On Testing the Extent of Noncircularity

Transcription

1 5632 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 11, NOVEMBER 211 On Testing the Extent of Noncircularity Mike Novey, Esa Ollila, an Tülay Aalı Abstract In this corresponence, we provie a multiple hypothesis test to etect the number of latent noncircular signals in a complex Gaussian ranom vector. Our metho sequentially tests the results of iniviual generalize likelihoo ratio test (GLRT) statistics with known asymptotic istributions to form the multiple hypothesis etector. Specifically, we are able to set a threshol yieling a precise probability of error. This test can be use to statistically etermine if a given complex observation is circular Gaussian, an if not, how many latent signals in the observation are noncircular. Simulations are use to quantify the performance of the etector as compare to a etector base on the minimum escription length (MDL) criterion. The utility of the etector is shown by applying it to a beamforming application using inepenent component analysis (ICA). Inex Terms Canonical coorinates, circularity, circularity coefficients, generalize likelihoo ratio test. I. INTRODUCTION Complex-value signals are intrinsic in many applications such as magnetic resonance imaging [1], raar [2], wireless communications [3], an in transforme ata such as the Fourier transform of image ata. Recently, the secon-orer statistics of complex ranom variables, namely the information in the commonly efine covariance an the more recently efine pseuocovariance matrices [4], [5], have been exploite in signal processing algorithms. These statistics are use to classify the signal as circular or noncircular, i.e., the signal is secon-orer circular, or proper, if the pseuocovariance matrix is zero. Depening on the circularity/noncircularity properties of the ata, ifferent classes of algorithms are use for signal processing. For noncircular ata, wiely linear minimum mean square estimation provies avantages over linear estimation [6]. Also in inepenent component analysis, specific algorithms exist to hanle the noncircular non-gaussian case such as shown in [7] an [9]. Consequently, circularity etectors have been uner active research in the recent literature [1] [13], [18] [2]. In [1] an [11], a generalize likelihoo ratio test (GLRT) statistic of circularity assuming complex normality was erive an further stuie in [11] [13]. Ajuste GLRT of circularity erive in [2] has the esirable feature of being robust to epartures from Gaussianity within complex elliptically symmetric (CES) istributions with finite fourth-orer moments. In the univariate case, circularity measures base on characteristic functions were propose [19], whereas [18] consiere GLRT of circularity uner complex generalize Gaussian istribution. Manuscript receive June 23, 21; revise January 22, 211 an April 29, 211; accepte July 16, 211. Date of publication July 29, 211; ate of current version October 12, 211. The associate eitor coorinating the review of this manuscript an approving it for publication was Dr. Arie Yereor. The work of T. Aalı an M. Novey is supporte by the NSF grants NSF-CCF an NSF-IIS M. Novey an T. Aalı are with the Department of Computer Science an Electrical Engineering, University of Marylan Baltimore County, Baltimore, MD 2125 USA ( t.aali@ieee.org; mnovey1@umbc.eu). E. Ollila is with the Department of Signal Processing an Acoustics, SMARAD CoE, Aalto University School of Science an Technology, Espoo, FI-76 Aalto, an also with the Department of Mathematical Sciences, University of Oulu, Oulu 76, Finlan ( esollila@wooster.hut.fi). Color versions of one or more of the figures in this corresponence are available online at Digital Object Ientifier 1.119/TSP Recall that a circular ranom variable (r.va.) is statistically uncorrelate with its complex conjugate. If z represents a noncircular complex ranom vector (r.v.) in, then besies the conventional covariance matrix C(z) [(z )(z ) H ], also the pseuo-covariance matrix [21] P(z) [(z )(z ) T ] nees to be calculate to obtain full secon-orer knowlege of correlations between the components. Above [z] enotes the mean of z. Ifz is circular, then P =. More generally, r.v. z with vanishing pseuo-covariance matrix is referre to as secon-orer circular [5] or proper [21]. This is equivalent to saying that circularity coefficients [7], [22], efine as the orere singular values of the coherence matrix C PC, vanish. As shown in [11], the circularity coefficients are canonical correlations between z an its conjugate z 3. Note also that circularity coefficients are invariant uner nonsingular linear transformations of the ata. Circularity etectors propose in the literature so far are for testing secon-orer circularity of z, the null hypothesis thus being H : 1 = 2 = 111 = =(,P = ) against the general (unrestricte) alternative H 1 : i 6=for at least one i(, P 6= ). Naturally, neither the circularity coefficients nor the pseuo-covariance matrix are ever exactly equal to zero for finite sample lengths an thus we seek for evience that they are statistically significantly ifferent from zero. The test statistic of the GLRT for circularity [1], [11], [13] assuming i.i.. sample z1;...; zn from a complex -variate Gaussian istribution is `n n ln (1 ^ 2 i ) (1) where ^ i ;i = 1;...; are the maximum-likelihoo estimators (MLEs) of the circularity coefficients, i.e., the orere singular values of ^C ^P(^C ) T, where ^C = 1 n n (z i z)(zi z) H an ^P = 1 n n (z i z)(zi z) T enote the MLEs of C an P an z enotes the sample mean. The test statistic has an asymptotic chi-square istribution with ( +1)egrees of freeom uner H [13], [2], i.e., (+1). 2 We note that it is recommene to use the multiplier (n ) instea of the conventional n in `n as a small sample ajustment [13, Sec. VII-B] an that more commonly `n in (1) is represente as `n = n ln[et(i ^%^% 3 )], where ^% = ^C 1 ^P. This latter form for `n is computationally simpler as it avois the explicit computation of the circularity coefficients. However, for the problem aresse in this corresponence, the former representation of the test statistic ientifie in (1) is more insightful. If the hypothesis of circularity H is rejecte, then it is of interest to stuy the extent of circularity (EOC), i.e., how many circularity coefficients are zero, or from a ifferent point of view, the extent of noncircularity (EONC), i.e., how many circularity coefficients iffer from zero. The iea is to test the null hypothesis that k circularity coefficients vanish or, that k circularity coefficients iffer from zero. The null hypothesis is given by : k > k+1 = 111 = = against the general alternative H 1 ; note that H () H. Our EONC etector sequentially forms k = ;...; 1 GLRT statistics with null hypothesis an alternative hypothesis H 1. Each GLRT statistic has a known limiting istribution, hence a confience level an threshol can be chosen. Then the first statistic that accepts the null hypothesis, correspons to the number of noncircular signals. Note that the EONC etector implements a sequential hypothesis etector X/$ IEEE

2 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 11, NOVEMBER similar to those use in etecting the imension of a signal subspace in [14] [16], the only ifference being the efinition of eigenvalues. Another approach, for etecting the number of sources in a noisy linear moel [17], forms a sequence of binary tests eciing between the two hypotheses: most likely of the ( k 1) assumptions having more than k sources an the most likely of the k assumptions to have a maximum of k sources. However in this metho, establishing the threshol is ifficult an must be one empirically. Because of this limitation, we i not pursue this metho further. In this corresponence, first we construct a GLRT for uner the complex Gaussian assumption in Section II by extening the work of [24], [25] for maximizing the likelihoo base on the rank of the crosscorrelation matrix in canonical correlation analysis. Secon, a multiple hypothesis test is forme using the GLRT to test for the number of noncircular sources in Section III-A. To quantify the performance of our metho, we compare its performance with a metho base on the minimum escription length principle [26] in Section III-D. We show empirically that the EONC etector provies better results over the MDL metho. In Section IV, the EONC etector is applie to a beamforming application using inepenent component analysis (ICA). Using the EONC etector to partition the ata into circular an noncircular subsets, we are able to apply ifferent ICA approaches to each ata set an increase the overall performance. II. GLRT FOR Let z 2 be a -imensional complex Gaussian ranom vector an y =[z T ; z