ROBUST ADAPTIVE BEAMFORMER WITH FEASIBILITY CONSTRAINT ON THE STEERING VECTOR

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1 ROBUST ADAPTIVE BEAMFORMER WITH FEASIBILITY CONSTRAINT ON THE STEERING VECTOR Wenyi Zhng nd Bhskr D. Ro Deprtment of Electricl nd Computer Engineering University of Cliforni, Sn Diego L Joll, CA , USA phone: , , emil: w3zhng@ucsd.edu, bro@ece.ucsd.edu ABSTRACT The stndrd MVDR bemformer hs high resolution nd interference rejection cpbility when the rry steering vector is ccurtely known. However, it is known to degrde if steering vector error exists. Motivted by recent work in robust dptive bemforg, we develop vrints of the constrined robust dptive bemformer tht ttempt to limit the serch in the underlying optimiztion problem to fesible set of steering vectors thereby chieving improved performnce. The robustness ginst steering vector error is provided through sphericl uncertinty set constrint, while set of mgnitude constrints is enforced on ech element of the steering vector to better constrin the serch in the spce of fesible steering vectors. By ppropritely chnging the vribles, the optimiztion problem is modified such tht the need for the mgnitude constrints re voided. The developed lgorithm is tested in the context of speech enhncement using microphone rry nd shown to be superior to existing lgorithms. 1. INTRODUCTION The stndrd MVDR bemformer hs high resolution nd interference rejection cpbility when the rry steering vector is ccurtely known [1]. However, the performnce of trditionl dptive bemformer cn degrde severely in prctice when the Signl Of Interest (SOI) steering vector errors exist, which my be due to look direction error, rry sensor position error, nd smll mismtches in the sensor responses. In such cses, the SOI might be mistken s n interference signl nd be suppressed. Mny robust bemforg lgorithms hve been proposed to ddress this problem. Derivtive constrint in the look direction is proposed in [2, 3]. Er nd Cntoni proposed robust bemforg lgorithm which restricts the error between the desired nd ctul bem pttern of the rry over smll sptil region round the rry s look direction, llowing for uncertinty in the look direction [4]. Norm constrined nd white noise gin constrined dptive bemformer is studied in [5, 6] nd widely used therefter. Recently some interesting robust dptive bemformers hve been proposed. Robust dptive bemforg using worst-cse performnce optimiztion is proposed in [7, 8]. The problem is formulted s imizing qudrtic function subject to infinitely mny qudrtic constrints. It is reduced to second-order cone progrmg problem which cn be solved by interior point methods. Li nd Stoic This mteril is bsed upon work supported by the RESCUE project, under Ntionl Science Foundtion Awrd Number proposed the robust Cpon bemformer (RCB) [9] where sphericl uncertinty set constrint is enforced on the rry steering vector. They lso developed doubly constrined robust Cpon bemformer (DCRCB) [10] bsed on RCB, wherein norm constrint on the bemformer steering vector is dded. A comprison of these two bemformers is given in [11] nd geometricl explntion is provided. In this pper, motivted by the constrined robust dptive bemformer developed by Li nd Stoic. We develop vrints tht ttempt to limit the serch in the underlying optimiztion problem to fesible set of steering vectors thereby chieving improved performnce.the robustness ginst steering vector error is provided through sphericl uncertinty set constrint, while set of mgnitude constrints is enforced on ech element of the steering vector to better constrin the serch to the spce of fesible steering vectors. By ppropritely chnging the vribles, the optimiztion problem is modified such tht the need for the mgnitude constrints re voided. The developed lgorithm is tested in the context of speech enhncement using microphone rry nd shown to be superior to existing lgorithms. 2. BACKGROUD 2.1 Stndrd MVDR Bemforg (MVDR) The MVDR bemforg is lso clled Cpon bemforg [1]. The problem is formulted s imizing the output energy of the bemformer while mintining constnt response in the look direction, i.e. w wh Rw, s.t. w H = 1. (1) where R is the signl correltion mtrix. is the SOI steering vector. w is the bemformer weight vector. The solution to this optimiztion problem is given by w = 2.2 Robust Cpon Bemforg (RCB) R 1 H R 1. (2) The Robust Cpon Bemforg (RCB) is proposed in [9]. Suppose 0 is the true SOI steering vector nd is the ssumed steering vector. 0 is ssumed to be in the vicinity of. This cn be expressed mthemticlly by the following inequlity 0 2 ε, (3) where ε is bound controlling the uncertinty in the ssumed look direction.

