User Determined Superdirective Beamforming

Transcription

1 IEEE -th Convention of Electrical an Electronics Engineers in Israel User Determine Superirective Beamforming Reuven Berkun, Israel Cohen Technion, Israel Institute of Technology Technion City, Haifa 3, Israel Jacob Benesty INRS-EMT, University of Quebec Montreal, QC H5A 1K, Canaa Abstract Superirective beamforming is a well-known metho for enhancement of reverberate speech signals. Nevertheless, it is very susceptible to errors in the sensor array characteristics, thermal noise, an white noise input, resulting in a low level of white noise gain, particularly at low frequencies. It is of great interest to evelop a beamformer with superior enhancement of reverberate signals, having a high irectivity factor, together with a relatively high white noise gain level. In this paper, a solution which controls both the irectivity factor an the white noise gain is examine. We propose a linear weighte combination of two conventional beamformers, the regularize superirective beamformer an the elay-an-sum beamformer. We analyze the beamformer gain responses, an consequently erive two user-etermine frequency-epenent white noise gain an irectivity factor beamformers, respectively. Simulation results approve our finings, an show of a robust user-controlle solution, with an effective traeoff between the performance measures of the beamformer. I. INTRODUCTION Microphone array processing for speech signals is a challenging task of great importance. Speech signals propagating in a close environment frequently suffer from reverberation an noise, istorting the quality an the intelligibility of the perceive signals. Beamforming represents a class of multichannel signal processing algorithms, that enable an extraction of esire source signals together with a suppression of unesire noise an reverberation 1 3. Superirective beamforming is a well-known approach that ensures high gain for reverberate signals, moele by a iffuse noise input. However, its high sensitivity to spatially white noise, significantly egraes its performance in practice 1,. Slight errors between the array characteristics, such as position errors an mismatches between the sensors, pass through the beamformer like spatially white noise or uncorrelate noise. Therefore, the white noise gain is an important measure for the robustness of the beamformer. Extensive research was conucte regaring the esign of such an aaptive robust beamformer, that woul have both high array gain an a satisfying level of white noise gain 1 9. In, Cox et al. formulate the problem an introuce few approaches for the optimization problem. They presente a conventional optimal constraine solution, an propose ifferent methos to implement it 5,. Others hanle ifferent types of beamforming mismatch errors, such as an arbitrary-type mismatch approach 1, or various moels of the problem, such as worst-case optimization 7 or other optimization methos 11. Recently, we propose an approach of a robust beamformer with fine control of the array gain response (i.e., the irectivity factor) an the white noise gain measure. In this paper, we expan our solution, ealing with an optimization problem of the irectivity factor maximization uner constraint of the white noise gain. The suggeste approach offers a simple optimize beamformer, as oppose to most of the familiar stateof-the-art solutions, which involve iterative solutions or linear This research was supporte by the Israel Science Founation (grant no. 113/11). programming methos 7. Similar to the analysis in, we propose a close-form solution of a beamformer with user-efine fine control on the white noise gain an the irectivity factor. We suggest a linear weighte combination of two conventional beamformers, enabling simple yet effective control of the filter response. Base on that, we erive filters which attain any esire frequency-epenent white noise gain