H ] T the augmente vector form an =[ T ; H ] T the augmente mean vector. The probability ensity function (pf) of z N (; C; P) is given by [4], [27], [28] where 1 p(y; C; P) = jrj exp 1 2 (y )H R 1 (y ) R = C P P 3 C (2) is the augmente covariance matrix. Assuming n inepenent an ientically istribute samples z 1 ;...; z n, the joint pf of the sample (maximize over ) becomes 1 p(y; C; P)= n jrj exp 1 2 n (y i ^) H R 1 (y i ^) = n jrj exp n 2 22n n 1 R ^R where Y 2 enotes the augmente sample matrix, ^ = [z T ; z H ] T is the augmente sample mean vector an ^R is the augmente sample covariance matrix, i.e., a matrix of the form (2) but with C an P replace by their MLEs ^C an ^P. Now recall that a GLRT ecies the general alternative H 1 against if the ratio of the likelihoo functions is greater than a threshol with the unknown parameters replace by their maximum likelihoo estimates uner each hypothesis [23]. In this case, a GLRT statistic can be written as L n(k) = p max C;P Y; ^C; ^P p Y; C; P (k) > (3) where P (k) enotes the pseuo-covariance matrix with rank r(p (k) )= k. Note that the enominator is maximizing the likelihoo function uner the restricte parameter space efine by the null hypothesis constraining the rank of the pseuo-covariance matrix equal to k. Since P is symmetric it has a special singular value ecomposition, enote Takagi factorization [8], such that P = S 1 3S T where 3 iag( 1;...; ) enotes an orere iagonal matrix with circularity coefficients i s as its iagonal elements an S is the strong-uncorrelating transform (SUT) [7], [22]. Writing P in this form emonstrates that any i = reuces the rank of P resulting in r(p) =k constraint uner. The enominator of (3) can now be formulate as max C;P p Y; C; P (k) = max P :r(p )=k = p(y; ^C; ^P) max p (Y; C; P) C i=k+1 (1 ^ 2 i ) (4) where ^ i are the MLEs for the circularity coefficients. Since the circularity coefficients are the canonical correlations [29] between z an z 3, the secon equality (4) follows using the results from [25]. For future reference we note that the numerator of (3) is simply p(y; ^C; ^P) = n j ^Rj e n = n e n j^cj n (1 ^ 2 i ) : (5) Substituting (4) into (3) an taking two times the log gives us our GLRT statistic `n(k) =2ln(L n (k)) = n ln i=k+1 (1 ^ 2 i ) > (6) where = 2 ln( ). Stanar likelihoo ratio testing theory concerning composite hypothesis testing states that the limiting istribution of the test statistic (4) uner the hypothesis is `n(k) a 2 p where 2 p enotes chi-square pf with p egrees of freeom [23]. The avantage of such a statistic is that the threshol can be set to yiel a probability of etection, an as will be shown in Section III-C, the probability of error. Since P is symmetric, there are (+1) 2 inepenent unknown complex parameters or ( +1)real parameters [2]. However, uner the null hypothesis, there are only k(2 k +1)inepenent unknown real parameters (see Appenix A). Therefore, the number of egrees of freeom is p = ( +1) k(2 k +1)=( k)( k +1)[23]. Thus, the GLRT for uner the Gaussianity assumption rejects the null hypothesis if `n(k) > 2 p;1 (7) where p;1 2 enotes the (1 )th quantile of the chi-square istribution with p egrees of freeom. The above test is asymptotically vali with level (type I error, probability of false alarm (PFA)) equal to. For example, with PFA =:5 (an =6;k =3), the rejection region is p;1 2 =21:26. Next, we investigate the valiity of the p 2 approximation to the finite sample istribution of the test statistic at small sample lengths. For this purpose, we generate N = 5 samples from the complex normal istribution N (; I; 3), = 6, with k = 3 nonzero circularity coefficients 1 =:98, 3 = :616 an 2 =:297. Let `n;[1] 111 `n;[n] enote the orere sample of `n(k) compute from N simulate samples of length n, i.e., the sample quantiles. Then, q [j] = F 1 j:5 N ; j=1;...; N, are the corresponing theoretical quantiles (where.5 in (j:5) ) is a commonly use continuity correction). Then a ( chi-square ) plot of points (q [j] ;`n;[j] ) shoul N resemble

