A simple tridiagonal loading method for robust adaptive beamforming

Signal Processing 157 (2019) 103–107

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Short communication

A simple tridiagonal loading method for robust adaptive beamforming Ming Zhang∗, Xiaoming Chen, Anxue Zhang School of Electronic and Information Engineering, Xi’an Jiaotong University, Xi’an 710049, China

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 12 July 2018 Revised 23 November 2018 Accepted 27 November 2018 Available online 28 November 2018

Various uncertainties exist in real sensor arrays, which can significantly deteriorate the performance of adaptive beamformers. In this communication, we propose a simple tridiagonal loading method, called automatic tridiagonal loading (ATL), to enhance the robustness of adaptive beamformers. The ATL uses a symmetric tridiagonal matrix to regularize the sample covariance matrix, which approximately imposes a smooth constraint on the weight vector of beamformer. We also present a parameter-free method to determine the loading level based on the output power of a low sidelobe beam in the presumed direction of the desired signal. Thus, ATL does not require user-defined parameters. Moreover, simulation experiments show that ATL has better robust performance than other widely used robust techniques, although the computational cost of ATL is almost the same as the standard Capon beamformer.

Keywords: Robust adaptive beamforming Tridiagonal loading Regularization

© 2018 Elsevier B.V. All rights reserved.

1. Introduction It is assumed that the covariance matrix and steering vector are accurately known in conventional adaptive beamformers [1]. However, both of them are imprecise in practice because of the mismatches between the presumed and actual signal models, such as direction of arrival (DOA) mismatch, sensor gain and phase errors, and sensor position perturbations. These uncertainties can severely deteriorate the performance of adaptive beamformers. Numerous robust adaptive beamforming (RAB) techniques have been developed to combat these mismatches, among which are weight norm constraint (WNC) [2–4], diagonal loading (DL) [5–7], subspace projection (SP) [8–10], uncertainty set constraint (USC) [11–13], steering vector estimation (SVE) [14–16], and covariance matrix reconstruction (CMR) [17–19]. Diagonal loading is one of the most popular methods in practical applications. Nevertheless, it is difficult to choose an appropriate loading level. The low computational complexity and small sample size make the subspace projection approaches attractive in real-time processing. However, the determination of a suitable dimension of the projection subspace is a difficult task, which will seriously impact the robustness of the beamformer. The methods of uncertainty set constraint and steering vector estimation are robust for many types of model errors. But their computational cost is very high and the algorithm structures are too complex to be

∗

Corresponding author. E-mail addresses: [email protected] (M. Zhang), [email protected] (A. Zhang).

[email protected]

https://doi.org/10.1016/j.sigpro.2018.11.019 0165-1684/© 2018 Elsevier B.V. All rights reserved.

(X.

Chen),

implemented in chips. Meanwhile, like most RAB techniques, they require user-defined parameters. The techniques based on covariance matrix reconstruction are very robust against DOA mismatch and have moderate computations. However, they are sensitive to sensor phase and location errors. Therefore, a parameter-free RAB method with low computational cost and good robust performance is still scarce. In this communication, we present a simple tridiagonal loading method for RAB to achieve this goal. The loading matrix is a tridiagonal Toeplitz matrix with 2’s on its diagonal and −1’s on its superdiagonal and subdiagonal. This loading technique approximates the Tikhonov regularization that imposes a smooth constraint on the weight vector of the beamformer. In addition, an automatic computation of the loading level is proposed. Because the performance of sample matrix inversion (SMI) beamformer will worsen with the increase of signal-to-noise ratio (SNR) [17], the loading level should increase with SNR too. Therefore, we use the output power of a low sidelobe beam, along the presumed direction of the desired signal, as the loading level. Considering that both loading matrix and loading level are automatically determined, the proposed method is called automatic tridiagonal loading (ATL). This method not only is very simple to be implemented, but also does not require parameters. Moreover, simulation results show that ATL outperforms other RAB techniques in terms of robustness and small sample support. We use lowercase boldface letters (a) and uppercase boldface letters (A) to represent vectors and matrices, respectively. The superscripts ()H and ()−1 stand for Hermitian transpose and matrix inversion, respectively. The symbol  ·  denotes the Euclidean norm and O( · ) means “on the order of”.

