in (42) which takes different values according to the choices of CES distributions. It follows that the following statisticσ 2 ε defined in (45) is an unbiased estimator of σ 2 Since the C-CCG distributions presented in section 4 form a subclass of the CES distributions, it follows from corollary 4 that, if ε ∼ CN r m (0, Σ), the quadratic from where q c = rank(H c Σ H H c ) = m − n. Therefore, the statistiĉ σ 2 ε in (45) remains unbiased estimator of σ 2 ε where here c 0 = E(τ ).

Singular NC-CES distributed error term
The r.v. z is assumed to have singular NC-CES distribution, i.e., z ∼ EC r m (Xα, Σ, Ω, g). Following the same reasoning as above, the ML estimator ofα, denotedα for the model (39) is obtained as follows.
whereΓ is defined in result 4 byΓ def = Σ Ω Ω * Σ * . The solution is given bŷ It is clear that this estimator is unbiased with covariance matrix where c 0 is positive real valued scalar such that Cov(z) = c 0 Σ and pcov(z) = c 0 Ω. It is easy to remark that for singular C-CES error where Ω = O, the non-circular ML estimator (46) reduces to the circular ML estimator (41). SinceΓ is (2m × 2m) structured block matrix, its Moore-Penrose pseudo-inverse has the same structure and can be expressed using eigenvalue decomposition (57) asΓ # def = G P P * G * where G def = (Σ − ΩΣ # * Ω * ) # and P def = −Σ # ΩG * are hermitian and complex symmetric matrices, respectively. It follows thatX HΓ#X has the same structure and by using the matrix inversion lemma [40], its inverse is given by and L def = −(X H GX) −1 (X H PX * )K * are hermitian and complex symmetric matrices, respectively. Therefore, from (46) and after simple algebraic manipulations, the non-circular ML estimator of α can be expressed as: where Sinceα is widely linear of multivariate singular NC-CES distributed vector z, the non-singular ML estimator (48) is singular NC-CES distributed where The augmented residuals vector for the model (39) can be defined asẽ where c 0 is a real positive constant defined in (47) which takes different values according to the choices of CES distributions. Therefore the following statisticσ 2 ε defined in (51) is an unbiased estimator of σ 2 Since the NC-CCG distributions presented in section 4 form a subclass of the CES distributions, it follows from result 9 that, if ε ∼ CN r m (0, Σ, Ω), the quadratic fromẽ has the following representatioñ where q c = rank(H) = 2(m − n). Therefore, the statisticσ 2 ε in (45) remains unbiased estimator of σ 2 ε where c 0 = E(τ ). Fig.1. illustrates the estimated of binary phase-shift keying (BPSK) and quadrature Phase shift keying (QPSK) signals, α, using (48) for the underlying complex-valued linear model (38) with error term ε following one of the three distributions: singular NC-CN distribution (CN 3 6 (0, Σ, Ω)), singular noncircular complex t-distribution (Ct 3 6,5 (0, Σ, Ω)) (singular circular complex t-distribution is obtained when Ω = O), parameter vector α consists of 2 identically independently distributed BPSK symbols, each out {+1, −1} or QPSK symbols, each out of {±1 ± j} and a 6 × 2 know matrix X of full column rank. The matrices Σ and Ω are defined as Σ = σ 2 ε AA H and Ω = σ 2 ε A∆ 3 A T with A def = (a 1 , a 2 , a 3 ) and ∆ 3 def = Diag(0.7, 0.6, 0.9) where a k def = (1, e jθ k , . . . , e j(M −1)θ k ) T . The two last distributions are normalized so that Cov(ε) = Σ and pcov(ε) = Ω (i.e., c 0 = 1). It can be seen from Fig.1 that the estimatesα are centered around the true constellation points in the presence of the three distributed error terms. Fig.2 compares the minimum square error (MSE) E((α − α) H (α − α)) associated with the circular ML estimate (41) and non-circular ML estimate (48) of α and the theoretical circular and non-circular bounds given respectively by (42) and (47). As can be seen in this figures, the MSE reaches the theoretical circular bound [resp. non-circular bound] for the three singular C-CES [resp. singular NC-CES] distributed error terms.

Conclusion
Absolutely continuous singular NC-CES distributions are presented by deriving explicit expressions for its p.d.f's. The stochastic representation of the singular NC-CES distributions and quadratic forms in NC-CES r.v. are proved. As special cases, explicit expressions for the p.d.f's of multivariate complex r.v.'s with singular NC-CN distribution and singular NC-CCG distribution are also presented. Some useful properties of singular NC-CES distributions and their conditional distributions are also derived. The singular C-CES distributions are presented as special cases of NC-CES distributions. Singular widely linear mean square estimators of a signal from singular non-circular or circular distributed measurement data vector are derived. The problem of estimating the parameters of a complex-valued non-circular multivariate linear model in the presence either of singular NC-CES or C-CES distributed error terms is presented and followed by deriving widely linear estimators.

A Proof of Lemma 1
Since Σ is singular with rank(Σ) = r, the matrix Σ can be decomposed via eigenvalue decomposition as where the columns of the complex matrixŪ r ∈ C m×m−r are the eigenvectors corresponding to the zero eigenvalues, therefore ΣŪ r = O. Let C be an (m × m) matrix defined as Since C is a complex symmetric matrix of rank p, by Takagi factorization [35] there exists a  Exploiting the block structure of the matrices given by (68), the parameter vector µz 1|z2 is a (r 1 × 1) vector of the conditional augmented (2r 1 ×1) vector (65) and using the matrix inversion lemma, and after some algebraic manipulations, Similarly, the matricesΣ 11.2 andΩ 11.2 are respectively the top left (r 1 ×r 1 ) submatrix and top right (r 1 ×r 1 ) submatrix of the conditional augmented scattered (2r 1 × 2r 1 ) matrix (66). Using the matrix inversion lemma, and after some algebraic manipulations, Using the fact that Σ # 22 = U r,2 Λ −1 r2 U H r,2 , Pz 2 can be expressed as Using (73)-(74) and P # z2 = U * r,2 (U T r,2 P z2 U * r,2 ) −1 U T r,2 , (68) can be written as We conclude, then, that