3 5634 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 11, NOVEMBER 211 (4) an (5) into (8) an omitting the irrelevant aitive constant terms yiels MDL(k) =n ln k (1 ^ 2 i ) + k(2 k +1)lnn: (9) Note that the MDL approach oes not have any threshol parameter but chooses the hypothesis that minimizes (9). Fig. 1. Chi-square plots when sampling from N (; I; 3) istribution with k =3nonzero circular coefficients. The vertical an horizontal lines inicate the corresponing values of =:5 an =:1 upper quantiles. a straight line through the origin with slope one. In particular, the theoretical (1 )th quantile 2 p;1 in (7) shoul be close to the corresponing sample quantile in orer for the test to retain the chosen PFA with high accuracy. For n =1an n = 5, the sample.95 (respectively,.99) quantile confirming to PFA = :5 (respectively, =:1) were an 2.84 (respectively, an 25.79), which can be contraste to theoretical quantile p;:95 2 =21:26 (respectively, p;:99 2 =26:217). Thus, when sample length increases, the sample istribution more accurately resembles the chi-square istribution which is supporte by chi-square plots epicte in Fig. 1. Goo fits to the straight line are obtaine even for n = 5, but with smaller sample length (n = 1) the fit is rather poor especially in the upper quantiles. The figure thus recommens the usage of = :5 rather than = :1 as it is better in maintaining the preetermine PFA (especially for small lengths). We wish to highlight that we can set in the simulations P =3 an C = I without loss of generality as the test statistic (an the null hypothesis) is invariant uner invertible linear transformations of the ata. A metho to generate ranom samples from N (;I; 3) is escribe in Appenix B. III. DETECTING THE NUMBER OF NONCIRCULAR SOURCES A. EONC Detector We now evise a sequential etector for the number k of noncircular sources that procees as follows: choose PFA, e.g., = :5, an iterate for k =; 1; 2;...: 1) Calculate `n(k) using (6). 2) Estimate ^k is the first value of k for which `n(k) 2 p ;1 where p k =(k)( k +1)i.e., until the null hypothesis is not rejecte. The etector above is referre to as the EONC etector. B. MDL-Base Detector For comparison purposes, we also implement a multiple hypothesis etector base on information theoretic criterion base on MDL [26], [3]. The MDL approach on moel selection is base on the moel that yiels the minimum coe length, an in the large-sample limit becomes MDL(k) =2ln max C;P p Y; C; P (k) + h ln n (8) where h = k(2k+1) is the number of free parameters of the moel; see Appenix A. The secon term in (8) is a penalty term that increases linearly with moel complexity h an sublinearly with n. Substituting C. Comments on the Selection of As shown in Section II, a choice of = :5 is able to provie an aequate false alarm rate with small sample sizes. However with large sample sizes, a lower value of may provie enhance accuracy, i.e., lower probability of error. Let us efine the probability of error as P e = P u + P o where P u is the probability of unerestimating an P o is the probability of overestimating the number of noncircular signals using the EONC etector. First let us assume the true number of noncircular signals is k, then the probability of selecting some r<kis P r u = Pr(E c \111\E c r1 \ E rj ) (1) where E i = f`n(i) 2 p ;1g enotes the event that the GLRT statistic `n(i) is below the threshol an thus woul lea to acceptance of H (i) ; the total unerestimating probability is P u = k1 i= P u.for i the unerestimating case, the alternate hypothesis is true, therefore the GLRT statistic has an asymptotic noncentral chi-square istribution `n(r) a 2 p () with noncentrality parameter [23] while the threshol is base on the central chi-square istribution. When the circularity coefficients are close to one, i.e., highly noncircular, the noncentrality parameter can be quite large as seen by examining (4). As shown in [31], the estimates ^ = [^ 1;...; ^ ] T are consistent, hence as n! 1;g(^)! p g() in probability provie that g() is a continuous function in. Thus (4), enote by g(), can be written as `n(r) nln (1 i=r+1 2 i ) for large n. Therefore, lim! 1 `n(r)! 