Robust adaptive beamforming using multi-snapshot direct data domain approach

Robust adaptive beamforming using multi-snapshot direct data domain approach

Int. J. Electron. Commun. (AEÜ) 75 (2017) 124–129 Contents lists available at ScienceDirect International Journal of Electronics and Communications ...

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Int. J. Electron. Commun. (AEÜ) 75 (2017) 124–129

Contents lists available at ScienceDirect

International Journal of Electronics and Communications (AEÜ) journal homepage: www.elsevier.com/locate/aeue

Regular paper

Robust adaptive beamforming using multi-snapshot direct data domain approach Lilong Qin a,b,⇑, Manqing Wu c, Zhen Dong a a

Institute of Space Electronics and Information Technology, School of Electronic Science and Engineering, National University of Defense Technology, China Department of Signal Processing and Acoustics, Aalto University, Finland c China Academy of Electronics and Information Technology, China Electronics Technology Group Corporation, China b

a r t i c l e

i n f o

Article history: Received 19 October 2016 Accepted 7 March 2017

Keywords: Direct data domain Array signal processing Robust adaptive beamforming Norm-constrained beamforming Probability-constrained beamforming

a b s t r a c t For the realistic case where there is no secondary snapshot that does not contain the desired signal and exhibits the statistical characteristics similar to the snapshot under test, direct data domain (D3) beamforming approaches have been proposed to estimate a desired signal in the presence of interference. However, the basic idea of the D3 methods is realized by making significant sacrifices with respect to the number of degrees of freedom (DoFs). In this paper, we present a multi-snapshot approach for D3 beamforming. Using the least-squares method with multiple snapshots, we can eliminate the interference without causing a severe reduction in the number of DoFs. In addition, to consider a mismatch between nominal and actual target steering vectors, we propose a D3 approach combined with a probability constraint to prevent the self-nulling effect, and the relationship between the probability constraint and norm constraint is discovered. The simulations verify that the proposed method provides better performance and robustness than the conventional D3 approaches. Ó 2017 Elsevier GmbH. All rights reserved.

1. Introduction In many wireless communication, biomedical imaging, sonar, and radar applications, adaptive beamforming approaches can be used to effectively separate the target from interference (jammer, clutter, and noise) [1]. Ignoring the detailed working principle, these approaches can be classified into two classes: stochastic beamformers and deterministic beamformers [2]. Regarding the conventional stochastic beamforming approach, the interference statistics (e.g., the covariance matrix) need to be estimated from secondary snapshot data. Conventional stochastic beamformers can achieve good performance when secondary snapshots do not contain the desired signal and exhibit homogeneous statistical characteristics similar to the snapshot under test. According to the Reed-Mallet-Brennan rule, the required number of snapshots must be twice the system degrees of freedom (DoFs) [3]. In many realistic cases, however, it is impossible to obtain sufficient secondary snapshots that satisfy these requirements (e.g., in passive localization, wireless communication, passive sonar, and

⇑ Corresponding author at: Institute of Space Electronics and Information Technology, School of Electronic Science and Engineering, National University of Defense Technology, China. E-mail address: [email protected] (L. Qin). http://dx.doi.org/10.1016/j.aeue.2017.03.004 1434-8411/Ó 2017 Elsevier GmbH. All rights reserved.

speech processing applications, signal-free data are unavailable), and the performance is significantly degraded [4]. Single-snapshot Direct data domain (D3) approaches have been proposed to overcome the drawbacks of the conventional statistical techniques [5–7]. In contrast to conventional stochastic beamforming approaches, single-snapshot D3 approaches do not require secondary snapshots to estimate the statistical characteristics. The snapshot under test is directly employed to design the beamformer. However, it is achieved by introducing a high side-lobe level and at the high cost of a reduction in the number of degrees of freedom (DoFs) [7]. In [8], a D3 beamforming method is proposed to increase the number of DoFs while decreasing the side-lobe level, and the proposed method can provide better performance than the conventional single-snapshot D3 approaches. However, the performance of the D3 approaches is significantly affected by the mismatch between the nominal and actual target steering vectors [9]. This imperfect steering vector mismatch will result in a target self-nulling effect that leads to severe performance degradations. Several approaches have been proposed to overcome this problem. The first approach is to establish multiple constraints that cover the region of uncertainty of the target parameters [10,11]. This solution can preserve the gain for the target at the expense of a reduced number of DoFs that are available to suppress the interference. The second approach aims to refine

