A Novel Beamformer Robust to Steering Vector Mismatch

Transcription

1 A Novel Beamformer Robust to Steering Vector Mismatch Chun-Yang Chen and P. P. Vaidyanathan Dept. of Electrical Engineering, MC California Institute of Technology, Pasadena, CA 9115, USA Abstract It is ell kno that the performance of the imum variance distortionless response MVDR beamformer is very sensitive to steering vector mismatch because of the signal cancellation. Such mismatches can occur as a result of direction-of arrival DOA errors, local scattering, near-far spatial signature mismatch, aveform distortion, source spreading, imperfectly calibrated arrays and distorted antenna shape. In this paper, an adaptive beamformer hich is robust against the DOA mismatch is proposed. This method imposes to quadratic constraints such that the magnitude responses of to steering vectors exceed unity. Then a diagonal loading method is used to force the magnitude responses ithin a range of the arrival angles to exceed unity. The diagonal loading factor can be computed systematically by a proposed algorithm. Numerical examples sho that this method has a significantly better SINR performance compared to previously published methods including the MVDR, and a complexity comparable to the standard MVDR beamformer. 1 I. INTRODUCTION Beamforg has long been used in many areas, such as radar, sonar, seismology, medical imaging, speech processing and ireless communications. Introduction to beamforg can be found in [18] [0] and the references therein. A datadependent beamformer as proposed by Capon in [1]. By exploiting the second order statistics of the array output, the method constrains the response of the signal of interest SOI to be unity and imizes the total variance of the beamformer output. This method is called imum variance distortionless response MVDR beamformer in the literature. The MVDR beamformer has very good resolution, and the SINR signal-to-interference-plus-noise ratio performance is much better than traditional data-independent beamformers. Hoever, the steering vector of the SOI can be imprecise. When the mismatched steering vector is used in the MVDR beamformer, the response of the SOI is no longer constrained to be unity and thus is attenuated by the MVDR beamformer hile imizing the total variance of the beamformer output []. This phenomenon is called signal cancellation or selfnulling. It ill dramatically degrade the output SINR. Many approaches, including [3] [17] and the references therein, have been proposed for improving the robustness of the MVDR beamformer. A good introduction to this topic can be found in [5]. Recently, some approaches based on an uncertainty set of steering vectors have been proposed [4] [9]. Instead of imposing a linear constraint, these approaches imize the output variances subject to the constraint that the magnitude responses of a set of steering vectors exceed unity. If the mismatched steering vector is still in this set, its magnitude response ill exceed unity and ill not be attenuated. In this paper, e focus on the steering mismatch caused by direction-of-arrival DOA mismatch. Inspired by these uncertainty based methods, e 1 Work supported in parts by the ONR grant N , and the California Institute of Technology. consider a simplified uncertainty set hich contains only the steering vectors ith a desired uncertainty range of DOA. To find a suboptimal solution for this problem, the constraint is first loosened to to non-convex quadratic constraints such that the magnitude responses of to steering vectors exceed unity. Then a diagonal loading method is used to force the magnitude responses at the arrival angles beteen these to steering vectors to exceed unity. Therefore this method can alays force the gains at a desired range of angles to exceed a constant level hile suppressing the interferences and noise. A closed form solution to the proposed imization problem is introduced, and the diagonal loading factor can be computed systematically by a proposed algorithm. Numerical examples sho that this method has an excellent SINR performance and a complexity comparable to the standard MVDR beamformer hich has poor performance. The rest of the paper is organized as follos. The analysis of steering vector mismatch and some previous ork are revieed in Section II. In Section III, e develop the theory and