1

A Complex Generalized Gaussian Distribution—Characterization, Generation, and Estimation Mike Novey, Member, IEEE, Tulay¨ Adalı, Fellow, IEEE, and Anindya Roy

Abstract— The generalized Gaussian distribution (GGD) provides a kurtosis where the scale factor is nonnegative and is a function of flexible and suitable tool for data modeling and simulation, however the noncircularity as shown in [14]. Since the kurtosis of the complex characterization of the complex-valued GGD, in particular generation of Gaussian is zero, as in the real-valued case, positive normalized samples from a complex GGD have not been well defined in the literature. In this study, we provide a thorough presentation of the complex-valued kurtosis values imply a super-Gaussian distribution, i.e., a sharper GGD by i) constructing the probability density function (pdf), ii) defining peak with heavier tails, and negative normalized kurtosis values imply a procedure for generating random numbers from the complex-valued sub-Gaussian distributions. GGD, and iii) implementing a maximum likelihood estimation (MLE) Recently, use of the full second-order statistics of complex random procedure for the shape and covariance parameters in the complex domain. We quantify the performance of the MLE with simulations and variables, namely the information in the commonly used covariance actual radar data. as well as the pseudocovariance matrices [12], [15], have proven Index Terms— MLE, complex-valued signal processing, generalized useful in signal processing. The second-order statistics are used to Gaussian distribution classify the variable as second-order circular or second-order noncir- cular, i.e., a complex-valued random variable is second-order circular if the pseudocovariance matrix is zero. Due to the recent interest in I.INTRODUCTION incorporating the circular/noncircular properties of complex-valued The generalized Gaussian distribution (GGD) has found wide use signals, the CGGD is not restricted to the circular case but is in modeling various physical phenomena in the signal processing also parameterized by the second moment thus providing a means community. For example, the GGD has been used to model synthetic of varying the noncircularity of the distribution. To enhance the aperture radar [1] and echocardiogram [2] images; features for face usefulness of the CGGD, we also provide a method for generating recognition [3]; load demand in power systems [4]; and subband samples from a CGGD as well as an MLE for its shape and signals in images [5]. The use of the GGD has also found utility in covariance parameters. We provide simulations to quantify the MLE’s independent component analysis (ICA) where it is used as a flexible performance and then test on actual radar data. class of source density models (see e.g., [6], [7], [8]). The GGD family of densities [9] is obtained by generalizing the Gaussian density to provide a variable rate of decay and is given by II.COMPLEXPRELIMINARIES “ ”c c − |x| pX (x; σ, c) = e σ A complex variable z is defined in terms of two real variables 2σΓ(1/c) √ zR and zI as z = zR + jzR where j = −1 and alternately as the T where Γ(·) is the Gamma function, σ is the scale parameter, and bivariate vector zb = [zR, zI ] . It is also convenient to work with ∗ T c is the shape parameter. What makes the GGD appropriate in so the augmented vector defined in [13], [16] as za = [z, z ] where many applications is its flexible parametric form which adapts to » 1 j – z = z . a large family of symmetric distributions, from super-Gaussian to a 1 −j b sub-Gaussian including specific densities such as Laplacian (c = 1) Similarly, a complex random variable is defined as Z = ZR +jZI T and Gaussian (c = 2). Although the GGD has found wide use, along with the bivariate Zb = [ZR,ZI ] and augmented Za = most applications employ the univariate version. A bivariate GGD [Z,Z∗]T vector forms. Assuming that E{Z} = 0, the bivariate is introduced in [10] and used in modeling a video coding scheme in covariance matrix is thus

