A linearly constrained minimum variance beamformer with a pre-specified suppression level over a pre-defined broad null sector

Signal Processing 109 (2015) 165–171

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A linearly constrained minimum variance beamformer with a pre-specified suppression level over a pre-defined broad null sector Alon Amar, Miriam A. Doron 1 Signal Processing Department, Acoustics Research Center, Rafael, Haifa, P.O. Box 2250, Israel

a r t i c l e i n f o

abstract

Article history: Received 23 July 2013 Received in revised form 2 September 2014 Accepted 13 November 2014 Available online 20 November 2014

We present a minimum variance beamformer with a pre-specified suppression level over a pre-defined angular null sector, which for example may be used when the interference moves across an a-priori known angular sector. To reduce the design computational complexity, we replace the original quadratic constraint of the optimization problem with a set of linear constraints. Given the suppression level, we determine the minimal number of linear constraints that ensures that the original quadratic constraint holds. Simulations show that the array gain of the proposed beamformer exceeds that of the classical minimum variance beamformer for a finite number of samples and coherent interference scenario, using the sample matrix inverse technique or the diagonal loading approach. & 2014 Elsevier B.V. All rights reserved.

Keywords: Minimum variance beamformer Interference suppression

1. Introduction Minimum variance beamforming is widely used in communications, radar, sonar, radio astronomy, tomography and seismology [1,2]. One of the challenges of designing such a beamformer arises when the interference direction may not be accurately known or may vary slightly with time. It is then desired to maintain a suppressed angular region (a trough) in the beampattern. Techniques to produce a trough include the use of covariance matrix tapering [3–5], multiple pattern nulls around the interference direction [2] or derivative constraints at the interference direction [6,7]. Another approach is using a generalized eigenanalysis technique [8] to design the minimum power distortionless response (MPDR) beamformer subject to the quadratic constraint that requires the output power in a pre-defined angular sector, Δθ, around the interference direction, will be E-mail addresses: [email protected] (A. Amar), [email protected] (M.A. Doron). 1 M.A. Doron was part of the Signal Processing Department during the research of the current paper. http://dx.doi.org/10.1016/j.sigpro.2014.11.009 0165-1684/& 2014 Elsevier B.V. All rights reserved.

below a pre-specified suppression level, η. In [9] the objective is to minimize the squared norm of the beamformer weight vector (and not the beamformer output power), subject to a similar quadratic constraint as in [8]. The optimal beamformer involves an iterative procedure which is needed to determine the Lagrange multipliers. (In [10] this optimal and iterative approach is extended to suppress an interference sector in case of a broadband adaptive beamforming.) To reduce the computational complexity, the quadratic constraint is approximated by linear constraints, obtained from an eigenvalue decomposition of the quadratic constraint [9]. This method results in a closed form expression for the weight vector without any need to perform iterative calculations. Using an additional pre-defined threshold, ξ, the number of linear constraints is determined as the largest number of eigenvalues, such that the ratio of their sum and the sum of all eigenvalues, is smaller than ξ [9]. Herein, we consider the beamformer design problem in [8] using the approximation technique in [9]. However, instead of using an additional threshold as in [9] to determine the number of linear constraints, we express the MPDR output power at the trough for a given number of linear

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constraints. We suggest to select the number of linear constraints as the smallest rank of the eigenvalue decomposition used for the quadratic constraint, which ensures that the output power is less than the pre-specified level, η. This results in a linear constrained sector suppressed (LCSS) MPDR beamformer. Another contribution of the current work is presented in the simulation results. We show that the LCSS MPDR beamformer has larger array gain compared to the classical MPDR (obtained by minimizing the output power such that the output power at the direction of interest is distortionless), at the non-asymptotic region, i.e., for a finite number of samples. We demonstrate another advantage of the LCSS beamformer over the classical MPDR beamformer in its ability to combat multipath scenario where the interference is correlated with the signal. In this case, constrained sector suppression of the interference prevents the unwanted signal suppression problem of the classical MPDR beamformer. It is worth to mention that there are similarities between the current work and robust adaptive beamforming techniques [11–14]. Robust adaptive beamforming is focused on minimizing the output power of the beamformer subject to the assumption that there are uncertainties regarding the source direction (due for example to calibration errors). To mitigate the uncertainties regarding the source direction, several robust adaptive beamforming techniques have been suggested including the recently suggested robust adaptive beamformer [13]. Specifically, the technique in [13] minimizes the output power of the beamformer subject to the constraint that the estimate of the steering vector of the source direction is forced not to converge to any interference steering vector outside the assumed sector of interest of the source direction. Herein, the direction of the source of interest is assumed to be exactly known, while the goal is to suppress the interference due to uncertainties regarding its direction, and as such, the current setup is focused on a different perspective of interference suppression. 2. Signal model and problem assumptions Consider an array with M sensors. A source of interest and an interference are located in the far field region of the array, and transmit narrowband signals, sD(t), and sI(t), respectively, which are observed by the array during an observation time, 0 rt r T. Assume that the direction of the source of interest, denoted by θD , is constant during the observation time. The interference direction, θI , on the other hand, is not constant during the observation time. As a first example, consider the case where the interference moves with respect to the array, such that its direction varies with time as