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2. Problem formulation For simplicity, consider a uniform linear array (ULA) consisting of M sensors located along the z-axis. Suppose D + 1 narrowband signals from far-field with DOAs θ i (i = 0, 1, . . . , D) impinge on the array. The M × 1 signal vector received at the time instant k can be modelled as [1]

x (k ) =

D 

s i ( k ) v ( θi ) + n ( k ) ,

(1)

i=0

where si (k) is the waveform transmitted by the ith source (s0 (k) is the desired signal), θ i is the ith polar angle with respect to the positive z-axis, v(θ i ) is the steering vector in the direction θ i , and n(k) is the additive noise vector. We assume that the signal, interferences, and noise are statistically independent of each other. To simplify the notation, v(θ 0 ) is hereinafter denoted by v0 . The output signal of the array is given by y(k ) = wH x(k ), where w is the weight vector of the beamformer, which is designed to maximize the signal-to-interference-plus-noise ratio (SINR) defined by

SINR =

 2 σs2 wH v0  wH Ri+n w

,

(2)

where σs2 is the power of desired signal and Ri+n is the covariance matrix of interference-plus-noise. The weight vector can be solved equivalently by the following constrained optimization problem

min wH Ri+n w

w∈C M

subject to wH v0 = 1 ,

(3)

which is known as the minimum variance distortionless response (MVDR) beamformer [1]. As the desired signal is present in the received data, obtaining an estimate of Ri+n is difficult in practice. Instead, we shall use the covariance Rx of the total received signal in the place of Ri+n , and the solution is given by [1]

wscb =

1 vH R−1 v0 0 x

−1 R−1 x v0 = γ Rx v0 ,



(4)



imposes a smooth constraint on the weight vector. Because the performance degradation of adaptive beamformer is mainly caused by noise eigenvalues and eigenvectors of  Rx [5,10], which are the high-frequency components of  Rx and constitute the noise subspace [1], the smooth constraint that plays the role of lowpass filtering can reduce the impacts of noise errors. Indeed, many RAB techniques can be explained from the viewpoint of lowpass filtering. For example, the noise eigenvectors are discarded in the method of subspace projection [8]. Similarly, the eigenvectors are damped by the factors λi /(λi + σl2 ) in the method of diagonal loading [10], where λ are the eigenvalues of  Rx and σ 2 is the loadi

l

ing level. Diagonal loading uses the identity matrix I to regularize the SCM  Rx . For tridiagonal loading, a Hermitian tridiagonal matrix T instead of I is employed. Thus, the loaded SCM is given by

 Rx,tl =  Rx + σl2 T ,

(6)

and the weight vector of the tridiagonal loading beamformer can  be expressed as (the scalar 1/ vH R−1  v is omitted because it does 0 x,tl 0 not affect the output SINR)



 wtl =  R−1 v =  Rx + σl2 T x,tl 0

−1

 v0 ,

(7)

where  v0 = v0 + ve is the presumed steering vector with ve being the model errors. √ Let A = XH / K , where X = [x(1 ), x(2 ), . . . , x(K )] is the data  smi =   matrix. Then  Rx = AH A and the SMI beamformer w R−1 x v0 can be expressed as the solution of

 smi = AH b , AH Aw



(8)

−1

where b = A AH A  v0 . When tridiagonal loading (6) is applied to SMI beamformer, (8) becomes





AH A + σl2 BH B wtl = AH b ,

(9)

where γ = 1 vH R−1 v0 . This solution is known as the minimum 0 x power distortionless response (MPDR) beamformer or the standard Capon beamformer (SCB) [1]. Because the scalar γ in (4) does not affect the output SINR, the key issue of SCB is to compute R−1 x v0 . However, both of Rx and v0 are not available in practice due to array model mismatches and limited number of snapshots. One commonly used estimation method for Rx is given by

which is equivalent to (7), where the Cholesky decomposition T = BH B is used. Recall that for a matrix C ∈ Cm×n of full column rank, the solution to the least squares problem min Cy − c, where y ∈ Cn and c ∈ Cm , is equal to the solution of its normal equations CH Cy = CH c [22]. It is easy to show that (9) is the normal equations of the following least squares problem

K 1  Rx = x ( k )xH ( k ) , K

min  

(5)

k=1

where K is the number of snapshots and  Rx is called the sample covariance matrix (SCM). The errors in Rx and v0 will lead to errors in wscb , and then deteriorate the performance of SCB. 3. The automatic tridiagonal loading method The most effective way to cope with model errors is regularization [20]. Two widely used methods in regularization are the norm constraint and smoothness constraint. Diagonal loading, which is known as ridge regression in statistics [21], is a method of norm constraint. Because the norm w of weight vector is related to array gain of the beamformer [3], diagonal loading has a wellestablished explanation. In contrast, the smoothness constraint is not directly related to some parameter of the beamformer. However, it is also a very effective way to reduce the problem of overfitting caused by noise and model errors [20,21]. The tridiagonal loading proposed in this paper is a method of regularization that



2  A b   . w− M σl B 0  w∈C

(10)

Therefore,



2  A b   wtl = arg min  w −  σ B 0  l w∈C M  = arg min Aw − b2 + σl2 Bw2 ,

(11)

w∈C M

which is the standard form of Tikhonov regularization. The loading level σl2 , called regularization parameter, controls the weight given to the regularization term Bw relative to the residual norm Aw − b. It plays an important role in regularization and must be chosen appropriately. The choice of matrix B is also very important because it represents what kind of additional constraint is imposed on the solution. When B = I, (7) becomes diagonal loading and (11) becomes weight norm constraint, which demonstrates the equivalence between the methods of DL and WNC.