2 The Cpon bemforg problem cn be reformulted s mx σ 2, s.t. R σ 2 H 0. (4) σ 2 where R is the signl correltion mtrix. σ 2 is the signl power to be estimted. Use the new formultion, one cn write the RCB problem s mxσ, s.t. R σ 2 H 0 nd 2 ε. (5) σ 2, Using the fct tht, for ny fixed, the solution to (4) with regrd to σ 2 is obtined by ˆσ 2 = 1/( H R 1 ) (6) the optimiztion problem (5) cn be written s H R 1, s.t. 2 ε. (7) The solution cn be found using Lgrnge multiplier method s â 0 = ā U(I + λγ) 1 U H ā (8) where R = UΓU H is the eigenvlue decomposition of R, nd λ is the Lgrnge multiplier. Once the SOI steering vector is estimted, the signl power cn be estimted s in (6) nd the bemformer weight vector is esily obtined s in MVDR bemforg (2). One difficulty with this pproch is tht it tends to overestimte the signl power σ 2, becuse both the SOI power nd the SOI steering vector re tken s unknowns in problem (5). Thus, (σ 2,) nd (σ 2 /α,α 1/2 ), α > 0 will give the sme item σ 2 H. Suppose (σ 2 0, 0) is the true solution to be found, the formultion of (5) will prefer the pir (σ 2 0 /α,α1/2 0 ) if only α < 1 nd α 1/2 0 is still in the uncertinty set. By the deduction bove, we cn be certin tht the solution to (5) will mke the inequlity constrint in (5) ctive, i.e. â 0 2 = ε. This problem is solved in [9] by normliztion step such tht â 0 2 = N, where N is the number of sensor elements. 2.3 Doubly Constrined Robust Cpon Bemforg (DCRCB) To void the signl power overestimtion problem discussed bove in section 2.2, the Doubly Constrined Robust Cpon Bemforg (DCRCB) is proposed [10]. The problem is formulted in similr wy s in (7) except tht n extr norm constrint on the steering vector is dded. The problem is formulted s H R 1, s.t. 2 ε nd 2 = N (9) The solution cn be found using the Lgrnge multiplier method â = (N ε 2 ) U(I + λγ) 1 U H ā ā H U(I + λγ) 1 U H (10) ā where R = UΓU H is the eigenvlue decomposition of R, nd λ is the Lgrnge multiplier. Then the bemformer weight vector is esily obtined s in MVDR bemforg (2). In both RCB nd DCRCB, the bound ε is chosen such tht ll possible SOI steering vectors 0 is included in the uncertinty set described by (3). 3. MAGNITUDE CONSTRAINED ROBUST MVDR BEAMFORMER The RCB (section 2.2) nd DCRCB (section 2.3) bemforg lgorithms my fil becuse the optimum solution â to the optimiztion problem (7) or (9) my not be vlid steering vector. A vlid steering vector is usully structured nd is not ny rbitrry element in the constrined set (3). We develop vrints of the constrined robust dptive bemformer tht ttempt to limit the serch in the underlying optimiztion problem to fesible set of steering vectors thereby chieving improved performnce. For n rry with identicl omnidirectionl sensors, vlid steering vector cn be expressed s = [e jωτ 1,e jωτ 2,...,e jωτ N ] T for the fr field sources. We observe tht ech element of the the steering vector hs mgnitude 1. Therefore n option is to enforce set of mgnitude constrints on ech element of the steering vector bsed on RCB (7) thereby mking the serch spce smller nd more fesible. The new optimiztion problem cn be formulted s R 1, s.t. 2 ε nd k = 1,k = 1..N (11) where k is the k th element of the steering vector, i.e. = [ 1, 2,..., N ] T. Unfortuntely, closed form solution to this optimiztion problem is not vilble nd n optimiztion routine hs to be utilized. 3.1 Time Dely Bsed Robust MVDR Bemformer (rob- MVDRtd) By mnipulting the vribles, we cn crete robust bemforg problem similr to problem (11). In prticulr, we use the form of the steering vector i for specific frequency ω i s i = [e jω iτ 1,e jω iτ 2,...,e jω iτ N ] T (12) As e jω iτ k 1, optimizing over the time dely vribles τ i ensures the mgnitude constrint in (11) is utomticlly stisfied nd thus need not be explicitly enforced. The new robust bemforg problem is formulted s H τ i R 1 i i, s.t. τ k τ k δ k, k = 1..N (13) where R i is the signl correltion mtrix for frequency ω i. τ = [τ 1,τ 2,...,τ N ] T, nd τ = [τ 1,τ 2,...,τ N ] T is the ssumed look direction time dely vector. δ k,k = 1..N is set of bounds controlling the uncertinty in the look direction. The new problem (13) cn be solved by using n pproprite optimiztion routine. We use subspce trust region method which is bsed on interior-reflective Newton lgorithm to find the solution to problem (13). We need the grdient nd Hessin of the objective function h(τ) = H i R 1 i i, where i is specified by (12). It is strightforwrd to obtin grdient s τ h = AR T i i + A R 1 i i (14) = rel(a R 1 i i ) (14b) where (.) denotes conjugte nd (.) T denotes trnspose A = ( jω i ) (15) 0 0 N