θI ðt Þ ¼ θI;0 þ

Δθ
T

 t  T=2 ;

0 rt r T;

where Δθ is the total direction change during the observation interval, and θI;0 is the direction at the center of the observation. As a second example, consider that the orientation of the interference fluctuates with time, or that the interference is located in a local time-varying scattering environment, such that during the observation interval, the interference direction, as observed by the array, is randomly

selected from the interval ðθI;0  Δθ=2; θI;0 þ Δθ=2Þ. The baseband signal at the output of the array at the sampling time ti, where 0 ¼ t 1 ot 2 o⋯ ot N ¼ T, is expressed by the M  1 vector, xðt i Þ ¼ ½x1 ðt i Þ; …; xM ðt i ÞT given by xðt i Þ ¼ aðθD ÞsD ðt i Þ þaðθI ðt i ÞÞsI ðt i Þ þ nðt i Þ;

ð1Þ

where aðθÞ is the M  1 array response vector, and nðt i Þ ¼ ½n1 ðt i Þ; …; nM ðt i ÞT is the M  1 vector of noises at the array output. We assume that the noises, fnðt i ÞgN i ¼ 1 , are zero mean white Gaussian random variables (r.v.) with 2 identical variance σn. Also, the signal samples fsD ðt i ÞgN i ¼ 1, and fsI ðt i ÞgN , are mutual independent zero mean Gaussian i¼1 2 2 r.v., with variances σD, and σI , respectively, and are also independent of the noise samples. In the sequel, we write aD , and aI instead of aðθD Þ; and aðθI ðtÞÞ, respectively. Let w ¼ ½w1 ; …; wM T be the M  1 beamformer weight vector. As the interference direction may vary in an angular sector, Δθ, around a nominal direction, θI;0 , the goal is to find a weight vector, w, that minimizes the beamformer output power, E½jwH xðtÞj2  ¼ wH Rw, where R ¼ E½xðt i ÞxH ðt i Þ is the covariance matrix of xðtÞ (In practice, we replace R with the sample covariance matrix, H R^ ¼ ð1=NÞ∑N i ¼ 1 xðt i Þx ðt i Þ.), not only subject to a distortionless constraint, wH aD ¼ 1, but also that the average power at the beamformer output in this sector, denoted by 1

Δθ

Z θI;0 þ Δθ=2 
  E wH a θ j2 dθ θI;0  Δθ=2

is suppressed below a pre-specified level, η. In other words, we want to find a weight vector, w, that solves the constrained optimization problem: min wH Rw w

s:t: wH aD ¼ 1;

wH Qw r η;

ð2Þ

where Q is an M  M matrix given by Q¼

1

Δθ

Z θI;0 þ Δθ=2

 a θ aH θ dθ: θI;0  Δθ=2

ð3Þ

We next present the optimal solution of the optimization problem, and an approximated solution based on linearizing the quadratic constraint.

3. A quadratic constrained sector suppressed MPDR We start by presenting the optimal solution of (2) (A similar approach is presented in [9,10]). Using the Lagrange multipliers technique we define H LðwÞ ¼ wH Rw þ ðwH aD 1Þμ1 þ μn1 ðaH D w  1Þ þ μ2 ðw Qw  ηÞ:

ð4Þ Taking the derivative with respect to w and equating it to zero yields w ¼  μ1 ðR þ μ2 Q Þ  1 aD . Substituting w in the distortionless response constraint yields μ1 ¼ ðaH D ðR þ μ2 Q Þ  1 aD Þ  1 . Finally, substituting μ1 in w results in the quadratic constrained sector suppressed (QCSS) MPDR weight vector wQCSS ¼