1 resulting in P u =since E r =. The overestimating case is quite ifferent however. An overestimating error will occur if there is no unerestimation an the correct null hypothesis is rejecte, thus the probability is P = Pr(E c \111\E c kj ) Pr(E c kj )= since Pr(Er) c = 1 for r < k. The probability of error becomes P e =, consequently a low value for is esire. However, for applications where there is a small number of samples or the expecte values of the circularity coefficients are not known, then a higher value of is warrante. We show this empirically in Fig. 2 by examining the performance of the EONC etector with hypothesis H (3) = true an n = 5 while varying the circularity coefficient values. With =6, the circularity coefficients become [; ; ; ; ; ] as is ajuste from zero to one. The EONC etector s performance is quantifie by the probability of etecting the true hypothesis efine as PD Pr(H (3) jh (3) ). As escribe above, when! 1 we woul expect PD = 1 P e =1. Fig. 2 highlights the performance increases as increases as seen by the PD curve shifting to the left, i.e., better etection at lower values. However, raising increases the P e as seen by a rop in PD, specifically with =1e1. Also note that PD approaches (1 ) at values much less than one, i.e., :4 <1. In conclusion, with no a priori knowlege of the circularity coefficient values or with small sample sizes, a value of =:5 is a statistically soun choice. Also note that the MDL etector has no such ajustment the probability of error was foun to be 1e 4 in this simulation showing that the EONC provie better performance with the same P e.

4 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 11, NOVEMBER TABLE I CONTINGENCY EONC(PFA = :5) DETECTOR FOR n = 1, 5, 15 OBTAINING COR = 464, 81, 94 AND ERR = 5293, 148, 436, RESPECTIVELY Fig. 2. Probability of etecting the correct number of noncircular signals versus circularity coefficients values for both the EONC an MDL etectors with n = 5. D. Simulations We generate 1 samples of lengths n = 1, 5, 2 from = 6 variate CN istribution N (; I; 3) with k largest circularity coefficients generate inepenently from the uniform istribution U(:1; :95) for each sample an k 2 f; 1;...;g. The PFA for the EONC etector was set to =:5. Table I reports the contingency tables of k versus the estimate number of noncircular sources ^k for the EONC etector. Note that each row sums to 1 (= number of samples for each fixe k). As expecte, for increasing n, we have larger numbers, i.e., correct estimation results, on the iagonals. Moreover, as n increases, the contingency matrix becomes closer to a iagonal matrix with smaller sprea. We use two accuracy measures to quantify the success of etection: COR = tr(q) ( +1) ERR = k= ^k= jk ^kjq k^k where Q enotes the contingency matrix an Q k^k its (k; ^k)th element. Note that COR gives the average correct estimation results whereas ERR measures the inaccuracy of the obtaine estimates; it gives more weight to those ^Q k^k the more incorrect the obtaine estimate ^k is from true k. Note that uner perfect etection, Q = I, an COR = 1 an ERR =. These figures for EONC (PFA =:5) etector for sample lengths n = 1, 5, 15 were COR = 464, 81, 94, an ERR = 5293, 148, 436, respectively, illustrating the higher accuracy as n increases. Table II reports the contingency tables of k versus the estimate number of noncircular sources ^k for the MDL etector (6). The results for sample lengths n = 1, 5, 15 were COR = 31, 67, 835, an ERR = 981, 3854, 135, respectively. The results inicate that the EONC etector provies better performance than the MDL etector as seen by comparing the COR an ERR results for both etectors. For example, the EONC provies 33% better performance than the MDL when n = 5. IV. AN APPLICATION TO ICA In this section, we emonstrate the utility of noncircularity etection with an application to blin beamforming [32] using ICA. Suppose that z follows ICA moel, z = As, where the unknown mixing matrix A is of full rank an the unobserve source vector s = [s 1 ;...;s ] T containing the statistically inepenent components is such that s 1 ;...;s k are noncircular an s k+1 ;...;s are circular. We wish to recall that the circularity coefficients i;i = 1;...; of z TABLE II CONTINGENCY MDL DETECTOR FOR n = 1, 5, 15 OBTAINING COR = 31, 67, 835, AND ERR = 981, 3854, 135, RESPECTIVELY in this case are (assuming that ICs are of finite variance) simply the (marginal) circularity coefficients of the sources s i;i =1;...;, that j [s ]j is, i = [js j ]. We can evise hybri ICA metho (HYBICA) to fin all (both the circular an noncircular sources) as follows: 1) Use the EONC etector to fin the number k of noncircular signals. 2) Pre-filter with the SUT algorithm such that y = Sz where S is the SUT. The SUT orers y by ecreasing circularity coefficients so that it can be partitione into noncircular y nc =[y 1 ;...;y k ] T an circular y c =[y k+1 ;...;y ] T vectors.