3 where k is the k th element of the steering vector i, i.e. i = [ 1, 2,..., N ] T. Also, the Hessin is obtined s 2 τh = rel(( jω i )dig( H i R 1 i )A + AR T i A ) (16) In the context of brodbnd signls, for ech frequency component ω i of the signl one hs to solve problem like (13). However, the objective imizer ˆτ is the true time dely from the SOI to ech microphone element, which doesn t depend on the frequency ω i. In other words, we wnt to find common imizer ˆτ tht is vlid for ll the frequencies. This is not utomtic nd hs to be enforced. It cn be chieved by combining the series of bemforg problems on individul frequency bins into single problem to provide robustness. The brodbnd bemforg problem cn be formulted s τ i R 1 i i, s.t. τ k τ k δ k, k = 1..N (17) i 3.2 Angle Bsed Robust MVDR Bemformer (robmv- DRngle) The RCB (section 2.2), DCRCB (section 2.3) nd robmv- DRtd (section 3.1) lgorithms ssume uncertinty in the steering vector, which tkes both the SOI look direction error nd the rry sensor s position error into considertion. The problem cn be simplified when only SOI look direction error exists. For instnce, in the cse of 2-dimensionl spce the sources incidence directions cn be represented by only one prmeter θ. Hence, we cn use v(θ) to substitute for the steering vector in (13). The new robust bemforg problem cn be written s θ v(θ)h R 1 v(θ), s.t. θ θ ε (18) where v(θ) = [e jωτ 1,e jωτ 2,...,e jωτ N ] T, nd τ i,i = 1,..,N is functions of θ bsed on the geometry of the rry. θ is the ssumed look direction. ε is bound controlling the uncertinty in the ssumed look direction. The problem (18) cn be solved by one dimensionl numericl optimiztion lgorithm such s the golden section serch method. 4. SIMULATION 4.1 Bemforg Algorithms Nottion We use the following nottion for ech bemforg lgorithm. OMVDR: the idel MVDR bemforg which ssumes we know the true SOI steering vector MVDR: stndrd MVDR bemforg DS: conventionl dely nd sum bemforg RCB: robust Cpon bemforg (section 2.2) DCRCB: doubly constrined robust Cpon bemforg (section 2.3) robmvdrtd: time dely bsed robust MVDR bemforg (section 3.1) robmvdrngle: ngle bsed robust MVDR bemforg (section 3.2) Figure 1: Cepstrl distnce between recovered signl s spectrum nd the SOI s spectrum, only look direction error exists. 4.2 Simultion Scenrio In this section, we provide numericl exmples on speech enhncement using microphone rry to compre the performnces of vrious bemformers. We ssume circulr microphone rry with 8 sensors. The 8 sensors re eqully distributed on 20cm dimeter circle nd indexed counter clockwise. The sources, both the SOI nd interference signls, re plne wves which exist in the sme plne s the circulr rry. We define the origin of the coordinte system to be the center of the circulr microphone rry, nd define ngle 0 to be the direction of the 8 th microphone. The ngle increses counter clockwise, which mens the 1 st microphone is t ngle 45, the 2 nd microphone is t ngle 90, nd so on. In the simultion, every source signl is single chnnel speech sentence, which is round 1s in durtion. The smpling rte is 8kHz. Short Time Fourier Trnsform (STFT) is used to trnsform the multichnnel dt into the frequency domin nd the nrrowbnd bemforg lgorithms re then pplied. The frme length is 0.25s (200 smples), with step length of 0.125s (100 smples). A 256 points FFT is used on ech frme. 4.3 Simultion Results The performnce of vrious bemformers is mesured by the cepstrl distnce between the recovered signl s spectrum nd the originl SOI s spectrum. The cepstrl distnce is used becuse it is perceptul metric commonly used in speech processing to mesure distortion. Fig.1 shows the bemformers performnce versus SNR, which is signl to white noise rtio. Only one SOI nd one interference signl exist in this experiment. The interference signl nd SOI hs the sme level of energy. The interference signl come from direction 90. The ssumed look direction is 180, while the true SOI direction is 178, which mens 2 look direction error. There s no sensor position error in this experiment. Fig.2 shows the bemformers performnce by cepstrl distnce when there exist not only the look direction error, but lso sensor position error. The displcement error for ech sensor is generted by n uniformly distributed rndom