ðR þ μ2 Q Þ  1 aD aH D ðR þ

μ2 Q Þ  1 aD

:

ð5Þ

A. Amar, M.A. Doron / Signal Processing 109 (2015) 165–171

ðCH R  1 CÞ  1 , and get that

Substituting (5) into the power constraint yields f






μ2 ¼

1 1 aD Þ2 ðaH D ðR þ μ2 Q Þ

1 aH Q ðR þ μ2 Q Þ  1 aD r η: D ðR þ μ2 Q Þ

Since f ðμ2 Þ 4 0, and ∂f ðμ2 Þ=∂μ2 o0, we conclude that f ðμ2 Þ is a monotonic decreasing function. We determine μ2 by an iterative procedure starting with an initial positive guess, and iteratively incrementing μ2 until f ðμ2 Þ r η. This involves a large number of computations as it requires a matrix inversion at each iteration. The optimization problem in (4) can be solved together with a robustness constraint such as J w J 2 r α which reduces the sensitivity to direction of arrival mismatch and array perturbations [2, Chapter 6]. The solution is similar to (5), where instead of R þ μ2 Q we write R þ μ2 Q þ αI, and adding αI is a version of the diagonal loading (DL) approach.

wLCSS ðr Þ ¼

R  1=2 PU?r R  1=2 aD ; H aD R  1=2 PU?r R  1=2 aD

0 0 0 qð r 0 Þ ¼ w H LCSS ðr ÞQ ðr ÞwLCSS ðr Þ;  1=2 ?  1=2 ? aH PUr0 R  1=2 UM  r0 ΛM  r0 UH PUr0 R  1=2 aD M  r0 R DR  1=2 ? ðaH PUr0 R  1=2 aD Þ2 DR

4. A linear constrained sector suppressed MPDR To reduce the complexity load of the QCSS MPDR beamformer design, we solve the optimization problem in (2) by approximating the quadratic constraint by a set of linear constraints, similar to [9]. Ideally, the goal is to obtain a zero power response at the desired sector, that is, wH Qw ¼ 0. Since Q is an Hermitian matrix we can factorize it as Q ¼ UΛUH , where U ¼ ½u1 ; …; uM  is an M  M matrix containing the orthonormal eigenvectors of Q, and Λ ¼ diagðλ1 ; …; λM Þ is an M  M diagonal matrix containing the eigenvalues of Q in a decreasing order. Assume that Q  M has a rank equal to r, that is, λn n ¼ r þ 1 ¼ 0. Let Ur ¼ ½u1 ; …; ur . We get that the quadratic constraint wH Qw ¼ 0 is satisfied if wH Ur ¼ 0. Combining this constraint with the distortionless constraint in (2) yields the linear constrained optimization problem: s:t: wH C ¼ eT1 ;

ð9Þ

1  1=2 where PU?r ¼ I  R  1=2 Ur ðUH Ur Þ  1 UH is the M  M r R r R projection matrix which spans the subspace orthogonal to the subspace spanned by R  1=2 Ur . To find the optimal rank, we consider a candidate rank r 0 A f1; 2; …; Mg. We then decompose Q as Q ðr 0 Þ ¼ H Ur0 Λr0 UH r 0 þ UM  r 0 ΛM  r 0 UM  r 0 where Ur 0 ¼ ½u1 ; …; ur 0 , UM  0 r ¼ ½ur0 þ 1 ; …; uM , Λr0 ¼ diagðλ1 ; …; λr0 Þ and ΛM  r0 ¼ diagðλr0 þ 1 ; …; λM Þ. For each rank r 0 we obtain a weight vector wLCSS ðr 0 Þ for which the output power at the trough is,

¼

min wH Rw

167

¼

M

∑

n¼

r0

þ1

λn

 1=2 ? jaH PUr0 R  1=2 un j2 DR

J PU?r0 R  1=2 aD J 2

;

ð10Þ

;

where in the second transition we use the result that PU?r R  1=2 Ur0 ¼ 0, and in the third transition we use the identity PU?r0 ¼ PU?r0 PU?r0 . We see that qðr 0 Þ decreases as r 0 increases. Following the constraint, qðr 0 Þ r η, we need to determine the smallest r 0 such that this constraint still holds. Hence, we select the optimal rank r 0opt as r 0opt ¼ arg min qðr 0 Þ r η:

ð11Þ

r 0 ¼ f1;2;…;Mg

We note that asymptotically the beamformer output power is P LCSS out

 3 2  

  1 2  6 MPDR 0  7 : N-1 ¼ E4wH LCSS r opt xðt Þ 5 ¼ H  1=2 ?  1=2  R P aD a Ur0 R D  opt

ð6Þ

ð12Þ

where C ¼ ½aD ; Ur , and e1 ¼ ½1; 0T T . Using the Lagrange multipliers we define

Recall, that the classical MPDR beamformer minimizes the output power subject to a distortionless response constraint, wH aD ¼ 1, and its output power is P MPDR ¼ out 1 E½jwH xðtÞj2  ¼ ðaH aD Þ  1 . The ratio between the output DR powers of the classical MPDR and the LCSS beamformer is   1=2 ?  aH PUr0 R  1=2 aD gH PU?r0 g DR  P MPDR opt opt out N-1 ¼ ¼ MPDR  1 gH g P LCSS aH aD  out DR

w

H H LðwÞ ¼ wH Rw þ ðwH C eH 1 Þμ þ μ ðC w  e1 Þ:

ð7Þ

The result of ∂LðwÞ=∂w ¼ 0 is w ¼  μ C R . Substituting w in the distortionless response constraint yields μ ¼  ðCH R  1 CÞ  1 . Finally, substituting μ in w results in the LCSS MPDR beamformer: H

wLCSS ðrÞ ¼ R  1 CðCH R  1 CÞ  1 e1 ;

H

1

ð8Þ

where we emphasize the dependency of w on the rank r. Similar to the QCSS MPDR beamformer, in practice, we ^ and then replace R with the sample covariance matrix, R, the sample matrix inversion (SMI) method can be applied 1 where we calculate R^ with or without DL. In practice, Q is a numerically rank deficient matrix, i.e.,  M the assumption that λn n ¼ r þ 1 ¼ 0 does not hold. We then need to determine the minimal number of linear constraints that ensures that the quadratic constraint holds. In [9] the rank is determined as the largest integer r, where r A f1; …; Mg, such that ∑ri ¼ 1 λi =∑M i ¼ 1 λi r ξ where ξ is another threshold. Instead of using additional threshold, we propose a different technique to determine the optimal rank based on the given pre-specified suppression level, η. Using the matrix inversion lemma [2] we express

¼ J PU?r0 g~ J 2 ¼ 1  J PUr0 g~ J 2 ; opt

ð13Þ

opt

where g ¼ R  1=2 aD and g~ ¼ g= J gJ . Since the ratio is always less than unity, we conclude that P MPDR out jN-1 MPDR jN-1 . Asymptotically, the MPDR provides the rP LCSS out minimal output power. Thus, the price of using the LCSS MPDR beamformer is an increase of the output power in the asymptotic region (i.e., as N-1). Still, the MPDR beamformer is optimal only when the source and the interference are uncorrelated, otherwise its output power decreases due to self-signal cancellation [2]. In practice, the benefits of applying the LCSS MPDR beamformer, as we show in the numerical examples, are twofold: (i) it outperforms the MPDR at the non-asymptotic region, i.e., for finite N, MPDR P LCSS jfinite N o P MPDR out out jfinite N and (ii) contrary to the MPDR, it overcomes the correlation between the source and the

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0

−10

−10

−20

−20

−30

−30 10[DEG]

10[DEG]

BEAM PATTERN [dB]

BEAM PATTERN [dB]

0

−40

−50

η=−60 [dB]

−60

−40

−50

−60

−70

−70

−80

−80

−90

−90

−100

−100 0

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180

SOURCE DIRECTION [DEG]

η=−60 [dB]

10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180

SOURCE DIRECTION [DEG]

45

45

40

40

35

35

ARRAY GAIN [dB]

ARRAY GAIN [dB]

Fig. 1. Beampatterns of the QCSS MPDR (left) and the LCSS MPDR (right), θI;0 ¼ 60○ , Δθ ¼ 10○ , η ¼ 10  6 .

30

25

20

30

25

20

15

15

10

10

5

5 10

2

N, NUMBER OF SAMPLES

10

3

10

2

10

3

N, NUMBER OF SAMPLES

Fig. 2. AG of the basic MPDR, QCSS MPDR, LCSS MPDR, eigen-based and robust adaptive beamformers versus N for the SMI without DL (left) and with DL of LNR ¼10 dB (right) for INR of 30 dB for each case.