[11] along with a maximum likelihood estimate (MLE) for the shape » 2 – T σR ρ parameter based on a chi-square test, however, the covariance matrix Cb = E{ZbZb } = 2 estimate presented is not an MLE. Complex-valued GGD models ρ σI have been described much less frequently and assume that the signal and the augmented covariance matrix is is circular, i.e., invariant to rotation, as in [1], [6]—both papers use an MLE for estimating the shape parameter. » 2 2 2 2 – H σR + σI (σR − σI ) + j2ρ In this paper, we extend the results for the complex normal distribu- Ca = E{ZaZa } = 2 2 2 2 (σR − σI ) − j2ρ σR + σI tion defined in [12], [13] to the GGD, by providing a fully-complex 2 distribution denoted as CGGD, given in (7). As in the univariate where σ is the variance and ρ is the correlation E{ZRZI }. It is case, the CGGD adapts to a large family of bivariate symmetric clear that the augmented covariance matrix will have real-valued distributions, from super-Gaussian to sub-Gaussian including specific diagonal elements Ca(0,0) = Ca(1,1) and complex-valued off- densities such as Laplacian and Gaussian distributions. Since the ∗ diagonal elements Ca(1,0) = Ca(0,1). For Z to be second-order CGGD is also a member of the elliptically symmetric distributions, ∗ circular, Ca(1,0) = Ca(0,1) = 0, i.e., the variance of ZR and ZI the normalized kurtosis values of the real and imaginary parts of are the same and ZR and ZI are uncorrelated. A measure of second- ˛ 2 ˛ ∗ a complex random variable are a scaled version of the complex order noncircularity [17], [18] is ˛E{Z }˛ /E{ZZ } with bounds ˛ 2 ˛ ∗ ˛ 2 ˛ 0 ≤ ˛E{Z }˛ /E{ZZ } ≤ 1 and ˛E{Z }˛ = 0 indicates circular Copyright (c) 2010 IEEE. The reference for the paper as originally published: M. Novey, T. Adalı, and A. Roy, “A complex generalized Gaussian data. A stronger definition of circularity, strict circularity, is based distribution- characterization, generation, and estimation,” IEEE Trans. Signal on the probability density function of the complex random variable Processing, vol. 58, no. 3, part. 1, pp. 1427–1433, March 2010. such that for any α, the pdf of Z and ejαZ are the same [15]. 2

Γ(2/c) We note the following identities from [13], [19] between the which results in a normalizing term η(c) = 2Γ(1/c) and the second bivariate and augmented forms: line follows from a rectangular to polar coordinate substitution— T H we can similarly show E{ZRZI } = 0 which is expected since the 2 zb zb = za za p p random variable Z is circular. We use this result to normalize the 2 |Cb| = |Ca| (1) variance of Z to unity through the linear transform Wb = NZb q 1 2 eigenvals(Cb) = eigenvals(Ca) where N = 2η(c) I, i.e., dividing by the square root of two times ∗ 2 where | · | is the determinant and eigenvals(·) are the eigenvalues. the standard deviation such that E{ZZ } = 1 and E{ZR} = 2 E{ZI } = 0.5. Applying this transform to (3), we obtain III.CGGD CONSTRUCTION 1 A. Pdf derivation p (w ) = p (N−1w ) Wb b |N| Zb b In this section, we derive the CGGD by first constructing the T c −[2η(c)(w wb)] bivariate pdf and then extending this distribution to the complex case = β(c)e b (5) with a general augmented covariance matrix, i.e., for noncircular cΓ(2/c) T where β(c) = πΓ(1/c)2 and E{WbWb } = 0.5I for c > 0. The data. We note that the bivariate GGD is defined in [11], however identities in (1) allow us to rewrite (5) in the complex-augmented the construction shown here provides insight into how the random form as variables are generated in Section III-B. H c p (w ) = β(c)e−[η(c)(wa wa)] (6) We begin by defining a complex random variable Z = RejΘ that Wa a is a function of two random variables R and Θ. The magnitude where pWa (wa) = pW (wb). Equation (6) is primarily notational 1/q b R, is a modified Gamma variate defined as R = G where since pdfs are defined with respect to real variables, however, this G ∼ Gamma(2/q, 1) is a gamma-distributed random variable with form allows us to work with probabilistic descriptions directly in shape parameter 2/q and unit scale and Θ ∼ U(0, 2π) has a uniform the complex domain as described in [20]. Our goal, however, is distribution. to have a form with a general augmented covariance matrix. We Before finding the pdf of Z, we first find the pdf of R starting tailor the covariance matrix by applying a linear transform Ta to the with the univariate gamma distribution defined as normalized data through Va = TaWa. The covariance matrix is (2/q−1) x −x now given by pG(x, 2/q) = e H H H H H Γ(2/q) Ca = E{VaVa } = TaE{WaWa }Ta = TaIaTa = TaTa . Due to the unique form of the augmented covariance matrix, the diag- where Γ is the gamma function. The Gamma random variable is then onal terms are real valued and the off-diagonal terms are conjugates, raised to the (1/q)th power resulting in the pdf of R given by H Ta = Ta. Now given any arbitrary augmented covariance,√ we find qr −rq pR(r) = e (2) the transform Ta using the matrix square root, i.e., Ta = Ca. The Γ(2/q) matrix square root can be found using the