5 5636 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 11, NOVEMBER 211 contain both the signal of interest an the interference, whereas HY- BICA performance increases ue to the partitioning of the ata. This improvement is ue to HYBICA using the optimal algorithm, C-QAM, for the BPSK sources only. Fig. 3. Performance of HYBICA, JADE2, an C-QAM algorithms versus number of BPSK (interference) sources where the total number of sources is six. 3) Estimate the noncircular sources using the ata y nc an an ICA algorithm suite to noncircular sources. If the circularity coefficients are istinct, then y nc contains the source estimates. 4) Similarly, estimate the circular sources using y c. In the blin beamforming context, A is the irection of arrival (DOA) matrix where the columns of A correspon to the spatial frequency associate with each source across the antenna array. To highlight the utility of noncircularity etection an the HYBICA algorithm, we assume that the signals of interest are binary phase shift keying (BPSK) moulate an hence noncircular. Along with the BPSK sources, we also assume the presence of several emitters of narrow-ban interference that are non-gaussian an circular in nature. Uner these assumptions, the source vector contains k BPSK an k interference sources each having a ifferent angle of arrival. Since the sources of interest are noncircular, we use the HYBICA algorithm first to partition the source vector an then process the noncircular BPSK sources separately. However since the circularity coefficients are the same for BPSK moulate signals, the complex quarature amplitue moulation algorithm (C-QAM) escribe in [33] is use instea of SUT in step 2. The C-QAM algorithm uses a cost function matching the probability mass function of BPSK signals an hence provies near-optimal performance. Simulations are run using 5 samples of = 6sources with the number of BPSK sources varie from k = ; 1;...; 5 an interfering sources k = 6; 5;...; 1. The interference signals use in the simulations are complex (bivariate) Laplacian an therefore super-gaussian. The angle of arrivals of each source is uniformly sprea between 63 egrees with the antenna element spacing 1 2 where is the wavelength. To evaluate the separation performance of the algorithms the interference to signal ratio (ISR) is use an average over the signals of interest, in this case the BPSK moulate source estimates. The ISR is calculate as ISR = 1 k p2d q=1 (v pq ) 2 max l (v pl ) 2 1 ^WA, ^W is the emixing matrix estimator, D where V =(v pq )= 1 is the set of BPSK inices foun from matching the columns of ^W with the irections of arrival of the BPSK sources. Fig. 3 epicts the ISR versus the number of BPSK an interference sources for the HYBICA, JADE2 [32], an C-QAM [33] algorithms. The figure is obtaine by averaging the results of 1 Monte Carlo runs. As seen in the figure, the HYBICA algorithm improves the performance, specifically when there are a large number of interfering signals. Both JADE2 an C-QAM performance egraes when the sources V. CONCLUSIONS AND FUTURE WORK In this corresponence, we erive a GLRT to etect the number of noncircular signals in a complex Gaussian ranom vector. We quantifie the performance of the etector empirically an showe that the etector outperforms a etector base on the MDL criterion. We then emonstrate the utility of the etector by applying it to a beamforming application using ICA. It is very likely that the propose GLRT will eteriorate severely uner violations of the Gaussian ata assumption whereas MDL may perform more reliably. However, as our ICA example illustrate, the EONC etector can provie reliable estimates even when the Gaussianity assumption oes not hol exactly. To aress this concern to some extent, we point out that also the erive GLRT for the hypothesis can be ajuste as in [2] to obtain a test that remains vali within the wie class of CES istributions