A. Amar, M.A. Doron / Signal Processing 109 (2015) 165–171

169

Example 1. We considered an interference located at a nominal direction θI;0 ¼ 60○ with INR ¼20 dB, and SNR¼0 dB. In Fig. 1 we show the beampatterns of the QCSS MPDR and the LCSS MPDR beamformers versus the source direction, when the beamformers were computed from the asymptotic covariance matrix. Both beamformers suppress the specified sector. The attenuation of the LCSS MPDR in this sector is slightly less accurate compared to the QCSS MPDR since it is based on optimizing a function over discrete values (see Eq. (10)).

interference and prevents the self-signal cancellation by explicit suppression of the interference angular sector.

5. Numerical examples We demonstrate the performance of the QCSS MPDR and LCSS MPDR beamformers via computer simulations. The proposed beamformers are compared with the following methods: (1) the classical MPDR, (2) the eigen-based beamformer [14], and the recently suggested robust adaptive beamforming [13] with a sector width around the source direction of 101 [13]. We consider a linear array with 20 sensors with a half wavelength spacing. The source of interest is at direction θD ¼ 90○ (broadside direction). We define the signal to noise ratio (SNR) and the interference to noise ratio (INR) as SNR ¼ σ 2D =σ 2n , and INR ¼ σ 2I =σ 2n , respectively. Unless stated otherwise, the sector width is Δθ ¼ 10○ , with a pre-specified level of η ¼ 10  6 .

Example 2. We evaluated the array gain (AG) of the different beamformers versus the number of samples N. We considered an interference located at nominal direction θI;0 ¼ 30○ . The SNR is 0 dB. Given N, we performed Nexp ¼ 100 independent trials, where in each trial the desired signal, the interference signal, and the noise were randomly generated according to their Gaussian distribution. Also, the interference direction was selected

Table 1 AG of the basic MPDR, QCSS MPDR, LCSS MPDR, eigen-based and robust adaptive MPDR beamformers for a correlated signals with a correlation coefficients, ρ¼ 0.95, and INR¼ 30 dB, SNR¼ 30 dB, LNR ¼10 dB, and θI;0 ¼ 30○ . 30

50

100

200

500

1000

AGEIGEN½14 AGRAB½13 AGLCSS AGQCSS AGMPDR

0.5199 1.5967 24.9858 24.9516 0.5221

0.4201 1.4450 26.9275 26.9081 0.4186

0.4614 1.4962 29.3617 29.3130 0.4621

0.4562 1.4898 32.4051 32.3441 0.4559

0.4416 1.4672 35.6597 35.5277 0.4417

0.4613 1.4909 37.8814 37.8083 0.4613

80

80

70

70

60

60

50

50

ARRAY GAIN [dB]

ARRAY GAIN [dB]

N

40

30

40

30

20

20

10

10

0

0 20

25

30

35

40

INR [dB]

45

50

55

60

20

25

30

35

40

45

50

55

60

INR [dB]

Fig. 3. AG of the basic MPDR, QCSS MPDR, LCSS MPDR, eigen-based and robust adaptive MPDR beamformers vesus INR for the SMI approach (left) and with DL (LNR ¼10 dB) (right).

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A. Amar, M.A. Doron / Signal Processing 109 (2015) 165–171

matrix. As can be seen, when using the SMI approach, the AG of both beamformers is higher than that of the basic MPDR beamformer for small N and decreases if we use DL. The difference reduces for both INRs as N increases and approaches the asymptotic performance of the basic MPDR. Still, for finite N, the beamformers consistently outperform the MPDR beamformer. Moreover, when no diagonal loading is used, the AG of the eigen-based beamformer [14] and the robust adaptive beamformer [13] outperforms that of the proposed beamformers. However, a switchover occurs when a diagonal loading is used, then the proposed beamformers have larger AG.

randomly in the sector around its nominal direction. We then evaluated the weight vector at the ith trial, wi, for each beamformer. We calculated the AG at the ith trial as AGðwi ; NÞ ¼

SINRout ðwi ; NÞ ; SINRin

where SINRout ðwi ; NÞ ¼

2 SNRjwH i aD j 2 2 INRjwH i aI j þ J wi J

and SINRin ¼

SNR INR þ 1

Example 3. We evaluated the AG of the different beamformers for N ¼ 30; 50; 100; 200; 500, and 1000, assuming that the source and the interference are correlated, as may occur in a multipath environment. We considered an interference located at nominal direction θI;0 ¼ 30○ . The SNR is 30 dB and the INR is 30 dB. We assumed that the correlation coefficient of the signals equals 0.95. We calculated the AG, given N, as detailed in Example 2. We used the DL technique with LNR¼ 10 dB. The results are given in Table 1 together with the asymptotic AG of each beamformer. The LCSS and the QCSS beamformers suppress the correlated interference and approach their asymptotic AG, and thus these beamformers are robust for such scenario. The MPDR beamformer, on

are the signal to interference and noise ratio (SINR) at the beamformer output and the array input, respectively. The final AG is AGðNÞ ¼