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the target parameter estimation, but an additional computational burden is introduced [12]. In addition, certain robust techniques that are used for the conventional Capon beamforming (e.g., the diagonal loaded technique [13,14], the beamformer optimizing the worst-case performance [15] and the signal-subspace projection method [16]) can be used to improve the robustness of the D3 beamformer. In contrast to the robust deterministic D3 beamforming approaches that utilize a single snapshot (e.g., [10–12]), in this paper, a robust stochastic D3 beamforming approach with multiple snapshots is proposed. Similar to conventional single-snapshot D3 approaches, the proposed multi-snapshot D3 beamforming algorithm does not require secondary snapshots, and snapshots under test can be directly employed. The proposed multi-snapshot D3 beamforming algorithm can provide a better output SINR with a slight loss of DoFs [17]. Moreover, the relationship between the probability constraint and the norm constraint is derived in this paper. Notation: In this paper, a variable, a column vector, and a matrix are represented by a lowercase letter, a lowercase bold letter, and a capital bold letter, respectively. The operations of transposition and conjugate transposition are denoted by ðÞT and ðÞH , respectively. The symbol kk denotes the ‘2 -norm operator, and jxj obtains the absolute value of x. xn denotes the nth entry of vector x, and xn1 :n2 , ½ xn1

xn1 þ1

T



xn2  .

2. Formulation The adaptive beamforming technique is known to be effective for suppressing the interference energy while receiving the useful signal. Consider an array system equipped with a uniform linear array (ULA) consisting of N omni-directional elements. The narrowband signal received at the fast time sk can be represented as [4]

xðsk Þ ¼ as ðsk Þaðhs Þ þ

P X

ai ðsk Þaðhi Þ þ tðsk Þ

ð1Þ

i¼1

where as ðsk Þ and ai ðsk Þ are the unknown complex amplitudes of the desired signal (if it exists) and the interference, respectively. P is the number of interferences impinging on the array, and tðsk Þ is the thermal noise vector with noise power r2n on each channel. h iT 2pdðN1Þ sin ðhÞ is the associated steering aðhÞ , 1 ej2pk d sin ðhÞ    ej k vector corresponding to the direction of arrival (DOA) h, where d is the inter-element distance between the elements of the array and k is the carrier wavelength. Note that (1) can be rewritten as

xðsk Þ ¼ as ðsk Þaðhs Þ þ

P X

X

i¼1

hj 2H

ai ðsk Þaðhi Þ þ

  bj ðsk Þa hj

ð2Þ

i¼1

þ



tn ðsk Þ  e

jfhs

tnþ1 ðsk Þ



~n ðsk Þ , ½~xn ðsk Þ;    ; ~xnþM1 ðsk ÞT x P h i X ¼ ai ðsk Þejðn1Þf hi 1  ejðf hi f hs Þ a1:M ðhi Þ i¼1

þ

X



" jðn1Þf h

bj ðsk Þe

j

j f h f hs

1e

#

ð4Þ

  a1:M hj

j

hj 2H

where M is the number of DoFs which should be less than N. As shown, the complex amplitude of the interferences has been modh i ulated by the coefficient ejðn1Þf hi 1  ejðf hi f hs Þ , whereas the associated steering information is unchanged, i.e., the subspace of interference a1:M ðhi Þ is unchanged. In the conventional singlesnapshot D3 approach, we form a cancellation matrix ~ 1 ðsk Þ; x ~2 ðsk Þ;    ; x ~ NM ðsk Þ, where the weighted sum of all Fðsk Þ ¼ ½x its column elements would be zero. Moreover, we fix the gain of the sub-array by forming the weighted sum xH v to a prespecified value C, where x denotes the designed filter coefficient  T is the pre-determined gain vector, and v , 1; ejfhs ;    ; ejðM1Þf hs direction, i.e., [7]

xH ½ v Fðsk Þ  ¼ ½ C 0 :

ð5Þ

In [12], it is shown that after the linear Eq. (5) is solved, an estimate of the signal complex amplitude can be derived as a^ s ðsk Þ ¼ xH x1:M ðsk Þ. In the single-snapshot D3 beamforming approaches, M ffi ðN þ 1Þ=2. Hence, the number of DoFs for suppressing interference is ðN  1Þ=2, which is considerably smaller than that in conventional stochastic beamforming (N  1). Moreover, the conventional D3 beamforming approach has a weak noise gain (due to the high side-lobe level of the beampattern). D3 beamforming approaches that use multiple snapshots have been proposed to overcome the drawbacks of the conventional single-snapshot approaches. The cancellation matrix in the multisnapshot D3 beamforming approach can be directly constructed as