the algorithm of our ne robust beamformer. The numerical examples are presented in Section IV. Section V gives the conclusions. Notations. Bold faced loer case letters such as y represent vectors, and bold faced upper case letters such as R y denote matrices. The element in ro n and column m of matrix R y is denoted by R y,n,m. The notation y denotes the conjugate transpose of the vector y. The notation E[x] denotes the expectation of the random variable x. The notation W z denotes the z-transform of the elements in vector. II. THE STEERING VECTOR MISMATCH AND SOME PREVIOUS WORK Consider a uniform linear array ULA of N omnidirectional sensors ith interelement spacing d. The signal of interest SOI is a narroband plane ave impinging from angle θ as shon in Fig. 1. The baseband array output yt N-1dsin Fig. 1. dsin dsin e j ft-kx The N-element uniform linear array. can be expressed as yt = xtsθ+vt, d

2 here vt denotes the sum of the interferences and the noises, xt is the signal of interest SOI, and sθ represents the baseband array response of the SOI. It is called steering vector and can be expressed as sθ 1 e j π λ d sin θ e jn 1 π λ d sin θ T,1 here λ is the operating avelength. The output of the beamformer can be expressed as yt, here is the complex eighting vector. The output SINR signal-to-interferencesplus-noise ratio of the beamformer is defined as SINR E xt sθ E vt = σ x sθ R v, here R v E[vtv t], and σx E[ xt ]. By exploiting the second order statistics of the array output yt, the output SINR can be maximized by imizing the total output variance hile constraining the SOI response to be unity. This can be ritten as the folloing optimization problem: R y subject to s θ =1, here R y E[yty t]. The solution to this problem is ell-knon and as first given by Capon in [1] as c = R y 1 sθ s θr 1 y sθ. 3 When there is a mismatch beteen the actual arrival angle θ and the assumed arrival angle θ m, this beamformer becomes R 1 y sθ m m = s θ m R 1 y sθ m. 4 It can be vieed as the solution to the imization problem R y subject to s θ m =1. 5 Because R y = R v +σx s θ, and s θ =1 is no longer valid due to the mismatch, the SOI is interpreted as an interference. The SOI magnitude response is thus suppressed as a part of the objective function. When the SOI is strong, the beamformer tends to place a zero around the SOI arrival angle θ to imize the total output variance. This ill almost destroy the SOI response. A small mismatch could cause a severe degradation in the SINR. Many approaches have been proposed for improving the robustness of the standard MVDR beamformer during the past three decades. In [10], the folloing modification of R y is performed before obtaining the MVDR beamformer by Eq. 3: R y R y + γi N. This approach is called diagonal loading in the literature. It increases the variance of the artificial hite noise by γ in R y. This modification forces the beamformer to put more effort on suppressing hite noise rather than interferences. When the SOI steering vector is mismatched, the SOI is attenuated as one of the interferences. As the beamformer puts less effort on suppressing the interferences, the SOI suppression problem addressed above is reduced. Hoever, hen γ is too large, the beamformer fails to suppress strong interferences because it puts most effort to suppress the hite noise. There is a trade-off beteen reducing SOI suppression and suppressing interferences. It is not clear ho to choose a good diagonal loading factor γ in the traditional diagonal loading methods. In [4], the folloing optimization problem is considered: here E is a sphere defined as R y subject to s 1, s E, E = s + e e 1}, 6 here s is the assumed steering vector. Instead of imposing a linear constraint, this method constrains the magnitude responses of a sphere set of steering vectors to exceed unity. If the mismatched steering vector is in this uncertainty set, its gain ill exceed unity and ill not be attenuated. Unlike the method in [10], this method directly protects the SOI response from being suppressed. In [5], [6], the uncertainty set in Eq. 6 has been generalized to an ellipsoid. This method takes into account any steering vector mismatch that keeps the vector in the sphere in Eq. 6. If the primary mismatch is knon to be angle of arrival mismatch, then the set in Eq. 6 can be replaced ith a much smaller set, thereby