eigenvalue decomposition H where we used the transform of a random variable to a power, i.e., of Ca, such that Ca = V ΛV, where V is the matrix of 1/q q−1 q if R = X then pR(r) = qr pX (r ). eigenvectors and Λ is the diagonal matrix of real-valued eigenvalues The complex variable Z = RejΘ can be rewritten in the bivariate due to the Hermitian symmetric properties√ of the covariance matrix. √ H p case as zR = r cos(θ) and zI = r sin(θ) with inverses r = |z| = It is easy to show that Ca = V ΛV and also |Ta| = |Ca|. p 2 2 zR + zI and θ = atan(zI /zR). The joint distribution of Zb = Applying the transform Ta to the pdf (6), we obtain T [ZR,ZI ] is found through the density transform as 1 −1 1 „q « pVa (va) = pWa (Ta va) 2 2 |Ta| pZb (zb) = p(R,Θ) zR + zI , atan(zI /zR) |J| 1 H −H −1 c = β(c)e−[η(c)(va Ta Ta va)] where p(R,Θ) is the joint distribution of R and Θ and the determinant |Ta| of the Jacobian is 1 H −1 c = β(c)e−[η(c)(va Ca va)] (7) ˛» cos(θ) −r sin(θ) –˛ p|C | |J| = ˛ ˛ = r. a ˛ sin(θ) r cos(θ) ˛ ˛ ˛ which defines the general CGGD distribution parameterized by the 1 Noting that p(R,Θ)(r, θ) = pR(|z|) 2π due to the independence of R shape c and augmented covariance matrix Ca. 1 and Θ and pΘ(θ) = 2π , our joint distribution becomes q 2 2 q/2 p (z ) = e−(zR+zI ) B. CGGD generation Zb b 2πΓ(2/q) c 2 2 c To generate CGG distributed samples with pdf (7) using Matlab = e−(zR+zI ) (3) πΓ(1/c) (www.mathworks.com), we use the same procedure for constructing the pdf as outlined in Section III-A, i.e., first generate the bivariate where we substituted c = q/2 in the last line so the pdf is Gaussian normalized random variable, then substitute the augmented form, when c = 1. The expression in (3) is a GGD that is circular due and lastly apply a transform to yield the desired covariance. Given to the invariance to θ with the variance a function of c. For the the shape parameter c, where c = 1 is Gaussian, and augmented variable to have unit variance (normalized), we first solve for the covariance C , the following procedure generates N independent second moment, E{Z2 } = E{Z2}, with the integral a R I complex variables: ∞ ∞ Z Z c 2 2 c 2 2 −(zR+ZI ) E{ZR} = zR e dxdy 1) Generate n = 1,...,N complex samples: −∞ −∞ πΓ(1/c) z(n) = gamrnd(1/c, 1)1/(2c)e(j2π rand); Z ∞ Z 2π 2 2 c −(r2)c 2) Normalize the complex variance: = r cos (θ) e r dθdr p Γ(2/c) 0 0 πΓ(1/c) w = z/ ηc(c) where ηc(c) = Γ(1/c) ; c » Γ(2/c) – Γ(2/c) 3) Form augmented vector: = = (4) T Γ(1/c) 2c 2Γ(1/c) wa = [w, conj(w)] ; 3

4) Calculate transform from Ca using matrix square root: IV. MLE PERFORMANCE Ta = sqrtm(Ca); Simulations, using data generated with the procedure in Section III- 5) Apply transform: B, are used to quantify the performance of the MLE method outlined va = Tawa in Section III-C; the results are the average of 500 runs. We then test where gamrnd, sqrtm, conj, and rand are Matlab functions. the MLE on actual sea clutter which is a good source of complex- valued data with a nonstationary distribution. The results of the shape parameter estimator are shown in Figures C. MLE estimator for the covariance C and shape parameter c b 1 and 2. In Figure 1, we plot the shape parameter estimate versus the Our approach for estimating the shape parameter and covariance true shape parameter with sample sizes of 128, 256, and 512 with matrix is to use a maximum likelihood approach. Since our parame- circular and noncircular data—the noncircular data has |E{Z2}| = ters are both real valued and complex valued, we choose to work in 0.9. The results show how well the MLE tracks the true value the real domain for our MLE, i.e., the bivariate vector form. Our with only a slight positive bias when c > 2 and N = 128. Also starting point is the log of the pdf (7) with N independent and note that the performance does not degrade with this high value of identically distributed samples which results in the likelihood function noncircularity. Figure 2 depicts the standard deviation of the shape N parameter estimate with the same data as the previous figure. As L(v ; φ) = Nln(β(c)) − ln(|C |) − b 2 b indicated, the standard deviation increases linearly with the shape N c parameter and is the same for both circular and noncircular data. c X “ T −1 ” η (c) vb (t)Cb vb(t) . (8) Figure 3 depicts the performance of the Cb estimate by plotting t=1 the mean square error (MSE) between the estimate and the true covariance matrix using circular and noncircular data. What the figure where our parameter vector is φ = [σ2 , σ2, ρ, c]T and the