F ¼ ½Fðs1 Þ; Fðs2 Þ;    ; FðsK Þ

ð6Þ

where K is the number of used snapshots. According to (4), the filter coefficient vector x designed to minimize (not to null the interfer 2 ence energy) xH F can not only suppress the interferences, but also decrease the side-lobe level to mitigate the thermal noise. It is known that the D3 beamforming approach is adversely affected by the mismatch between the nominal and actual target steering vectors [11]. In the next section, we propose a novel D3 beamforming approach that uses multiple snapshots and a probability constraint to overcome the drawbacks of conventional approaches. 3. Solutions

where H is the set of whole spatial angles, with the exception of the angles of signal and interferences, and bj ðsk Þ are the corresponding complex amplitudes. In (2), the thermal noise is essentially regarded as a summation of the noises from all spatial directions [18]. Let us define f h , 2pd sin ðhÞ=k as the spatial frequency. It is clearly observed that

~xn ðsk Þ ¼ xn ðsk Þ  ejfhs xnþ1 ðsk Þ P   X ai ðsk Þ 1  ejðf hi f hs Þ an ðhi Þ ¼

contains no components of the signal of interest (SOI). Then we have

Consider the possibility of mechanical vibrations, calibration errors, or atmospheric refractions of the incident electromagnetic waves. The assumed DOA may not be very accurate [19–22]. Hence, the filter pulse response should be designed such that it minimizes the output interference power while robustly maintaining a pre-determined array gain. 3.1. Norm-constrained optimization

ð3Þ

The array gain can be defined as [23]



G,

2





a2s xH v~ a2s xH vvH x þ xH Cv~ x ¼ : xH FFH x xH FFH x

ð7Þ

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In (7), the mismatch vector is established as an unknown random vector abided by the complex circularly symmetric Gaussian ~  N C ðv ; Cv~ Þ [24,25]. As proposed by some applicadistribution v   tions, Cv~ is assumed to be a diagonal matrix of the form r2 =M I for the sake of simplicity, where I is the M  M identity matrix and r2 is the variance of the steering vector error. Even though Cv~ is actually non-diagonal, it can typically be approximated by the scaled identity matrix for simplicity [26]. The sensitivity of the array gain relative to the random mismatch is [23]

Prf1 þ 1 P n P 1  1g ¼ Prfn 6 1 þ 1g  Prfn 6 1  1g 0 1 H v 11 B x  C  ¼ Q @ pffiffiffi ;   1=2  pffiffiffiA 1=2  Cv~ x= 2 Cv~ x= 2 0 1 H x v 1 þ 1 B C     Q @  1=2  pffiffiffi ;  1=2  pffiffiffiA: Cv~ x= 2 Cv~ x= 2

ð13Þ

ð8Þ

We define a useful function ~ ðx; 1Þ ¼ Q ðx; ð1  1ÞxÞ  Q ðx; ð1 þ 1ÞxÞ. Moreover, we define an Q ~ 1 ðx; 1Þ with regard to the first parameter when inverse function Q

To decrease the sensitivity of the array gain in terms of (8), the optimization problem subject to S < d can be formulated as a norm-constrained approach

~ ðx; 1Þ is monotonously increasing if we fix 1. Then, the optimizaQ tion problem can be rewritten as



xH x : xH vvH x

min

xH R F x

H x v¼1 s:t: kxk2 6 d min

ð9Þ

where d is a positive constant, and RF ¼ FFH . Since the problem (9) is convex and Slater’s condition holds, strong duality is obtained, and the solution of (9) can be solved using the Lagrange multiplier method [21,27].