resulting in better performance. In this paper, e take such an approach. III. NEW ROBUST BEAMFORMER In this paper, e focus on the direction-of-arrival DOA mismatch. When there is a DOA mismatch, the imization in Eq. 5 suppresses the magnitude response of the actual SOI. To avoid this, e should force the magnitude responses at a range of arrival angles to exceed unity hile imizing the total output variance. This optimal robust beamformer problem can be expressed as d = arg R y subject to s θ 1 for θ 1 θ θ, 7 here θ 1 and θ are the loer and upper bounds of the SOI arrival angle respectively, and sθ is the steering vector defined in Eq. 1 ith the arrival angle θ. The folloing uncertainty set of steering vectors is considered: s = 1 e jω e jn 1ω T ω1 ω ω }, 8 here ω 1 π sin θ 1 /λ, and ω π sin θ /λ. Unlike other uncertainty set based approaches using ellipsoid, spherical, or polyhedron sets, the uncertainty set e considered in Eq. 8 is a curve. This constraint protects the signals in the range of angles θ 1 θ θ from being suppressed. A. To-point quadratic constraint There is no standard tool for solving the optimal beamformer d in Eq. 7 because the constraint does not fit any of the existing standard optimization tools. The constraint s θ 1 for θ 1 θ θ can be vieed as infinite number of quadratic constraints. Other uncertainty steering vector based methods solve this problem by adding more constraints so that the problem fits some standard optimization tools such as SOCP. Our approach does this in an opposite ay. We start looking for the solution by loosening the constraint. We first loosen the constraint by choosing only to constraints s θ 1 1 and s θ 1 from

3 the infinite constraints s θ 1 for θ 1 θ θ. The corresponding optimization problem can be ritten as 0 = arg R y subject to s θ 1 1, and s θ 1. 9 Because the constraint is loosened, the imum to this problem is a loer bound of the original problem in Eq. 7. In order to obtain an analytic solution, e recast the problem to the folloing equivalent form:,φ,ρ R y 0 1,ρ 1 1 subject to S = ρ 1 e jφ, here S = sθ 1 sθ, and ρ 0, ρ 1 and φ are real numbers. To solve this problem, e first assume φ, ρ 0 and ρ 1 are given and solve. The solution 0 can be obtained by Lagrange multiplier method and can be expressed as 0 = R 1 y SS R 1 y S 1 ρ 1 e jφ. 10 Given φ, ρ 0 and ρ 1, 0 can be found from the above equation. The task no is to solve for φ, ρ 0 and ρ 1. Write S R 1 y S 1 r = 0 r e jβ r e jβ, r 1 here r 0, r 1 and r are real nonnegative numbers. Substituting 0 in Eq. 10 into the objective function, it becomes 0 R y 0 = ρ 0 ρ 1 e jφ S R 1 y S 1 ρ 1 e jφ = r 0 ρ 0 + r 1 ρ 1 +Rer ρ 0 ρ 1 e jβ+φ } r 0 ρ 0 + r 1 ρ 1 r ρ 0 ρ To imize the objective function, φ can be chosen as φ = β + π 1 so that the last equality in Eq. 11 holds. No φ and 0 are obtained by Eq. 1 and Eq. 10, and the objective function becomes Eq. 11. To further imize the objective function, ρ 0 and ρ 1 can be found by solving the folloing optimization problem: r 0ρ 0 + r 1 ρ 1 r ρ 0 ρ 1. ρ 0 1, ρ 1 1 This can be solved by using the Karush-Kuhn-Tucker KKT condition. The folloing solution can be obtained: 1, r /r ρ 0 = 0 1 r /r 0, r /r 0 > 1, 1, r /r ρ 1 = 1 1 r /r 1, r /r 1 > Summarizing Eq. 1, Eq. 13 and Eq. 10, the folloing algorithm for solving the beamformer ith the to-point quadratic constraint in Eq. 9 is obtained. Algorithm 1: Given θ 1, θ and R y, 0 can be computed by the folloing steps: 1. S sθ 1 sθ.. V R y 1 S. r 3. R 0 r e jβ r e jβ S V 1. r 1 4. φ β + π. 1, r /r ρ r /r 0, r /r 0 > 1. 1, r /r ρ VR r /r 1, r /r 1 > 1. ρ 1 e jφ. The matrix inversion in Step contains most of the complexity of the algorithm. Therefore the algorithm has the same order of complexity as the MVDR beamformer. Because the constraint is loosened, the feasible set of the to-point quadratic constraint problem in Eq. 9 is a superset of the feasible set of the original problem in Eq. 7. The imum found in this problem is a loer bound of the imum of the original problem. If the solution 0 in the to-point quadratic constraint problem in Eq. 9 happens to satisfy the original constraint s θ 0 1 for θ 1 θ θ, then 0 is also the solution to the original problem in Eq. 7. Unfortunately, in general the original constraint s θ 1 for θ 1 θ θ is not guaranteed to be satisfied by the solution to Eq. 9. This problem ill be overcome