bivari- R I shows is that the MLEs performance increases with both sample ate vectors are substituted for the augmented vectors. Setting the size and shape parameter with near identical performance using both derivative of (8) to zero does not yield a closed form solution to the circular and noncircular data. Figure 4 compares the performance of parameter vector, and hence a numerical solution is warranted. Our the MLE and the sample covariance estimator with N = 256 by method is the three step procedure: depicting the MSE of both estimators versus shape parameter. What 1) The initial covariance matrix is estimated using the sample we glean from the figure is that the MLE shows better performance ˆ 0 1 P T covariance Cb = N t vb(t)vb (t); then the sample covariance estimator as expected, however both 0 2) The initial shape parameter cˆ is estimated using a moment estimators perform the same when c = 1 since the sample covariance estimator as suggested in [5], [21]; is the MLE for the Gaussian case. Figure 5 depicts the number of 3) A Newton-Raphson iteration is used to find the final estimated steps for the Newton-Raphson iteration to converge versus shape ˆ values φ. parameter. As seen in the figure, the MLE converges in about five We show in the simulations section that this three step procedure iterations on average. provides fast convergence, typically in five steps with an accuracy of Next, we test the MLE on complex-valued sea-clutter data with a 10−5, over a wide range of parameter values. small target collected with the McMaster University IPIX radar off In step two, we implement a method of moments estimator prior the coast of Canada, http://soma.crl.mcmaster.ca/ipix/, see [22] for to the Newton-Raphson iteration to aid in convergence. The moment more details. The data that we are using is from file 19 with radar used in the estimator is the scale-invariant fourth moment term parameters: X-band, 30 m range resolution, horizontal polarization, defined as and pulse repetition time of 10−3 seconds. The data is collected in E{Z4 } E{Z4} κ(v) = R + I . blocks of 256 time samples with adjacent blocks overlapping by 128 2 2 2 2 E {ZR} E {ZI } samples. Each block is then transformed to the frequency domain which is then tested with the MLE. We test two range gates, one Using a procedure similar to the one given in (4), we find with a small target in clutter and one with clutter only. Figure 6 3Γ(1/c)Γ(3/c) depicts the shape parameter estimate for each block with and without κ(v) = . (9) [Γ(2/c)]2 a target. What we glean from the figure is that when a target is present the distribution becomes more super-Gaussian as seen by the Our moment estimator then solves for the root of smaller shape parameter values. This is expected since a target in E{Z4 } E{Z4} 3Γ(1/c)Γ(3/c) f(c) = R + I − the frequency domain is a line component causing a heavier tail. 2 2 2 2 2 E {ZR} E {ZI } Γ (2/c) The clutter-only data is closer to a Gaussian distribution but still shows areas of low c values, around block 150 for example. This resulting in the estimator demonstrates the non-stationary nature of sea clutter due to the wind cˆ = arg min |f(c)| (10) and wave interactions with the sea and the ability of our MLE to c follow these fluctuations. This demonstrates the utility of modeling over the domain c ∈ [0.1, 4] for the simulations. sea clutter with the CGGD. The moment estimator given in equation (10) and the sample covariance are not MLEs, however they provide an accurate initial V. CONCLUSION value to the Newton-Raphson iteration defined as We introduced a complex-valued generalized Gaussian probability φn = φn−1 − H−1∇ distribution along with a procedure to generate samples from the CGGD. Also presented is a maximum likelihood estimator for the ∂L ∂2L where ∇ = ∂φ is the gradient and H = ∂φ∂φT is the shape parameter and covariance matrix using a a Newton-Raphson Hessian matrix evaluated at φn−1. Both ∇ and H are derived iteration. We show empirically the performance of the MLE on in the appendix. The MLE coded in Matlab can be found at circular and noncircular simulated data and then on complex-valued http://mlsp.umbc.edu/resources. radar data. 4

(a) Circular (b) Noncircular |E{Z2}| = 0.9 (a) Circular (b) Noncircular |E{Z2}| = 0.9

Fig. 1. Shape parameter estimate versus true shape parameter with circular Fig. 3. Mean square error between Cb estimate and the true covariance and noncircular data with sample sizes of 128, 256 and 512. versus shape parameter with circular and noncircular data.