In this section, we will first show that the norm-constrained problem (9) is equivalent to problem (10):

s:t:

(

xH R F x xH v ¼ 1 ~ P 1  1 P p Pr 1 þ 1 P xH v

ð10Þ

where Prfg represents the probability operator, and 1 is a small positive constant determined by the actual demand. Second, based on this relationship, we will show that the robustness performance using the probability-constrained optimization in [28] should outperform that using norm-constrained optimization. The optimization problem (10) is subject to a stochastic constraint, where the probability of the signal response in the range ½1  1; 1 þ 1 is greater than or equal to p. Problem (10) is mathematically intractable, and we will directly convert the problem to a mathematically tractable problem. ~  N C ðv ; Cv~ Þ. Hence, the The mismatch vector is established as v ~ have the folreal and imaginary parts of the random variable xH v lowing real Gaussian distribution:

  1=2  2 ~  N R Re xH v ;  Re xH v Cv~ x =2

  1=2  2 ~  N R Im xH v ;  Im xH v Cv~ x =2 :

ð11Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ~ ¼ RefxH v ~ g2 þ ImfxH v ~ g2 is the so-called Hence, n ¼ xH v Rice distributed random variable [29]. Then, the cumulative distribution function of n can be expressed as

 8 jxH vj pffiffi <1  Q x pffiffi xP0 ; 1=2 kC1=2 F n ðxÞ ¼ v~ xk= 2 kCv~ xk= 2 : 0 x<0 R1

2 2 z þa 2

ð12Þ

where Q ða; bÞ , b ze I0 ðazÞdz denotes the generalized Marcum’s Q-function, and I0 ðxÞ denotes the modified Bessel function of the first kind of order 0. Then, we have

(

s:t:

xH R F x xH v ¼ 1  pffiffi  1=2  Cv~ x 6 Q~ 1 ð2p;1Þ :

ð14Þ

Remark 1: As mentioned, Cv~ can be assumed to be Hence, (14) can be represented as

min s:t:

3.2. Probability-constrained optimization

min

the second parameter and the probability are given, i.e., if ~ 1 ðy; 1Þ ¼ x. In addition, note that ~ ðx; 1Þ ¼ y, then we have Q Q

xH R F x 8 < xH v ¼ 1





r2 =M I.

ð15Þ

2 2M : kxk 6 rQ~ 1 ðp;1Þ 2 : ½ 

Hence, it is clear that if d ¼

2M 2 , then problem (10) is ½rQ~ 1 ðp;1Þ equivalent to the norm-constrained problem (9). The reason for why the norm-constrained approach can effectively avoid the self-nulling effect with a high likelihood can be presented in a natural and intuitive way. Remark 2: It can be observed from (10) that the constraint H x v ~ 6 1 þ 1 provides no benefit for improving the robustness and reduces the range of feasible solution. Hence, it can be concluded that the performance of the optimization problem

min s:t:

(

xH R F x xH v ¼ 1 ~ P 1  1 P p Pr xH v

ð16Þ

should outperform that using norm-constrained optimization (9) if p and 1 are selected to make d ¼ ~ 12M 2 . ½rQ ðp;1Þ It has been proven in [25] that the constraint ~ P 1  1 P p can be rewritten as a strengthened Pr xH v constraint:

  r 1 pffiffiffiffiffi kxk 6 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xH v  1 þ 1 : M  ln ð1  pÞ

ð17Þ

Consequently, we have

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H ln ð1  pÞ x v P 1  1 þ r  p ffiffiffiffiffi kxk: M

ð18Þ

2 ~ 0 6 kxk2 kv ~ 0 k2 ¼ M kxk2 : ð1  1Þ2 6 xH v

ð19Þ

~ P 1  1 P p implies that a steerNote that the term Pr xH v ~ 0 exists that can satisfy ing vector v

~ 0 k ¼ M is employed, which is reasonIn (19), the assumption kv able for many scenarios. This assumption is violated when the array response vector also has gain perturbations. However, if 2