by a method provided in the next section. B. To-point quadratic constraint ith diagonal loading Substituting Eq. 8 into the constraint in Eq. 7, e can rerite the robust condition into W e jω 1 for ω 1 ω ω. If the robust condition is not satisfied, W z must have a zero hose angle is beteen ω 1 and ω.ifθ 1 and θ are close, the zero is also close to both of the quadratic-constraint points. Thus, it attenuates the gain at these points. Hoever, the magnitude responses at these points are constrained to exceed unity. To satisfy the constraints, the overall energy of must be adjusted to a certain high level. Therefore, if the robust condition s θ 1 for θ 1 θ θ is not satisfied, the norm of the eighting vector ill become very large. By using this fact, e can impose some penalty on to avoid this situation from happening. This can be done by the diagonal loading approach mentioned in Sec. II. The corresponding optimization problem can be ritten as γ = arg R y + γ subject to s θ 1 1, and s θ 1, 14 here γ is the diagonal loading factor hich represents the amount of the penalty put on. The solution γ can be found by perforg the folloing modification on the output covariance matrix: R y R y + γi N and then applying Algorithm 1. When γ, the solution converges to = arg subject to s θ 1 1, and s θ 1. 15

4 The folloing lemma gives the condition for hich satisfies the constraint sθ 1 for all θ in θ 1 θ θ. Lemma 1: s θ 1 for θ 1 θ θ if and only if sin θ sin θ 1 λ/dn. Proof: According to Eq. 15, substituting R y = I N and applying Algorithm 1, one can obtain = here 1 N + sincd ω ω1 sθ 1+sθ e j ω ω 1 N 1, ω 1 π λ d sin θ 1, ω π λ d sin θ, and sincdω sinωn sin ω. By direct substitution, one can obtain s θ = sincd ω1 ω +a sincd ω ω N + sincd ω ω1, 16 here ω π λ d sin θ and 1, if sincd ω ω 1 a = > 0 1, otherise. By Eq. 16, it can be verified that s θ 1 for ω 1 ω ω if and only if ω ω 1 π N hich can also be expressed as sin θ sin θ 1 λ/dn. If the condition sin θ 1 sin θ λ/dn is satisfied, there exists a γ>0 such that the condition s θ γ 1 for θ 1 θ θ is satisfied. Hoever, introducing the diagonal loading changes the objective function R y to R y + γi N. The modification of the objective function affects the suppression of the interferences. To keep the objective function correct, γ should be chosen as small as possible hile the condition s θ 1 for θ 1 θ θ is satisfied. For finding such a γ, e propose the folloing algorithm: Algorithm : Given θ 1, θ, R y, an initial value of γ, a search step size α>1 and a set of angles, ζ i, i =1,,,n hich satisfies θ 1 <ζ i <θ for all i, γ can be computed by the folloing steps: 1. R y R y + γi N.. Compute γ by Algorithm If s ζ i γ 1 for all i =1,,,n then stop. else γ αγ, and go to 1. This algorithm keeps increasing γ by multiplying α until s ζ i γ 1for all i =1,,,n is satisfied. This is an approximation for s θ γ 1 for θ 1 θ θ. In the numerical results in next section, the number n can be very small. The choice n =3orks ell in all of the examples. Also, the SINR is not sensitive to the choice of α. IV. NUMERICAL EXAMPLES For the purpose of design examples, the same parameters used in [6] are used in this section. An uniform linear array ULA of N = 10 omnidirectional sensors spaced halfavelength apart i.e. d = λ/ is considered. There are three signals impinging upon this array: 1 the signal of interest SOI xt ith angle of arrival θ, an interference signal x int1 t ith angle of arrival θ int1 =30 and σint1 =104 40dB above noise and 3 another interference signal x int t ith angle of arrival θ int =75 and σint =10 0dB above noise. The received noise variance σ =1. The actual arrival angle θ is 43 but the assumed arrival angle θ m is 45. Example 1: SINR versus diagonal loading factors γ. In this example, the SNR is 10dB. The SINR defined in Eq. is compared for different diagonal loading factors γ. The folloing methods are considered: 1 Algorithm 1 ith θ 1 =4 and θ =48. General-rank method [9] ith the parameter ɛ = max 48 θ 4 sθs θ s45 s 45 F Diagonal loading method [10]. 4 Directional LCMV linear constrained imum variance [13] ith to linear constraints hich forces the responses of the signals from 4 and 48 to be unity. 