M. Zhang, X. Chen and A. Zhang / Signal Processing 157 (2019) 103–107

is used to compute σl2 , where wdc is the Dolph-Chebyshev weights for a specified SLL (0.05 (−26 dB) in this paper) and  denotes the entrywise product. The loading level in (6) is given by

In our method T takes a very simple form as follows

⎡

2

⎢−1 ⎢ 0 ⎢ T=⎢ . ⎢ .. ⎣ 0 0

−1 2 −1 .. . 0 0

0 −1 2 .. . 0 0

··· ··· ··· .. . ··· ···

0 0 0 .. . 2 −1

⎤

0 0⎥ 0⎥ ⎥ ..⎥ . .⎥ ⎦ −1 2

(12)

Now let us explain why (12) is used. Since T = BH B is a real symmetric tridiagonal matrix, B is a real upper bidiagonal matrix, i.e.,

⎡ α1 ⎢0 ⎢0 B=⎢ ⎢. ⎣ .. 0

β2 α2 0 .. . 0

0

β3 α3 .. . 0

··· ··· ··· .. . ···

0 0 0 .. .

⎤ ⎥ ⎥ ⎥. ⎥ ⎦

(13)

αM

By comparing the elements of T and BH B, we obtain

α = 2, αi βi+1 = −1, α + β = 2 . 2 1

2 i

2 i



2 − 1/αi2−1 ,

(14)

i = 2, 3, . . . , M ,

(15)

i = 2, 3, . . . , M .

(16)

and

βi = −1/αi−1 ,

Because the sequence α i is decreasing and bounded, it has a limit α [23]. Let αi = αi−1 = α , we have

α=



2 − 1/α 2 ⇒ α = 1 .

(17)

Hence, limi→∞ αi = 1 and limi→∞ βi = −1, which indicates that B approximates the first-order derivative operator, i.e.,

⎡

1 ⎢0 ⎢0 B≈⎢ ⎢. ⎣ .. 0

−1 1 0 .. . 0

0 −1 1 .. . 0

··· ··· ··· .. . ···

⎤

0 0⎥ ⎥ 0⎥ . .. ⎥ .⎦ 1

(18)

Because the coefficients 1 and −1 constitute a highpass filter [24], Bw contains the high-frequency components of w. Since (11) minimizes Aw − b2 + σl2 Bw2 , the high-frequency components of w are suppressed. In other words, the loading matrix T given by (12) imposes a smooth constraint on w. Therefore, the weight vector wtl given by (7) can reduce the impacts caused by noise and model errors, and thus increase the robustness of the beamformer. The final step of the proposed method is to compute the loading level σl2 . Because the higher the SNR, the more serious the performance degradation of SMI beamformer, the loading level σl2 should increase as SNR increased. In a previous work we proposed a very simple approach to choose the loading level σl2 : a coarse estimate of the desired signal power, i.e., the output power of the presumed steering vector  v0 [10]. Numerical experiments illustrated the superiority of this approach over the conventional methods. However, the beampattern of steering vector has a high sidelobe level (SLL), which will affect the estimation of the desired signal power when strong interferences appear at sidelobes. An intuitive improvement is to use a low sidelobe beam such as the Dolph-Chebyshev or Taylor weighted beam [25]. Here we use the Dolph-Chebyshev beam which can provide the lowest SLL for a given specified beamwidth. Therefore, the weighted steering vector

 v0 = wdc   v0

 σl2 =  v0 −1 vH 0 Rx v0 .

(20)

Here we use  v0  instead of  v0 2 to normalize the loading level because simulation experiments show that it has slightly better performance. The additional computations of the proposed method (ATL) as compared with SCB are 2M2 + 6M flops (a flop is a floating point add, subtract, multiply, or divide [22]), coming from the computations of (19), (20), and (6). This additional cost is negligible because the computations involved in SCB are 2KM2 + M3 /3 + O(M2 ) flops, coming from the computations of (5) and (4). However, simulation results show that ATL has better performance than many RAB techniques reported in the literature. Therefore, the proposed method is of much interest in practical applications. 4. Simulation experiments

Thus,

αi =

105

(19)

A ULA with M = 10 isotropic sensors spaced half-wavelength is used in the simulation experiments. Four types of model uncertainties are considered: DOA mismatch, sensor gain and phase errors, and sensor location perturbations. The mth element of the actual steering vector v0 can be modelled as