or irectivity factor. The paper is organize as follows. In Section II, we escribe the signal moel an formulate the problem. In Section III, we present conventional fixe beamformers, one that maximizes the white noise gain an another that maximizes the irectivity factor. In Section IV, the propose beamformer is introuce. Base on that, we erive beamformers with user-etermine white noise gain or user-etermine irectivity factor. This approach is base on a combination of the aforementione conventional beamformer with the regularize aapte beamformer. This metho provies a useretermine management of both white noise gain an irectivity factor, an attains an effective traeoff between the two measures. Finally, simulation results emonstrating the beamformer properties are presente in Section V. II. SIGNAL MODEL AND PROBLEM FORMULATION Let us consier a source signal (plane wave), in the farfiel, propagating in an anechoic acoustic environment at the spee of soun, i.e., c 3 m/s, an impinging on a uniform linear sensor array consisting of M omniirectional microphones, the istance between two successive sensors is equal to δ. The irection of the source signal to the array is parameterize by the azimuth angle θ. In this context, the steering vector (of length M) is given by (ω,θ) 1 e jωτcosθ e j(m 1)ωτcosθ T, (1) the superscript T is the transpose operator, j 1 is the imaginary unit, ω πf is the angular frequency, f > is the temporal frequency, an τ δ/c is the elay between two successive sensors at the angle θ. We consier fixe beamformers with small values of δ, like in superirective 5, or ifferential beamforming 1, 9, the main lobe is at the angle θ (enfire irection) an the esire signal propagates from the same angle. Then, our goal is to esign linear array beamformers, which are able to achieve supergains at the enfire with a better control on white noise amplification. For that, a complex weight, H m(ω), m 1,,...,M, is applie at the output of each microphone, the superscript enotes complex conjugation. The weighte outputs are then summe together to form the beamformer output. Putting all the gains together in a vector of length M, we get h(ω) H 1(ω) H (ω) H M(ω) T. ()

2 The mth microphone signal is given by Y m(ω) e j(m 1)ωτ X(ω)+V m(ω), m 1,,...,M, (3) X(ω) is the esire signal an V m(ω) is the aitive noise at the mth microphone. In a vector form, (3) becomes y(ω) Y 1(ω) Y (ω) Y M(ω) T x(ω)+v(ω) (ω)x(ω)+v(ω), () x(ω) (ω)x(ω), (ω) (ω, ) is the steering vector at θ (irection of the source), an the noise signal vector, v(ω), is efine similarly to y(ω). The beamformer output is then Z(ω) M Hm(ω)Y m(ω) h H (ω)y(ω) m1 h(ω)(ω)x(ω)+h H (ω)v(ω), (5) Z(ω) is suppose to be the estimate of the esire signal, X(ω), an the superscript H is the conjugate-transpose operator. We constrain the solution to be istortionless, i.e., h H (ω)(ω) 1. () If we take microphone 1 as the reference, using the efinitions for input signal-to-noise ratio (SNR) an output SNR, the gain in SNR is efine as Gh(ω) osnrh(ω) isnr(ω) h H (ω)γ. (7) v(ω)h(ω) Γ v(ω) Φv(ω) φ V1 (ω) an Φv(ω) E v(ω)v H (ω) are the pseuo-coherence an correlation matrices of v(ω), respectively, an φ V1 (ω) E V 1(ω) is the variance of V 1(ω). In the fiel of superirective beamformers, we are usually intereste in two types of noise. The temporally an spatially white noise with the same variance at all microphones 1. In this case, Γ v(ω) I M, I M is the M M ientity matrix. Therefore, the white noise gain (WNG) is efine as Wh(ω) h H (ω)h(ω). () We can easily euce that the maximum WNG is W max M which is frequency inepenent. The white noise amplification is the most serious problem with superirective beamformers, which prevents them from being wiely eploye in practice. The iffuse noise, Γ v(ω) ij Γ (ω) ij sinω(j i)τ ω(j i)τ sincω(j i)τ. (9) In this scenario, the gain in SNR is calle the irectivity factor (DF) an it is given by Dh(ω) h H (ω)γ (ω)h(ω). (1) It is easy to verify that the maximum (frequency epenent) DF is D max(ω) H (ω) (ω)(ω). We refer to Dmax(ω) as supergain when it is close to M, which can be achieve with 1 This noise moels appropriately the sensor noise. This situation correspons to the spherically isotropic noise fiel. a superirective beamformer but at the expense of white noise amplification. Then, one of the most important issues in practice is how to compromise between Wh(ω) an Dh(ω). Ieally, we woul like Dh(ω) to be as large as possible with Wh(ω) 1. III. CONVENTIONAL BEAMFORMERS In this section, we review two conventional fixe beamformers; one that maximizes the WNG an another that maximizes the DF. Afterwars we relate to a regularize version of the secon approach. The most well-known beamformer is the elay-an-sum (DS), which is erive by maximizing the WNG () subject to the istortionless constraint (). We get h DS(ω) (ω) H (ω)(ω) (ω) M. (11) Therefore, with this filter, the WNG an the DF are, respectively, an Wh DS(ω) M W max () Dh DS(ω) M 1. (13) H (ω)γ (ω)(ω) This beamformer maximizes the WNG an never amplifies the iffuse noise since Dh DS(ω) 1. However, in reverberant an noisy environments, our aim is to obtain high DF for goo speech enhancement (i.e., ereverberation an noise reuction). This unfortunately oes not happen with the DS beamformer, that malfunctions in reverberant rooms, even with a large number of microphones. The secon important beamformer is obtaine by maximizing the DF (1) subject to the istortionless constraint (). We get the wellknown superirective beamformer 5: h S(ω) (ω)(ω) () H (ω) (ω)(ω). Its WNG an DF are, respectively, H (ω) Wh S(ω) (ω)(ω) H (ω)γ (ω)(ω) (15) an Dh S(ω) H (ω) (ω)(ω) D max(ω). () While the DS beamformer has maximal an constant WNG response, but suffers from low DF, the superirective beamformer, on the other han, maximizes the DF but has a negative WNG. We can express the WNG as Wh S(ω) W maxcos ϕ(ω), (17) cosϕ(ω) cos (ω), (ω)(ω) H (ω) (ω)(ω) H (ω)(ω) H (ω)γ (ω)(ω) () is the cosine of the angle between the two vectors (ω) an (ω)(ω), with cos ϕ(ω) 1. At low frequencies, cos ϕ(ω) can be very close to. As a result, W h S(ω) can be smaller than 1, which implies white noise amplification. While the superirective beamformer attains maximum

3 irectivity factor, which is goo for speech enhancement in reverberant rooms, it amplifies the white noise to intolerable levels, especially at low frequencies. Since () is sensitive to the spatially white noise, Cox et al. 5, propose to maximize the DF subject to a constraint on the WNG. Uner the istortionless constraint (), the obtaine optimal solution is 5, h S,ǫ(ω) Γ (ω)+ǫi M 1 (ω) H (ω)γ (ω)+ǫi M 1 (ω), (19) ǫ is a Lagrange multiplier. This is a regularize version of (), ǫ can be seen as the regularization parameter. This parameter tries to fin a compromise between a supergain an white noise amplification. A small ǫ leas to a large DF an a low WNG, while a large ǫ gives a low DF an a large WNG. Two interesting cases of (19) are h S,(ω) h S(ω) an h S, (ω) h DS(ω). We can express (19) as anǫ-regularize superirective beamformer: h S,ǫ(ω) ǫ (ω)(ω) H (ω) ǫ (ω)(ω), () Γ ǫ(ω) Γ (ω)+ǫi M is a regularize version of the pseuocoherence matrix of the iffuse noise. The corresponing WNG an DF for this beamformer are, respectively, H (ω) ǫ (ω)(ω) Wh S,ǫ(ω) (1) H (ω)γ ǫ (ω)(ω) an Dh S,ǫ(ω) H (ω) ǫ (ω)(ω) H (ω) ǫ (ω)γ (ω) ǫ (ω)(ω). () Similarly to (17), we can express the WNG (1) as Wh S,ǫ(ω) W maxcos ϕ ǫ(ω), (3) cosϕ ǫ(ω) cos (ω), ǫ (ω)(ω) H (ω) ǫ (ω)(ω) H (ω)(ω) H (ω)γ ǫ (ω)(ω) () is the cosine of the angle between the two vectors (ω) an ǫ (ω)(ω), with cos ϕ ǫ(ω) 1. For small ǫ, cosϕ ǫ(ω) woul be similar to cosϕ(ω). Large ǫ woul enlarge Wh S,ǫ(ω), so that cos ϕ ǫ(ω) woul be closer to 1. While h S,ǫ(ω) has some control on white noise amplification, it is certainly not easy to fin a close-form expression for ǫ given