5 Derivative LCMV [15] ith to linear constraints hich forces the responses of the signals from 45 to be unity and the derivative of the beampattern on 45 to be zero. One can verify that the choice of θ 1, θ, d and N satisfies sin θ 1 sin θ λ/dn. Therefore, by Lemma 1, there exist a γ>0so that sθ γ 1 for θ 1 θ θ. The SINR of the MVDR beamformer ithout mismatch is also plotted. This is an upper bound on the SINR. Fig. shos the result for SNR = 10dB. One can observe that there is SINR db MVDR, no mismatch Algorithm 1 General rank ε= Diagonal loading Directional LCMV Differential LCMV Diagonal loading level γ Fig.. Example 1: SINR versus γ ith SNR = 10dB. a huge jump in the SINR of Algorithm 1 around γ = 3. When this happens, the SINR of Algorithm 1 increases significantly and becomes very close to the upper bound provided by the MVDR beamformer ithout mismatch. This jump happens hen the robust condition sθ γ 1 for θ 1 θ θ is satisfied. After the jump, the SINR decays

5 sloly as γ increases because of the over-suppression of hite noise. Observing Fig., e can see hy Algorithm orks. Algorithm increases γ by repeatedly multiplying α until γ satisfies γsζ i 1 for i =1,,,n. This happens as γ crosses the jump in SINR. Also, the SINR is not sensitive to the choice of α because the SINR decays very slo after the jump. By Algorithm, e can find a suitable γ ith only a fe iterations. For other approaches involving diagonal loading, it is not clear ho to find a good diagonal loading factor γ. Example : SINR versus SNR. In this example, the SINR in Eq. are compared for different SNRs ranged from -0dB to 30dB. The folloing methods are considered: 1 Algorithm ith θ 1 = 4, θ = 48, ζ 1 = 43.5, ζ =45, ζ 3 =46.5, initial γ =1and step size α =. General-rank method. Same as in Example 1. 3 Extended diagonal loading method [4] [6] ith the parameter ɛ = max sθ 48 θ 4 s Directional LCMV [13], [14] ith to linear constraints at the angles 4, Directional LCMV ith three linear constraints at the angles 4, 45 and Derivative LCMV ith to linear constraints [15]. Same as in Example 1. 7 Derivative LCMV ith three linear constraints hich force the responses of the signals from 45 to be unity and both the first and second derivatives of the beampattern on 45 to be zero. 8 The standard MVDR beamformer in Eq. 4. Again, the SINR of the MVDR beamformer ithout mismatch is also plotted. It provides the upper bound on the SINR of these robust beamformers. The results are shon in Fig. 3. The SINR of the standard MVDR beamformer SINR db MVDR, no mismatch Algorithm Extended diagonal loading ε= General rank ε= Directional LCMV to points Directional LCMV three points Differential LCMV first order Differential LCMV second order MVDR SNR db Fig. 3. Example : SINR versus SNR ith γ =10. is seriously degraded ith only of mismatch. When the SNR increases, the MVDR beamformer tends to suppress the strong SOI to imize the total output variance. Therefore, in the high SNR region, the SINR decreases hen SNR increases.the performances of the uncertainty based methods are better than the LCMV methods. This is because the linear equality constraints are too strong compared to the quadratic inequality constraints. Among the uncertainty based methods, Algorithm has the best SINR performance because it has an uncertainty set hich focuses on the DOA mismatch only. The SINR is very close to the upper bound provided by the MVDR beamformer ithout mismatch. V. CONCLUSIONS In this paper, a ne beamformer ich is robust against DOA mismatch is introduced. This robust beamformer quadratically constrains the magnitude responses of to steering vectors and then uses a diagonal loading method to force the magnitude response in a range of arrival angles to exceed unity. Therefore this method can alays force the gains in a desired range of angles to exceed a constant level. The analytic solution to the quadratic constraint imization problem has been proposed in Algorithm 1, and the diagonal loading factor γ can be detered by a simple iteration method proposed in Algorithm. The complexity required in Algorithm 1 is approximately about the same as in the MVDR beamformer. The overall complexity depends on the number of iterations in Algorithm hich depends on the SNR. In our numerical examples, hen SNR < 10dB, the number of iterations is less than three. The numerical examples also sho that our approach has a significantly better SINR performance compared to previously published methods. REFERENCES [1] J. 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