2 (a) Circular (b) Noncircular |E{Z2}| = 0.9 (a) Circular (b) Noncircular |E{Z }| = 0.9 Fig. 2. Shape parameter estimator’s standard deviation versus shape param- Fig. 4. Mean square error of Cb estimate using MLE and sample covariance eter with circular and noncircular data. estimator versus shape parameter with N = 256.

N ∂2L N(σ2 )2 APPENDIX R c X ˆ c−2 2 2 2 = 2 − η (c)c (c − 1) (m(t)) my(t) ∂(σ ) 2|Cb| I t=1 A. Derivation of gradient of likelihood function c−1 ˜ + (m(t)) myy(t) , We begin with the gradient of equation (8) with respect to the parameter vector φ given by 2 2 N ∂ L |Cb| + 2ρ c X ˆ c−2 2 2 = N 2 − η (c)c (c − 1) (m(t)) mρ(t) N ∂ρ |Cb| 2 t=1 ∂L −NσI c X c−1 2 = − η (c)c (m(t)) mx(t), c−1 ˜ ∂σ 2|Cb| + (m(t)) mρρ(t) , R t=1 2 N ∂L −NσR c X c−1 2 2 2 N 2 = − η (c)c (m(t)) my(t), ∂ L |Cb| − σRσI c X ˆ c−2 ∂σ 2|Cb| = −N − η (c)c (c − 1) (m(t)) × I t=1 2 2 2 ∂σR∂σI 2|Cb| N t=1 ∂L Nρ c X c−1 c−1 ˜ = − η (c)c (m(t)) mρ(t), and mx(t)my(t) + (m(t)) mxy(t) , ∂ρ |Cb| t=1 0 N 2 2 N ∂L Nβ (c) c 0 X c ∂ L −NσI ρ c X ˆ c−2 = − η (c)ln(η(c))η (c) m (t) 2 = 2 − η (c)c (c − 1) (m(t)) mx(t)mρ(t) ∂c β(c) ∂σ ∂ρ |Cb| t=1 R t=1 N c−1 ˜ + (m(t)) mxρ(t) , c X c −η (c) m (t)ln(m(t)) t=1 2 2 N ∂ L −NσRρ c X ˆ c−2 2 2 = − η (c)c (c − 1) (m(t)) my(t)mρ(t) T −1 ∂m(t) y −σ m(t) 2 2 I ∂σI ∂ρ |Cb| where m(t) = vb (t)Cb vb(t), mx(t) = 2 = |C | , t=1 ∂σR b 2 2 c−1 ˜ ∂m(t) x −σRm(t) ∂m(t) 2ρm(t)−2xy + (m(t)) myρ(t) , my(t) = 2 = |C | , mρ(t) = ∂ρ = |C | , ∂σI b b β0(c) = ∂β(c) = β(c) + 2β(c) (ψ(1/c) − ψ(2/c)), η0(c) = ∂η(c) = ∂c c c2 ∂c 2 » „ 0 « – N η(c) ∂ L c cη (c) X c−1 2 (Ψ(1/c) − 2Ψ(2/c)), and Ψ(·) is the digamma function. = −η (c) c ln(η) + + 1 (m(t)) m (t) c ∂σ2 ∂c η x R t=1 N c X c−1 B. Derivation of Hessian of likelihood function −η (c)c (m(t)) ln(m(t))mx(t), t=1 The terms of the Hessian are the second and cross derivatives of equation (8) with respect to the parameter vector φ resulting in 2 » „ 0 « – N ∂ L c cη (c) X c−1 = −η (c) c ln(η) + + 1 (m(t)) my(t) 2 2 2 N ∂σ2∂c η ∂ L N(σ ) c X c−2 2 I t=1 = I − η (c)c ˆ(c − 1) (m(t)) m (t) 2 2 2 x N ∂(σ ) 2|Cb| R t=1 c X c−1 −η (c)c (m(t)) (m(t))m (t), c−1 ˜ ln y + (m(t)) mxx(t) , t=1 5

and 1 η00(c) = ˆ(η0(c)c2 − 2cη(c))(Ψ(1/c) − 2Ψ(2/c))− c4 η(c)(Ψ(1, 1/c) − 2Ψ(1, 2/c))] .

