A Robust Beamformer For Suppressing the Moving Interferences

Yang Jun

Li Jiani

School of Computer Science and Information, Guizhou University, Guiyang , China; e-mail: yjadams@gmail.com

Luoyang Institute of Electro-Optical Equipment, Aviation Industry Corporation of China, Luoyang, China; e-mail: jianili@tom.com

Abstract—The beamformer cannot work well when the interferences move quickly, and the performance will degrade sharply. So a robust beamforming algorithm with broadening null based on the statistical model of interferences position was presented in this paper. It can suppress the moving interferences and work robustly with few snapshots or in the case of the array perturbations existed, so the interferences will be suppressed more efficiently. The simulation results verified the efficiency of this algorithm. Keywords-Robust Beamformingt; Diagonal Loading; NullBroadening

I.

INTRODUCTION

Sensor arrays are widely used in radar, sonar and the speech signal processing systems. This technique combines the signal received in each array element, can form a beam pattern to suppress the interferences and increased the signal to noise ratio, which is an efficient spatial filter[1]. Because of its high array gain and low side-lobe, the Capon beamformer based on the minimum variance distortionless response principle was widely used in many places. But this beamformer was very sensitivity to the perturbations of array, and its performance will degrade sharply when the perturbations exited. And in the case of insufficient snapshots, the beam pattern will not form well. All these problems restrained the application of this beamformer, so how to improve the robustness of Capon was studied by researchers in recent years. The eigenspace method was brought by Feldman[2], and only need the number of interferences to form a robust beam pattern. But this method needs to calculate the eigenspace, so it has a high algorithm complexity. And if the number of interferences was changing, this robust method will not work well. The diagonal loading method was presented by Carlson[3] was a efficient method to improve the robustness of Capon beamformer. It only need a simple modification in the covariance matrix to get the robust beamformer. So this method was commonly used. But how to choose the diagonal loading value is still a problem unsolved. At the beginning, this diagonal loading value was chosen as a constant, as the SNR increase, the SINR of output will decrease. Elnashar[4,5] proposed an algorithm with a quadratic constraint on the weight vector norm, enhanced the performance of robust beamformer at low SNR. Recently stoica[6,7] proposed a robust Capon beamformer based on

the model of uncertainties in array manifold. And they proved the diagonal loading method was equivalent to solve the optimum beamforming problem when there exit uncertainties. This method was also belongs to a class of diagonal loading methods, and the diagonal loading value will vary with different SNR levels. So this algorithm has a good performance in improving the robustness. The SINR of array output may degrade seriously when the interferences move quickly or the platform is vibrating, for the array vector cannot update fast enough. An efficient method is to broaden the width of the null, which can keep the moving interferences in the null, and suppressed it. In this paper, based on the robust Capon beamformer, we presented a robust beamforming algorithm with broadening null based on the statistical model of interferences position, which can suppress the moving interferences and work robustly with few snapshots or in the case of the array perturbations existed. The simulation result verified efficiency of this algorithm. II.

BACKGROUND

Wherever Times is specified, Times Roman or Times New Roman may be used. If neither is available on your word processor, please use the font closest in appearance to Times. Avoid using bit-mapped fonts if possible. True-Type 1 or Open Type fonts are preferred. Please embed symbol fonts, as well, for math, etc. We consider an-M element linear antenna array with half of the wavelength between each element. The signal received in each element is narrowband plane wave. The input vector to the array can be written as

x (t ) = As (t ) + n(t ) Where

(1)

s (t ) represent the waveforms of the desired

signal and interference signals, n(t ) is the background noise which is assumed to be spatial white, and

A = [a(θ s ), a (θi1 ),⋯ , a (θ ip )] Where

(2)

a (θ s ) and a(θ i1 ),⋯ , a (θip ) represents the

phase vectors of the desired signal and interference signals. The correlation matrix of the array input vector is

−1 ɵ = R as w H aɵ s R −1 aɵ s ( R + λ −1 I ) −1 = H a s ( R + λ −1 I ) −1 R ( R + λ −1 I ) −1 a s

H

R = E[ x(t ) x (t )] = APAH + σ n2 I (3)

p

= σ 12 as asH + ∑ σ i2 ai aiH + σ n2 I i =1

Where E[i] denotes expectation and H denotes the complex conjugate transpose, and P is the correlation matrix of s (t ) . And the signals number is less than the array element number are assumed. i.e., ( p + 1) ≤ M . We can get the estimation of correlation matrix by N H N = 1 R x(i) x (i) ∑ N i =1 H N −1 1 = R N −1 + x(k ) x (k ) N N

We can see that the RCB is a class of diagonal loading method. Let (12) R = U ΓU H Where the columns of U contain the eigenvectors of R , we let

z =U H a (4)

And as

, s.t.w H a = 1 min wH Rw s

(5)

w

a

Where s is the phase vectors of the desired signal calculated from DOA estimation which may have some perturbations. Then by using Lagrange multipliers method we can get the Capon array vector −1

a R s −1 H a R a s

(6) s

The beam pattern which is an important parameter of the array can be written as

p (θ ) = wH a(θ ) III.