a esire value of the WNG. A. Derivation IV. PROPOSED BEAMFORMER As we saw, the DS an the regularize superirective beamformers achieve maximum WNG an high DF, respectively. Therefore, we suggest to linearly combine the aforementione beamformers into the following beamformer: h α,β,ǫ (ω) α(ω)h S,ǫ(ω)+β(ω)h DS(ω) (5) ǫ (ω)(ω) α(ω) H (ω) ǫ (ω)(ω) +β(ω)(ω) M α(ω)i M +β(ω)h DS(ω) H (ω) h S,ǫ β(ω)i M +α(ω)h S,ǫ(ω) H (ω) h DS, α(ω) an β(ω) are two real numbers with α(ω)+β(ω) 1. () It is easy to verify that with the conition (), this beamformer (5) is istortionless, i.e., h H α,β,ǫ(ω)(ω) 1. This beamformer controls the regularization with ǫ, an the DS or regularize superirective influence with α(ω) an β(ω). It is not har to show that the WNG of h α,β,ǫ (ω) is Wh DS(ω)Wh S,ǫ(ω) Wh α,β,ǫ (ω) α (ω)wh DS(ω)+1 α (ω)wh S,ǫ(ω) W maxcos ϕ ǫ(ω) Wmax, (7) α (ω)+1 α (ω)cos ϕ ǫ(ω) which epens on the WNGs of the DS an regularize superirective beamformers. We see that for α(ω) in this case β(ω) 1, we have Wh,1,ǫ(ω) Wh DS(ω), an for α(ω) 1 so β(ω), we have Wh 1,,ǫ(ω) Wh S,ǫ(ω). Also, we have Wh α,β,ǫ (ω) W h S,ǫ(ω), α (ω) 1, () suggesting that we shoul always choose 1 α(ω) 1. If we efine D max,ǫ(ω) H (ω) ǫ (ω)(ω), it can be verifie that the inverse DF corresponing to h α,β,ǫ (ω) is D 1 h α,β,ǫ (ω) α (ω)d 1 h S,ǫ(ω)+ α(ω)β(ω) H (ω)γ (ω) ǫ (ω)(ω) +β (ω)d 1 h M H (ω) ǫ (ω)(ω) DS(ω) α (ω)d 1 h S,ǫ(ω)+α(ω)β(ω) Dmax,ǫ(ω) ǫ 1 1 M +β (ω)d 1 h DS(ω), (9) which epens on the DFs of the DS an regularize superirective beamformers. We observe that for β(ω), i.e., α(ω) 1 we have Dh 1,,ǫ(ω) Dh S,ǫ(ω), an for β(ω) 1 so α(ω), we have Dh,1,ǫ(ω) Dh DS(ω). These results are consistent with the ones obtaine for the WNGs. Also, we have Dh α,β,ǫ (ω) Dh S,ǫ(ω) (3) Dh α,β,ǫ (ω) Dh DS(ω), β (ω) 1, (31) suggesting that we shoul always take 1 β(ω) 1. From all of the above we euce that α(ω),β(ω) 1. Examples of the WNG an the DF of h α,β,ǫ (ω) beamformer are escribe in Fig. 1(a)-(b). Our goal is to esign a beamformer with aequate WNG level an relatively high DF. When we esign the filter parameters, first we set the regularization factor ǫ. It will etermine the maximal DF Dh S,ǫ(ω), an the minimal WNG Wh S,ǫ(ω). Setting the parameters α(ω) an β(ω) is epene on what we esire. Next, we iscuss two interesting approaches. In the first approach, we woul like to fin the value of α(ω) in such a way that Wh α,β,ǫ (ω) W (ω), W (ω) is a useretermine frequency-epenent WNG response, withwh S,ǫ(ω) < W (ω) < M, ω. Using (7), we fin that ( ) α Wmax cos (ω) W 1 ϕ ǫ(ω) (3) (ω) 1 cos ϕ ǫ(ω) from which we euce two possible solutions for α(ω): W max α ±(ω) ± 1 cotϕǫ(ω). (33) W (ω) The corresponing values for β(ω) are β 1(ω) 1 α +(ω), β (ω) 1 α (ω). (3)

4 From the two pairs of solutions {α +(ω),β 1(ω)} an {α (ω),β (ω)}, we obviously choose the first one; therefore the obtaine beamformer is h α+,β 1,ǫ(ω) α +(ω)h S,ǫ(ω)+β 1(ω)h DS(ω). (35) with two possible solutions for β(ω), marke by β ±(ω). Therefore, the corresponing values for α(ω) are α 1(ω) 1 β +(ω), α (ω) 1 β (ω), (37) from which we take the positive one, an obtain the user-etermine DF beamformer: h α1,β +,ǫ(ω) α 1(ω)h S,ǫ(ω)+β +(ω)h DS(ω). (3) B. Justification First, we can rewrite the robust superirective beamformer (19) as h S,ǫ(ω) S ǫ(ω)γ (ω)+ǫi M 1 (ω), (39) S ǫ(ω) 1 H (ω)γ (ω)+ǫi M 1 (ω) () is a scaling factor which ensures that h S,ǫ(ω) is istortionless. Therefore, S ǫ(ω) has no effect on the robustness of the filter an only the term ǫi M in Γ (ω) + ǫi M 1 (ω) has, since it is the linear system that we want to solve. Let us assume that α(ω) ; Using Woobury ientity we can express the propose beamformer as h α,β,ǫ (ω) α(ω)i M +β(ω)h DS(ω) H (ω) h S,ǫ(ω) 1 α(ω) I M β(ω) (ω)h (ω) H ǫ (ω)(ω) (ω)(w) H (ω) ǫ (ω)(ω) S α(ω) Γ (ω) β(ω) Γǫ(ω)(ω)H (ω) 1(ω), H (ω)(w) (1) S α(ω) α(ω) H (ω) ǫ (ω)(ω) () is a scaling factor which ensures that h α,β,ǫ (ω) is istortionless. We can say that (1) is also a regularize superirective beamformer with a rank-one complex matrix. Let us assume that β(ω). In this case, the propose beamformer can be rewritten as h α,β,ǫ (ω) β(ω)i M +α(ω)h S,ǫ(ω) H (ω) h DS(ω) β(ω) I M α(ω) Γ 1 ǫ (ω)(ω) H 1 (ω) (ω) H (ω) ǫ (ω)(w) H (ω)(ω) 1(ω), S β (ω) I M α(ω) Γ 1 (3) S β (ω) ǫ (ω)(ω) H (ω) H (ω) ǫ (ω)(w) β(ω) H (ω)(ω) () is a scaling factor which ensures that h α,β,ǫ (ω) is istortionless. Again, we can consier (3) as a regularize form of the superirective beamformer. V. SIMULATIONS In the secon approach, we woul like to esign the parameters In general, both the filter-esign parameters an the physical in such a way that Dh α,β,ǫ (ω) D (ω), D (ω) is a properties of the microphone array etermine the filter response. First, esire frequency-epenent DF with Dh DS(ω) < D (ω) < the regularization factor ǫ controls the range of the WNG an DF Dh S,ǫ(ω), ω. We can express (9) as a secon egree polynomial we can get, as note in () an (3). To emboy this, we ae of β(ω): β (ω) { D 1 h DS(ω)+D 1 h the response of h S,ǫ(ω) to the illustrate simulations. Clearly, the } S,ǫ(ω) Dmax,ǫ(ω) ǫ 1 1 array physical properties, such as the number of elements M an the M +β(ω) { Dmax,ǫ(ω) ǫ 1 1 M D 1 h } microphone spacing δ, affect the response as well 5,. S,ǫ(ω) First, we simulate the user-etermine WNG beamformer +D 1 h S,ǫ(ω) D 1 (ω), (3) h α+,β 1,ǫ(ω) (35). In Fig. 1(c)-(), we show an example of the WNG an the DF response of such beamformer. The etermine WNG example of W (ω) + 5sin( 1 ω) B provies a tolerable π white noise amplification, above the minimal WNG of Wh S,ǫ(ω), together with a satisfying DF, between the DF of the superirective beamformer an the DF of the regularize superirective beamformer. Of course, any esire WNG can be set, within the allowe range. Next, we simulate the user-etermine DF beamformer h α1,β +,ǫ(ω) by solving (3) for a linear equation example: D (ω) + 1 π 1 3 ω B. The corresponing WNG an DF responses are illustrate in Fig. 1(e)-(f). We observe that the receive WNG is between the WNG levels of the regularize superirective an the DS beamformers, as expecte. Likewise, the etermine D (ω) is limite within the range of Dh DS(ω) < D (ω) < Dh S,ǫ(ω), ω. Finally, following 5, to emonstrate the influence of the filter parameters on the WNG DF traeoff, we show in Fig. the DF curve vs. WNG for increasing α(ω), for ifferent ǫ values. The parameter α(ω) α, ω, varies from to 1 along the curves, β(ω) 1 α corresponingly. This example inicates of a monotonic relationship between α (given a specific regularization factor ǫ) an the gains of the beamformer. Increase of the WNG from its minimal value Wh S,ǫ(ω) at α to its maximum, at α 1, causes monotonic ecrease in the DF from its maximal value (of Dh S,ǫ(ω)) to the low DF of the DS beamformer. One can see that setting ifferent regularization factor ǫ changes the WNG DF traeoff vastly, hence there is a major importance of choosing an appropriate value for this parameter as well. VI. CONCLUSION We have extene our previously propose approach to robust regularize beamforming by using a linear combination of a regularize version of the superirective beamformer an the DS beamformer. The propose solution allows the user to ajust each of these beamformers influence, by setting the beamformer parameters α(ω) an β(ω), in favor of the esire application. Using the propose beamformer enables us to achieve any user-etermine WNG or DF (such as the sine an linear equation shown here), within a wie allowe range. We examine the WNG DF traeoff, an analyze the influence of the filter parameters α, β an ǫ on the WNG DF relationship. The presente approach opens a winow to a wie family of combinations of known beamformers, with management of the beamformer frequency response. By setting the filter parameters, the user etermines the weight of each factor, hence controlling the behavior of the beamformer WNG an DF.