1 Nexp ∑ AGðwi ; NÞ: N exp i ¼ 1

1 We considered two options to compute R^ : (1) no loading (SMI approach)and (2) using DL by defining the loading to noise ratio (LNR) as LNR ¼ σ 2L =σ 2n , to be equal to 10 dB. For each case we considered an INR of 30 dB. The results are plotted in Fig. 2. As a reference, we also plot the AG of the MPDR obtained using the asymptotic covariance

45

ARRAY GAIN [dB]

40

35

30

ANALYTIC MPDR QC SS MPDR LC SS MPDR EIGEN−BASED [14] ROBUST [13]

25

20 −20

−15

−10

−5

0

5

10

15

20

25

SNR [dB] Fig. 4. AG of the basic MPDR, QCSS MPDR, the LCSS MPDR, the eigen-based [14], and the robust adaptive [13] beamformers versus SNR with DL of LNR ¼10 dB. Table 2 AG of the basic MPDR, QCSS MPDR, LCSS MPDR, eigen-based and robust adaptive MPDR beamformers versus the actual gain of the sensors with N ¼100, INR ¼ 30 dB, SNR¼10 dB, LNR ¼10 dB, and θI;0 ¼ 30○ . Gain

1

1.05

1.1

1.15

1.2

1.25

1.3

AGEIGEN½14 AGRAB½13 AGLCSS AGQCSS AGMPDR

36.9390 36.1590 42.0092 42.0187 32.9136

36.9944 35.6758 41.1358 41.1275 32.5468

36.7177 36.1687 39.9188 39.9040 32.1134

37.4091 36.0516 37.9071 37.8883 29.7827

37.0391 35.7028 36.8018 36.8171 30.2547

36.4206 36.3345 35.8555 35.8015 28.9439

35.5602 36.7581 35.9668 35.9895 26.7695

A. Amar, M.A. Doron / Signal Processing 109 (2015) 165–171

the other hand, does not perform well since it cancels the desired signal in this case. We conclude that both beamformers can overcome the signal cancellation problem of the basic MPDR beamformer. Example 4. We compared the AG of the different beamformers versus the INR. We considered an interference located at nominal direction θI;0 ¼ 30○ . The number of samples is N¼100. We varied the INR from 20 dB to 60 dB with a step of 5 dB. Also, SNR¼INR 20 dB. We considered two cases: (1) SMI approach and (2) LNR¼10 dB. The AG, for a given INR, was computed using Nexp ¼ 100 independent trials. The results are plotted in Fig. 3. As a reference, we also plot the asymptotic AG of the basic MPDR. As can be seen, when using the SMI approach or the diagonal loading method, the difference between the AG of both beamformers and the MPDR beamformer is large for all values of INR. Moreover, similar to the results in Example 2, while the eigen-based beamformer [14] and the robust adaptive beamformer [13] outperform the suggested beamformers when using the SMI technique, the opposite occurs when the diagonal loading method is used. Example 5. We compared the AG of the various beamformers versus the SNR, which varied between  20 dB and 25 dB with a step of 5 dB. The LNR equals 10 dB. We considered an INR equal to 30 dB and the number of samples is 30. The results are plotted in Fig. 4. As can be seen, the proposed beamformers outperform the other beamformers for all values of SNR. Example 6. We compared the AG of the various beamformers versus errors in the actual gain of the sensors. We assumed that the actual gain equals to 1þ γ, where γ is randomly selected in the range ð  α=2; α=2Þ, where α is varied between 0.05 and 0.03 with a step of 0.05. The LNR equals 10 dB. We considered an INR equal to 30 dB and the SNR equal to 10 dB. The results are shown in Table 2. As can be seen, the proposed beamformers outperform the other beamformers when the gain error is small, while as the gain errors increase, the robust adaptive beamformers have better array gain. 6. Conclusion We proposed modified MPDR beamformers which incorporate an attenuation constraint on a pre-defined sector in their beampatterns. Simulations show that the array gain of the beamformers outperforms that of the

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