(1 + gm )e jφm e j(2π /λ)(md+z) cos(θ0 +θ ) ,

(21)

√ where j = −1 is the imaginary unit, λ is the wavelength of the incident waves, and d = λ/2 is the interelement spacing. The DOA mismatch θ is assumed to be uniformly distributed in [−3◦ , 3◦ ]. The sensor gain errors gm are modelled as independent and identically distributed zero-mean Gaussian random variables with standard derivation σg = 0.1. The sensor phase errors φ m are assumed to be independent and uniformly distributed in [−10◦ , 10◦ ]. And the sensor locations are perturbed randomly by z in the range of [−0.05λ, 0.05λ]. Preliminary experiments were carried out to determine the parameters for the parameterdependent methods to produce the best performance. All of the performance curves are averaged over 10,0 0 0 independent trials. In the first example we consider the effect of input SNR on the output SINR. The number of snapshots is fixed at 100. One desired signal impinges on the array from 90° and two interferences imping on the array from 60° and 135°, respectively. The interferenceto-noise ratio (INR) are INR1 = INR2 = 20 dB. First we compare the performance of ATL with diagonal loading techniques, including fixed DL (FDL) [5] (loading level is σl2 = 10 dB), variable DL (VDL) [3] (forgetting factor is λ = 0.99 and constraint value on weight norm is β = 0.5), general linear combination (GLC) [7], and automatic DL (ADL) [10]. The results are shown in Fig. 1. It can be observed that ATL has the best performance in all of the compared methods. Then we compare the performance of ATL with other widely used robust beamformers, including WNC beamformer [4] (white noise array gain is AWN = 0.9M), robust Capon beamformer (RCB) [6] (uncertainty parameter is  = 3.2), modified eigensubspace projection (MSP) beamformer [9] (energy parameter is ρ = 0.8), SVE beamformer [14] (angular sector is [θ0 − 8◦ , θ0 + 8◦ ]), and CMR beamformer [19] (angular sector is [θ0 − 10◦ , θ0 + 10◦ ]). The results are shown in Fig. 2. We see that ATL also outperforms these methods except for CMR in high SNR region. This is because the desired signal has been removed during the computation of SCM in the method of CMR, which avoids the signal self-cancellation. In the second example we consider the effect of snapshots on the output SINR. The input SNR is fixed at 15 dB. One desired signal appears at 90° and three interferences appear at 45°, 75°, and 120°, respectively. The INRs of the interferences are INR1 = 10 dB,

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Fig. 4. Output SINR versus snapshots in presence of mismatches. Fig. 1. Output SINR versus input SNR in presence of mismatches.

INR2 = 20 dB, and INR3 = 30 dB. We first compare the performance of ATL with DL techniques, and then with WNC, RCB, MSP, SVE, and CMR. The results are shown in Figs. 3 and 4. It can be seen from Fig. 3 that ATL provides at least 2 dB performance improvement when compared with DL techniques. The ATL approach with 20 snapshots can obtain the same performance of DL approaches with 100 snapshots. Meanwhile, we observe from Fig. 4 that ATL is also the best performing one as compared with WNC, RCB, MSP, SVE, and CMR. It should be noted that there is a constraint on the methods of WNC, GLC, RCB, SVE, and CMR that the number of snapshots K must be greater than or equal to the number of sensors M. In contrast, the methods of ATL, ADL, VDL, and FDL can work in the situation of K < M. The most efficient algorithms of the 10 compared methods are FDL, ADL, ATL, and VDL, followed by WNC, RCB, GLC, and MSP, CMR and SVE are the most time consuming algorithms. 5. Conclusions

Fig. 2. Output SINR versus input SNR in presence of mismatches.

A new robust adaptive beamforming method based on tridiagonal loading has been proposed in this paper, which can be viewed as a generalization of the diagonal loading technique. This method employs a simple tridiagonal Toeplitz matrix to regularize the sample covariance matrix. Meanwhile, an interpretation of tridiagonal loading from the viewpoint of Tikhonov regularization was also provided. In addition, an automatic method for computing the loading level has been developed based on a coarse estimation of the desired signal power. Thus, the proposed approach is parameter-free and easy to be implemented, which makes it very useful in practical applications. Simulation results demonstrated the superiority of the proposed method in terms of robustness and sample size as compared with other commonly used robust adaptive beamforming techniques. Acknowledgment This work was supported in part by the National Natural Science Foundation of China under Grant no. 61801368 and in part by the China Postdoctoral Science Foundation under Grant no. 2018M640990. Supplementary material

Fig. 3. Output SINR versus snapshots in presence of mismatches.

Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.sigpro.2018.11.019.

M. Zhang, X. Chen and A. Zhang / Signal Processing 157 (2019) 103–107

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