5 WNG B WNG B WNG B 1 Wh DS (ω) Wh α,β,ǫ (ω) Wh S (ω) 1 Wmax Wh α+,β 1,ǫ (ω) Wh S (ω) Wh S,ǫ (ω) 1 Wmax Wh α1,β +,ǫ (ω) Wh S (ω) Wh S,ǫ (ω) f KHz (a) f KHz (c) f KHz (e) DF B 1 DF B 1 DF B 1 Dh S (ω) Dh α,β,ǫ (ω) Dh DS (ω) f KHz (b) Dmax Dh α+,β 1,ǫ (ω) Dh DS (ω) Dh S,ǫ (ω) f KHz () Dmax Dh α1,β +,ǫ (ω) Dh DS (ω) Dh S,ǫ (ω) f KHz Fig. 1: Examples of the array gain of each beamformer, with M 1 microphones, δ 1 cm, an ǫ 1 1. The top figures illustrate the propose beamformer WNG (soli line) versus frequency. As a reference, W max (ashe line), Wh S(ω) (otte line), an Wh S,ǫ(ω) (ot-ash-ot line) are plotte. The bottom figures illustrate the propose beamformer DF (soli line). As a reference, D max(ω) (ashe line), Dh DS(ω) (otte line), an Dh S,ǫ(ω) (ot-ash-ot line) are plotte. (a)-(b) WNG an DF of h α,β,ǫ (ω), with α β.5. (c)-() WNG an DF of h α+,β 1,ǫ(ω) (35). The esire WNG is set to W (ω) +5sin( 1 ω) B. (e)-(f) WNG an DF of h π α 1,β +,ǫ(ω) as a solution of (3). The esire DF is set to D (ω) + 1 π 1 3 ω B. (f) DF B ǫ ǫ 1 1 ǫ ǫ WNG B 5 1 Fig. : The DF curve versus WNG, of h α,β,ǫ (ω) for increasing α, for ifferent values of ǫ: ǫ (soli line), ǫ 1 1 (ashe line), ǫ (otte line), an ǫ (ot-ash-ot line). M 1 microphones, δ 1 cm, an α is monotonically increase from to 1 (with β 1 α corresponingly). REFERENCES 1 G. W. Elko an J. Meyer, Microphone arrays, in Springer Hanbook of Speech Processing, J. Benesty, M. M. Sonhi, an Y. Huang, Es., Berlin, Germany: Springer-Verlag,, Chapter, pp J. Benesty, J. Chen, an Y. Huang, Microphone Array Signal Processing. Berlin, Germany: Springer-Verlag,. 3 B. D. Van Veen, an K. M. Buckley, Beamforming: A versatile approach to spatial filtering, IEEE ASSP Magazine, vol. 5, no., pp., April 19. M. Branstein, an D. War, Microphone arrays: Signal processing techniques an applications. Berlin, Germany: Springer-Verlag, 1. 5 H. Cox, R. M. Zeskin, an T. Kooij, Practical supergain, IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-3, pp , June 19. H. Cox, R. M. Zeskin, an M. M. Owen, Robust aaptive beamforming, IEEE Trans. Acoust., Speech, Signal Process., vol. ASSP-35, pp , Oct S. A. Vorobyov, A. B. Gershman, an Z.-Q. Lou, Robust aaptive beamforming using worst-case performance optimization: A solution to the signal mismatch problem, IEEE Trans. Signal Process., vol. 51, no., pp , Feb. 3. R. Berkun, I. Cohen, an J. Benesty, Combine beamformers for robust regularize superirective beamforming,, submitte to IEEE Trans. Ausio, Speech, an Language Process., July. 9 J. Benesty an J. Chen, Stuy an Design of Differential Microphone Arrays. Berlin, Germany: Springer-Verlag,. 1 S. Shahbazpanahi, A. B. Gershman, Z. Q. Luo, an K. M. Wong, Robust aaptive beamforming for general-rank signal moels, IEEE Trans. Signal Process., vol. 51, no. 9 pp. 57 9, Sep S. Doclo, an M. Moonen, Superirective beamforming robust against microphone mismatch, IEEE Trans. Auio, Speech, an Language Process., vol. 15, no. pp , Feb. 7. G. A. F. Seber, A Matrix Hanbook for Statisticians. Hoboken, NJ: John Wiley & Sons, Inc.,.