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and the variance of the steering vector error r2 is assumed to be 0:3M [25,30]. Hence, we should choose d ¼ ~ 12M 2  0:2267. ½rQ ðp;1Þ The same values of p ¼ 0:7 and 1 ¼ 0:2 are taken for the probability-constrained D3 beamformer (21). The diagonal loading factor is 10 dB with respect to the noise power for the diagonal loading sample matrix inversion (LSMI) beamformer [26], the parameter p ¼ 0:9 for the probability-constrained beamformer [28], and the integral sector is ½90 ; 4 Þ [ ð4 ; 90  for the interference-plus-noise covariance matrix reconstruction (IPNCM) beamformer [31]. Example 1: Mismatch due to Signal Direction Error: In the first simulation, we assume that the actual signal impinges the array from the DOA of 1 , i.e., there is a 1 mismatch in the signal look direction. The beampattern and the performance curves versus the number of snapshots are drawn in 1. As shown in Fig. 1, the sample matrix inversion (SMI) beamformer trained by signaldependent snapshots suppress the signal energy because of the self-nulling effect, and the output signal-interference-to-noise ratio (SINR) performance is severely deteriorated. Moreover, we can observe that the robust D3 multi-snapshot beamformers maintain the pre-determined gain in the desired direction while suppressing energy from the DOAs of interferences, and the output SINR significantly improves. However, the probabilityconstrained D3 beamformer has a lower side-lobe level and a better SINR performance than the norm-constrained D3 beamformer, which verifies the conclusion in Remark 2. In Fig. 2, the proposed D3 beamformers are compared to the LSMI beamformer, the probability-constrained beamformer, and the IPNCM beamformer. It can be found that the output SINR performance of the probability-constrained multi-snapshot D3 beamformer is similar to that of the beamformer trained by the signalfree snapshots and IPNCM beamformer, and outperforms that of the other beamformers. Example 2: Mismatch due to Incoherent Local Scattering: In the sencond simulation, we consider the case of coherent signal scattering in the Ricean propagation channel. The actual vector has PL ju dffi l aðh þ h Þis been modeled as aðhÞ þ eðhÞ, where eðhÞ ¼ prffiffiffiffi l l¼1 e

the gain perturbations are small, then this assumption still approxpffiffiffiffiffi imately holds [21]. Hence, we have kxk P ð1  1Þ= M. According H ~ P 1  1 P p implies that to (18), it can be observed that Pr x v

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi H x v P 1  1 þ rð1  1Þ  ln ð1  pÞ : M

ð20Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

rð11Þ

 ln ð1pÞ

Since 1 is a small positive constant, 1  1 þ  1. M Thus, it is clear that xH v ¼ 1 in (16) yields little to no robustness improvement. In addition, xH v ¼ 1 is achieved by sacrificing the spatial DOFs, which reduces the ability to suppress the interferences significantly. Hence, it is better to drop the constraint xH v ¼ 1 and reformulate the optimization problem as

xH R F x ~ P 1  1 P p: s:t: Pr xH v min

ð21Þ

In general, we can conclude that the performance using probability-constrained optimization (21) should outperform that using norm-constrained optimization (9). The solution to (21) can be efficiently obtained using a sequential quadratic programming technique [28]. 4. Results Our simulations were perfermed as follows: An array system equipped with a ULA consisting of N ¼ 10 omni-directional elements was employed, and the elements were spaced half a wavelength apart, i.e., d ¼ k=2. We modeled additive noise as spatially and temporally independent complex Gaussian noise with zero mean and unit variance. Let us consider two narrowband jams (P ¼ 2) with the DOAs equal to 10 and 10 , and the interference-to-noise ratio (INR) is set to be 20 dB for both interfer

ences. The DOA of the desired signal is assumed to be 0 . The desired signal is always present in the training snapshots and all the results are obtained by the average of 100 Monte Carlo simulations. In beamforming applications, the DoFs is an important parameter that will decide the desired performance. In our proposed D3 beamforming approaches, the DoFs is N  1, i.e., M ¼ N  1 [17]. In both of the simulations below, the actual requirement of robustness for the norm-constrained D3 beamformer (9) is that ~ 6 1:2 should be no less than 0.7, the probability of 0:8 6 xH v

ML

the mismatch vector [15,24,25]. The parameter r2d characterizes the power of the scattered non-line-of-sight (NLOS) signal components received by the antenna array, L is the number of NLOS components, and ul and hl are the phase shift and the angular shift of the lth NLOS component, respectively. In each Monte-Carlo

30

0

25

-5 -10

Output SINR (dB)

Output SINR (dB)

20 15 SMI beamformer with signal-free data SMI beamformer with signal-dependent data Probability-constrained D3 beamformer Norm-constrained D3 beamformer

10 5 0 -5

-20 -25 -30 -35 SMI beamformer with signal-free data SMI beamformer with signal-dependent data Probability-constrained D3 beamformer Norm-constrained D3 beamformer

-40

-10 -15 2

-15

-45 4

6

8

10

12

14

Number of snapshots

16

18

20

-50

-80

-60

-40

-20

0

20

40

60

80

Angel (Degrees)

Fig. 1. Case where there is a 1 mismatch in the signal look direction in the situation when the SNR is 20 dB. (left) Plot of the output SINR vs. the number of snapshots used. (right) Beampatterns of three different beamformers for K = 20.