(7)

PREESNTED METHOD

It is inevitable to have some perturbations of the array systems in application. The direction of arrival is usually computed from the insufficient snapshots, so the precise steering vector can’t always be obtained. With high array gain and low side-lobe, the Capon beamformer was very sensitivity to the perturbations. In literature[6], the upper bound of the uncertainty of the steering vector was used as a constraint to get the robust Capon array verctor

min asH R −1as , s.t. as − a s = ε 0

(8)

as

Where the a s is the imprecise steering vector. By Lagrange multipliers

F (as , λ ) = asH R −1as + λ ( as − a s

2

− ε0 )

g (λ ) = U ( I + λΓ) −1U H a s

Then the Capon array vector is

=∑

zm

(14)

2

= ε0 (1 + λγ m ) Note that g (λ ) is a monotonically decreasing function of λ ≥ 0 . g (0) > ε 0 and lim g (λ ) = 0 < ε 0 , hence there λ →∞

is a unique solution

λ >0

to (9). Using

0 < λ < 100 / ε 0

by experience, we can get the value of λ by bisection method. The RCB was a good method to obtain the diagonal loading value, but when there exited interferences moving quickly or the array platform is vibrating, because of the complexity of algorithm and snapshots is insufficient, the SINR of array output will degrade sharply. An efficient method is to broaden the width of the null, which can keep the moving interferences in the null. So we introduced a robust beamforming algorithm with broadening null based on the statistical model of interferences position, which can suppress the moving interferences and work robustly with few snapshots or in the case of the array perturbations existed. So the interferences will be suppressed more efficiently. The null-broadening algorithm was implemented by covariance matrix taper[10], which has a low the algorithm complexity, and was convenient to realize. First we can calculate each element of T by

1 [T (θ p , σ p2 )]mn = exp{− σ p2 [(m − n)π cos θ p /180]2 } 2 (15) And the direction of interferences was θ p

θp

= θ p + ∆θ p ,

is mean of θ p , and ∆θ p is a zero mean gauss 2

(10)

2

m =1

where

−1

R aɵ s = ( + I ) −1 a s = a s − ( I + λ R )−1 a s

M

( 9)

We get

λ

(13)

zm denote the mth element of z , (9)can be written

By using the minimum variance distortionless response principle, array beamforming can be converted into following optimization problem

wopt =

(11)

white random variable with variance σ p . Then calculate the hadamard product as below

CMT = R T R

(16) Then by using RCB aogorithm, we can get the robust beamformer for suppressing moving interferences, which can suppress the moving interferences and work robustly with few snapshots or in the case of the array perturbations existed. IV.

NUMERICAL EXAMPLES

A. The Case with Small Snapshots The template is designed so that author affiliations are not repeated each time for multiple authors of the same affiliation. Please keep your affiliations as succinct as possible (for example, do not differentiate among departments of the same organization). This template was designed for two affiliations. We assume a uniform linear array (ULA) with N = 10 and half-wavelength spacin is used. The sources are narrowband plane wave and is spatial white. The interested signal is from vertical direction, and the SNR is 10dB. The DOA of interference is 0.35 in 2

wavenumber, and σ p

From Fig1, we can see with the perturbations of small eigenvalue of covariance matrix, the beam pattern degraded sharply. But by the algorithm presented in this paper, the beam pattern was very robust to the perturbations, and had formed a broadened null which can suppress the moving interference efficiently. B. The Case with DOA Estimated Errors The antennas array and source signals was the same as in example 4.1. The DOA of interested signal has errors of 0.1 in wavenumber. The value of

ε0

2

is 3.1, and σ p

= 4 . The

beampattern was also plot every 25 snapshots. By the numerical experiment, we can see that the beam pattern forms well, and the array has a good gain with DOA errors and moving interference.

= 8 . The power of the interference is

30dB. We plot the beam pattern every 25 snapshot, so as to observe the robustness of the beamformer against the small snapshot.

Fig3. Beam Pattern

Fig1. Insufficient Snapshots

Fig4. Array Gain V.

Fig2. Null-broadening

CONCLUSION

To suppress the moving interferences more efficiently, we presented a robust beamformer with broadening-null, which can form broaden the null based on the statistical model of interferences position, and work robustly with few snapshots or in the case of the array perturbations existed, so it can suppress the interferences more efficient. And this beamformer has a low algorithm complexity in application,

so it has a good practicability. By numerical examples, we verified the efficiency of this beamformer. REFERENCES [1]

[2]

[3]

[4]

B.D.V Veen, K.M. Buckley. Beamforming: A versatile approach to spatial filtering[J]. Acoust. Speech Signal Processing, 1988, 5, (2): 424. D.D. Feldman, L. J. Griffiths. A projection appraoch to robust adaptive beamforming[J]. IEEE Trans. Signal Processing, 1994, 42, (4): 867-876. Carlson Blair D. Covariance matrix estimation errors and diagonal loading in adaptive arrays[J]. IEEE Trans. on Aerospace and Electronic Systems, 1988, 24, (4): 397-401. Ayman Elnashar, S. Eloubi, H. Elmikati, Robust Adaptive Beamforming with Variable Diagonal Loading[J]. 6th IEEE International Conference on 3G & Beyond 3G (2005), 1-5.

[5]

Elnashar A., Elnoubi S. M., Elmikati H. A., Futther Study on Robust Adaptive Beamforming With Optimum Diagonal Loading[J], IEEE Transactions on Antennas and Propagation, 2006,54,(12):3647-3658. [6] Petre Stoica, Zhisong Wang, Jian Li. Robust Capon Beamforming[J]. IEEE Signal ProcessingLetters, 2003, 10(6): 172-175. [7] Jian Li,Petre Stoica, Zhisong Wang.On Robust Capon Beamforming and DiagonalLoading[J]. IEEE Trans. on Signal Processing, 2003, 51(7): 1702-1715. [8] Mailloux R J. Covariance Matrix augmentation to produce adaptive array pattern troughs. Electronics Letters, 1995, 31(10): 771-772. [9] Zatman M. Production of adaptive array troughs by dispersion synthesis. Electroncs Letters, 1995,31(25): 2141-2142. [10] Zetterberg P, Ottersten B. The spectrum efficiency of a basestation antenna array system for spatial selective transmission. IEEE Trans. on Vehicular Technology, 1995, 44(3): 651-660.

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