4 bemformer. This cn be explined by the choice of uncertinty bound ε. The bound ε is chosen such tht ll possible SOI steering vectors 0 re included in the prescribed uncertinty set. This usully brings on big vlue of ε t high frequency, which results in mny infesible steering vectors being included in the uncertinty set (3). Thereby the imizer to the optimiztion problem (7) nd (9) is no longer vlid steering vector in the high frequency rnge. A close observtion of the steering vector which imizes the optimiztion problem (7) nd (9) t high frequency rnge confirms the bove resoning. The element mgnitudes of those steering vectors hve been found to be fr wy from 1. REFERENCES Figure 2: Cepstrl distnce between recovered signl s spectrum nd the SOI s spectrum, both look direction error nd sensor position error exist. vrible whose mximum vlue is 3mm. All the other settings re the sme s those of the forementioned experiment. The OMVDR bemformer gives the optiml performnce nd bounds the performnce tht cn be ttined by these clss of dptive bemformers. Our simultion results clerly demonstrte tht the proposed robmvdrtd bemformer consistently performnce well nd is very close in performnce to the OMVDR bemformer. The robmvdrtd bemformer outperforms the conventionl fixed DS bemformer nd other dptive bemformers such s MVDR, RCB nd DCRCB. The proposed robmvdrngle bemformer works extremely well when only look direction error exists. It hs the sme performnce s the optiml OMVDR bemformer in this condition. However, it deteriortes nd hs performnce comprble to the stndrd MVDR bemformer when sensor position error exists. This highlights the sensitivity of ngle bsed formultion. Although the norm constrint on steering vector is introduced in the DCRCB method to prevent overestimtion of signl power in the RCB method, our simultions show tht the DCRCB method hs worse performnce thn the RCB method. This cn be explined by noting tht even when n extr norm constrint on steering vector is dded, the imizer to the optimiztion problem (9) is still not vlid steering vector. This phenomenon cn be observed by exing the bemptterns in Fig.3. Listening to the reconstructed speech indictes the output of the RCB/DCRCB to be better thn the DS bemformer even though it is not evident from the cepstrl distnce mesure employed. Fig.3 shows the mgnitude bem pttern of vrious bemformers on one smple dt. This smple dt is selected from the dt set used to generte Fig.1. The SNR is 43dB. It is evident tht the bem pttern of the robmvdrtd method is close to tht of the OMVDR bemformer. The MVDR bemformer forms two deep nulls, one in the interference direction, the other in the SOI direction. The RCB nd DCRCB method cn steer null in the interference direction (90 ) t low frequency rnge, while t middle to high frequency rnge, their bem ptterns re similr to tht of DS [1] J. Cpon, High resolution frequency-wvenumber spectrum nlysis, Proc. IEEE, vol. 57, pp , Aug [2] K. M. Buckley nd L. J. Griffiths, An dptive generlized sidelobe cnceller with derivtive constrints, IEEE Trnsctions on Antenns nd Propgtion, vol. AP-34, pp , Mr [3] M. H. Er, Adptive ntenn rry under directionl nd sptil derivtive constrints, Microwves, Antenns nd Propgtion [see lso IEE Proceedings- Microwves, Antenns nd Propgtion], IEE Proceedings H, vol. l35, Issue 6, pp , Dec [4] M. H. Er nd A. Cntoni, A unified pproch to the design of robust nrrow-bnd ntenn rry processors, Antenns nd Propgtion, IEEE Trnsctions on, vol. 38, Issue 1, pp , Jn [5] H. Cox, R. M. Zeskind, nd M. Owen, Robust dptive bemforg, IEEE Trns. on ASSP, vol. 35, pp , [6] J. E. Hudson, Adptive Arry Principles. London, U.K.: Perter Peregrinus, [7] S. Vorobyov, A. B. Gershmn, nd Z-Q. Luo, Robust dptive bemforg using worst-cse performnce optimiztion: solution to the signl mismtch problem, IEEE Trns. Signl Processing, vol. 51, pp , Feb [8] R. G. Lorenz nd S. P. Boyd, Robust imum vrince bemforg, Signl Processing, IEEE Trnsctions on, vol. 53, Issue 5, pp , My [9] J. Li, P. Stoic, nd Z. Wng, On robust Cpon bemforg nd digonl loding, IEEE Trns. Signl Processing, vol. 51, pp , Jul [10] J. Li, P. Stoic, nd Z. Wng, Doubly constrined robust Cpon bemformer, IEEE Trnsctions on Signl Processing, vol. 51, pp , Sep [11] J. Wrd, H. Cox, nd S. M. Kogon, A comprison of robust dptive bemforg lgorithms, Signls, Systems nd Computers, Conference Record of the Thirty-Seventh Asilomr Conference on, vol. 2, pp , Nov

5 14th Europen Signl Processing Conference (EUSIPCO 2006), Florence, Itly, September 4-8, 2006, copyright by EURASIP () OMVDR bemformer (b) stndrd MVDR bemformer (c) DS bemformer (d) RCB bemformer (e) DCRCB bemformer (f) robmvdrtd bemformer Figure 3: Bem pttern of vrious bemformers over ngle θ nd frequency bins. The look direction is 180, the true SOI direction is 178, nd the interference direction is 90.