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30

Output SINR (dB)

20

25

20

Output SINR (dB)

25

30 SMI beamformer with signal-free data LSMI beamformer Norm-constrained D3 beamformer IPNCM beamformer Probability-constrained D3 beamformer Probability-constrained beamformer

15

10

5

0

-5 -10

SMI beamformer with signal-free data LSMI beamformer Norm-constrained D3 beamformer IPNCM beamformer Probability-constrained D3 beamformer Probability-constrained beamformer

15

10

5

0

-5

0

5

10

15

20

-5 -10

-5

0

Output SINR (dB)

SMI beamformer with signal-free data LSMI beamformer Norm-constrained D3 beamformer IPNCM beamformer Probability-constrained D3 beamformer Probability-constrained beamformer

15

10

20

5. Conclusion

5

0

-5 -10

15

that the spectrum integral in the IPNCM beamformer method only takes the direction estimation error into account, and other potential errors are ignored. Since the prior knowledge about the steering vector is imprecise, the IPNCM beamformer based on direction information is no longer applicable [30]. However, the proposed probability-constrained multi-snapshot D3 beamformer is more robust to the signal-dependent data and the random steering vector mismatch. As shown in Fig. 4, the probability-constrained multi-snapshot D3 beamformer substantially outperforms all of the beamformers that were tested, including the IPNCM beamformer.

30

20

10

Fig. 4. Output SINR vs. SNR for K ¼ 20.

Fig. 2. Output SINR vs. SNR for K ¼ 20.

25

5

SNR (dB)

SNR (dB)

-5

0

5

10

15

20

SNR (dB) Fig. 3. Output SINR vs. SNR for K ¼ 20.

simulation, the parameters ul and hl are independent. In this simulation, the parameters are the same as those in [25]. Fig. 3 displays the output SINR vs. the SNR with K ¼ 20. As shown in this figure, the probability-constrained multi-snapshot D3 beamformer and IPNCM beamformer outperform the other beamformers that were tested. Example 3: Mismatch due to Steering Vector Random Error: In the third simulation, we take the comprehensive and arbitrary-type error into consideration (e.g., direction error, calibration error, and local scattering). In this case, the actual steering vector has   e been modeled as aðhÞ þ eðhÞ, where eðhÞ ¼ peffiffiffi ej/1    ej/M is M the random error vector [30]. The parameter ee characterizes the norm of eðhÞ and is uniformly distributed in the interval h pffiffiffiffiffiffiffii 0; 0:3 in each Monte-Carlo simulation, and the parameter /m is the phase of mth coordinate of the random error vector eðhÞ, which are independently and uniformly drawn in each MonteCarlo simulation from the interval ½0; 2pÞ. It can be observed from Fig. 4 that the IPNCM beamformer suffers from a more obvious performance degradation compared to Fig. 2 and Fig. 3. The reason is

In this paper, we considered the problem of robustness for multi-snapshot D3 beamforming with an additional probability constraint, and we studied the relationship between the norm constraint and the probability constraint. The results demonstrated that the proposed method can effectively enhance the robustness and has a better output SINR with the slight loss of DoFs. The proposed method can be widely used in wireless communications, biomedical imaging, sonar, and radar applications, and it can be further extended to two-dimensional (2-D) beamforming applications, such as the D3 STAP for ground moving target indication. Acknowledgments The authors thank the National Natural Science Foundation of China under grant 61101178 and the China Scholarship Council for their support, and the suggestions presented by the reviewers are sincerely appreciated. References [1] Chakrabarty S, Habets EAP. On the numerical instability of an LCMV beamformer for a uniform linear array. IEEE Signal Process Lett 2016;23 (2):272–6. [2] Wicks MC, Rangaswamy M, Adve R, et al. Space-time adaptive processing: a knowledge-based perspective for airborne radar. IEEE Signal Process Mag 2006;23(1):51–65. [3] Wu R, Sua Z, Lu W. Space-time adaptive monopulse processing for airborne radar in non-homogeneous environments. AEÜ Int J Electron Commun 2011;65(3):258–64. [4] Rahmani M, Bastani MH. Robust and rapid converging adaptive beamforming via a subspace method for the signal-plus-interferences covariance matrix estimation. IET Signal Process 2014;8(5):507–20.

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