REFERENCES [1] M.S. Davis, P. Bidigare, and D. Chang, “Statistical modeling and ML parameter estimation of complex SAR imagery,” in Proc. Asilomar Conference on Signals, Systems and Computers, Nov. 2007, pp. 500– 502. (a) Circular (b) Noncircular |E{Z2}| = 0.9 [2] O. Bernard, J. D’Hooge, and D. Fribouler, “Statistical modeling of the radio-frequency signal in echocardiographic images based on Fig. 5. Number of steps for MLE to converge versus shape parameter with generalized Gaussian distribution,” in 3rd IEEE International Symposium circular and noncircular data. on Biomedical Imaging: Nano to Macro, Arlington, VA, April 2006, pp. 153–156. [3] D. Gonzalez-Jimenez, F. Perez-Gonzalez, P. Comesana-Alfaro, L. Perez- Freire, and J.L. Alba-Castro, “Modeling Gabor coefficients via general- ized Gaussian distributions for face recognition,” in IEEE International Conference on Image Processing, San Antonio, TX, Oct. 2007, vol. 4, pp. 485–488. [4] M.O. Mohamed Mahmoud, M. Jaidane-Saidane, J. Souissi, and N. Hiza- oui, “Modeling of the load duration curve using the asymmetric generalized Gaussian distribution: Case of the tunisian power system,” in IEEE Power and Energy Society General Meeting - Conversion and Delivery of Electrical Energy in the 21st Century, Pittsburgh, PA, July 2008, pp. 1–7. [5] M. Do and M. Vetterli, “Wavelet-based texture retrieval using general- ized Gaussian density and Kullback-Leibler distance,” IEEE Trans. on Image Proc., vol. 11, no. 2, pp. 146–158, Feb. 2002. [6] M. Novey and T. Adalı, “Adaptable nonlinearity for complex max- imization of nongaussianity and a fixed-point algorithm,” in Proc. IEEE Workshop on Machine Learning for Signal Processing (MLSP), Maynooth, Ireland, Sept. 2006, pp. 79–84. Fig. 6. Shape parameter estimate using the frequency domain sea clutter [7] Jianting Cao and N. Murata, “A stable and robust ICA algorithm based data with and without a target versus block number. on t-distribution and generalized Gaussian distribution models,” in IEEE Signal Processing Society Workshop on Neural Networks for Signal Processing, Madison, WI, Aug. 1999, pp. 283–292. [8] Z. Koldovsk, Y. Tichavsk, and E. Oja, “Efficient variant of algorithm 2 » „ 0 « – N fastICA for independent component analysis attaining the Cramer-Rao ∂ L c cη (c) X c−1 = −η (c) c ln(η) + + 1 (m(t)) mρ(t) lower bound,” IEEE Trans. Neural Networks, vol. 17, no. 5, pp. 1265– ∂ρ∂c η 1277, 2006. t=1 [9] N. Johnson and S. Kotz, Distributions in Statistics–continuous univariate N distributions–2, Houghton Mifflin Co., Boston, MA, 1970. c X c−1 −η (c)c (m(t)) ln(m(t))mρ(t), [10] T. Taguchi, “On a generalization of Gaussian distribution,” Annals of t=1 the Institute of Statistical Mathematics, vol. 