with finite fourth-orer moments. To be more specific, the ajuste GLRT statistic for the hypothesis is obtaine simply by iviing the GLRT statistic `n(k) by an ajustment factor n, efine as n 1 ^ j 2+j^% j j 2 where ^% j an ^ j are the conventional sample estimators of the circularity coefficient [12]%(z j ) an the (z ) stanarize [jz j ] [jz j ] ( [jz j ]) fourth-orer moment (z j ) of the jth marginal variable z j, respectively (j =1;...;). Then, the ajuste GLRT statistic, ` `n;aj, also possess the same asymptotic 2 p-istribution uner as the nonajuste GLRT (but uner the more general assumption of sampling from an unspecifie CES istribution with finite fourthorer moments). This result follows as in [2] ue to the fact that the Corollary 1 of [34, pp. 415] applies also for hypothesis (an the respective GLRT `n(k)). For example, if we repeat the simulations of Section III-D using the exact same setting with an exception that the ata is generate from centere complex multivariate t-istribution with = 6egrees of freeom having covariance matrix I an pseuo-covariance matrix 3, enote z t 6 (; I; 3), then for n = 15, we obtaine COR an ERR values 88 an 865 (respectively, 837 an 1278) for the ajuste EONC etector (respectively, for the MDL etector). That is, the ajuste EONC etector ha more correct estimation results (i.e., larger COR value) an better accuracy (i.e., smaller ERR value) than the MDL etector. Although the propose GLRT an the EONC etector can be robustifie against non-gaussianity ata assumption, they still suffer from being nonrobust uner heavy-taile noncircular moels or in the face of outliers. One approach to obtain a robust etector is to erive a GLRT for the hypothesis uner the assumption of sampling from the multivariate extension of the generalize Gaussian ensity (GGD); thus the iea is to exten the metho in [18] to the multivariate setting. Such tests will not be pursue herein but are a subject of a separate paper. APPENDIX A NUMBER OF MODEL PARAMETERS UNDER The number of free moel parameters are etermine by the egrees of freeom of pseuo-covariance matrix P (k) with rank k. Noting that P (k) is symmetric, it can be iagonalize using Takagi factorization

6 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 11, NOVEMBER [8], such that P (k) = VDV T, where V is unitary an D is the iagonal matrix of k nonzero singular values. Thus, the total number of parameters is 2k real parameters of V an k real parameters of D resulting in 2k + k total parameters. However, the unitary constraint of V will reuce the number of free parameters. There are k constraints ue to v H i v i =1for i =1;...;k. Also each v H i v j =for i 6= j results in two constraints, one for the real part an one for the imaginary part the result is an aitional k(k 1) constraints. Therefore, the total number of moel parameters is h =2k + k (k + k(k 1)) = k(2 k +1). APPENDIX B GENERATION OF A SAMPLE FROM CN DISTRIBUTION A complex Gaussian r.v. z with mean =, covariance C = I an pseuo-covariance matrix P = iag( 1;...; ) (i.e., z N (; I; 3)) is generate as follows: 1) Choose circularity coefficients i s from [; 1]. 2) Generate a r.v. z from circular stanar (( = ; C = I) CN istribution. 3) Then a r.v. z following N (; I; 3) istribution is obtaine as an -linear transform: z = 1 2 A + B z A B z3 where A = iag( p 1+ i ) an B = iag( p 1 i ). Naturally, to obtain n samples, one simply repeats steps 2 3 n times. To obtain a ranom eviate from N (; C; P) with P = A3A T an C = AA H where 3 is the iagonal matrix of circularity coefficients an A 1 is the strong-uncorrelating transform (SUT), then a single aitional step is neee: 4) A r.v. z following N (; C; P) istribution is obtaine as a -linear transform from z as z = + Az. Step 4 is neee, for example, in simulations with ICA applications. REFERENCES [1] V. Calhoun, T. Aali, L. Hansen, J. Larsen, an J. 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