30, pp. 211–242, 1978. [11] M. Coban and R. Mersereau, “Adaptive subband video coding using bivariate generalized Gaussian distribution model,” in Proc. ICASSP, and Atlanta, GA, May 1996, vol. 4, pp. 1990–1993. [12] B. Picinbono, “Second-order complex random vectors and normal ∂2L » −(β0(c))2 β00(c) – distributions,” IEEE Transactions on Signal Processing, vol. 44, no. = N + − 10, pp. 2637–2640, Oct. 1996. ∂c2 (β(c))2 β(c) [13] A. van den Bos, “The multivariate complex normal distribution—a ˆηc(c) 2(η(c)) − η0(c)(η(c))c−1 − η00(c)ηc(c) (η)˜ × generalization,” IEEE Trans. Information Theory, vol. 41, no. 2, pp. ln ln 537–539, March 1995. N N [14] E. Ollila and V. Koivunen, “Adjusting the generalized likelihood ratio X c c 0 X c m (t) + η (c)ln(η)η (c) m (t)ln(m(t)) − test of circularity robust to non-normality,” in Proc. IEEE International Workshop on Signal Proc., Perugia, Italy, June 2009. t=1 t=1 [15] B. Picinbono, “On circularity,” IEEE Trans. Signal Processing, vol. 42, " N N # pp. 3473–3482, Dec. 1994. c X c X c 2 η (c) ln(η(c) m (t)ln(m(t)) + m (t)ln (m(t)) [16] P. J. Schreier and L. L. Scharf, “Second-order analysis of improper t=1 t=1 complex random vectors and processes,” IEEE Trans. Signal Processing, vol. 51, pp. 714–725, March 2003. [17] J. Eriksson and V. Koivunen, “Complex random vectors and ICA models: 2 2 2 2 2 ∂mx(t) −y σI −σI mx(t)|Cb|+(σI ) m(t) Identifiability, uniqueness, and separability,” IEEE Trans. on Information where mxx(t) = 2 = |C |2 , ∂σR b Theory, vol. 52, no. 3, pp. 1017–1029, March 2006. 2 2 2 2 2 ∂my (t) −x σR−σRmy (t)|Cb|+(σR) m(t) [18] P. Schreier, “The degree of impropriety (noncircularity) of complex myy(t) = 2 = |C |2 , ∂σI b random vectors,” in Proc. ICASSP, Las Vegas, Nevada, April 2008, pp. ∂mρ(t) 2 ˆ 2 ˜ 3909–3912. mρρ(t) = = 2 |Cb|(m(t) + ρmρ(t)) + 2ρ m(t) − 2xyρ , ∂ρ |Cb| 2 2 2 2 2 [19] A. van den Bos, “Complex gradient and Hessian,” in IEE Proc. Image ∂mx(t) −y σR−|Cb|(m(t)+σI my (t))+σRσI m(t) Signal Processing, Dec. 1994, vol. 141, pp. 380–382. mxy(t) = 2 = |C |2 , ∂σI b 2 2 [20] T. Adalı, H. Li, M. Novey, and J.-F. Cardoso, “Complex ICA using ∂mx(t) 2y ρ−σI (mρ(t)|Cb|+2ρm(t)) nonlinear functions,” IEEE Trans. on Signal Processing, vol. 56, pp. mxρ(t) = = 2 , ∂ρ |Cb| 2 2 4536–4544, Sept. 2008. ∂my (t) 2x ρ−σR(mρ(t)|Cb|+2ρm(t)) myρ(t) = = 2 , [21] M. Varanassi and B. Aazhang, “Parametric generalized Gaussian density ∂ρ |Cb| estimation,” J. Acoustical Soc. Amer., vol. 85, pp. 1404–1415, Oct. 1989. [22] S. Haykin, Adaptive Radar Signal Processing, John Wiley and Sons, ∂β0(c) β0(c)c − β(c) Hoboken, NJ, 2007. β00(c) = = + ∂c c2 2 ˆΨ(1/c)(β0(c)c2 − 2cβ(c)) − β(c)Ψ(1, 1/c)˜ − c4 2 ˆΨ(2/c)(β0(c)c2 − 2cβ(c)) − 4β(c)Ψ(1, 2/c)˜ , c4