/09/$ IEEE 472

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1 Bai Puruit for Robut Paive Acoutic Beamforming Ben Shapo and Chri Kreucher Integrity Application Incorporated 900 Victor Way, Suite 220 Ann Arbor, MI Abtract 1 Beamforming i a proce that upplie directional gain to enor array proceing. One modality where beamforming add great value i paive onar, where real-aperture array receive ignal emitted by acoutic ource. In paive onar ytem, the beamformer i the backbone of a proceing tructure that detect, localize, and claifie external target. Conventional beamformer ue determinitic time-delay (often implemented a phae hift) to arrange coherent addition of plane-wave ignal at each enor. Recently, adaptive beamformer take advantage of ignal time hitory by impoing a model on the environment. Bai Puruit i another recontruction approach ued in Compreed Sening that alo enforce a phyic-baed model in thi cae a model of cene parity. Thi paper decribe an application of thi technique to the beamforming problem. The main benefit of the Bai Puruit beamforming approach i that it i robut to miing array element, providing nearly full-aperture performance in a reduced enor environment. Thi reult i advantageou in the cae of proceing with inoperative hydrophone. It may alo provide cot aving by allowing array deign with fewer hydrophone. I. INTRODUCTION Recently, the ignal proceing community ha hown great interet in Compreed Sening ( CS ) [1]. In CS, fewer (ometime ignificantly fewer) ample of a phyical phenomenon than traditional Nyquit bound would mandate recontruct the original ignal via a technique uch a Bai Puruit (BP). Good recontruction i poible under the aumption that the ampled ignal are actually pare. Paive onar often obey thi aumption, with 180- (or 360)-degree field of view but few target. Thi paper decribe a beamforming method that exploit the parity contraint ued in BP. Like the Conventional Beamformer ( CBF ), we aume each ignal impinging on a paive array of enor arrive a a planewave [2]. In addition, we alo aume the number of planewave ignal i mall compared to the number of arrival angle conidered ( beam ). Our method produce reult comparable to CBF in the cae of a full-aperture array. However, when the number of enor decreae (either by hyrdrophone becoming unexpectedly inoperable or by deign), our method produce higher-fidelity reult than the CBF approach operating on the ame reduced input data. 1 Thi work wa partially funded by the Office of Naval Reearch contract N C The author would like to thank Dr. John Tague for hi upport, and Mr. Scott Spencer and Dr. Charle Choi for their aitance. Thi paper proceed a follow. Section II review a tandard model of plane-wave phyic and derive the CBF. A imulated example demontrate the power of thi claic approach, and highlight the benefit of uing data in the wide frequency range available with broadband enor. Section III build on the model preented in Section II, and give a brief decription of our BP approach to beamforming. Thi ection how reult from a BPbeamformer operating on the ame imulated data input from Section II. Section IV demontrate the benefit of the methodology on Navy collected at-ea data by comparing CBF and BP beamformer output in the cae where the number of enor ha been ignificantly reduced. II. PLANE WAVE MODEL AND CBF A. Plane Wave Model Thi ection content are baed on material from [2]. We aume an array of acoutic enor with energy from acoutic ource impinging on the element in the enor array. The array aperture refer to the fact that each element reide at a different patial location. Thi diverity caue propagation time from a target ource to each element to differ. In the frequency domain, thee propagation time manifet themelve a different phae value. Figure 1 illutrate the ituation for the pecial cae of a linear array. A plane wave (red line) arrive at an angle θ with repect to the array endfire direction. If the ditance from the ource to the array center element i r, then the ditance from the ource to an element located at a ditance D from the center element i: r Δr, where Δr = D co(θ ). For a narrowband acoutic ignal, or for a ingle Fourier component of a broadband ignal (i.e., a plane wave), thi relation provide the differential phae for a ignal received at the nth enor element: 2π Δϕ ( n) = Δr( n), (1) where λ i the wavelength of the plane wave. λ B. CBF The received ignal would add coherently acro element if their arrival time (or phae) were forced to align. The idea of the Conventional Beamformer ( CBF ) i to hypotheize a large et of candidate arrival angle ( beam ) and, on a per-beam bai, impart appropriate phae delay at /09/$ IEEE 472 Ailomar 2009

2 each element to force any ignal arriving from that direction to add coherently. Thi proceing caue ignal in each hypotheized direction to add contructively, while ignal emanating from other direction do not. [ a u ) a( u ) a( ] T A = ) (4) allow the operation ( 1 2 u N θ H b = A z (5) Figure 1: Senor Array Geometry. From the time-delay relationhip at the bottom of Figure 1, time delay i a function of the coine of arrival angle. Thu, in the figure that follow, the independent axi repreentation of arrival angle i u = co(θ) (rather than θ). The quantity that decribe a narrowband, plane-wave ignal arriving from an angle u a it impinge on each element i a N 1 vector (where N i the number of element in the array). Each term in the vector repreent the phae rotation of the ignal a it arrive at the correponding element. Thi quantity i called a teering vector or a manifold vector. Denote the teering vector for a 1D array a a. We have (from Figure 1) thi expreion for a plane wave ignal arriving from angle pecified by u for the nth element in the vector: a ( u) n jkp( n) u = e (2) where p(n) i the location of the nth element in 1D, and the wavenumber k i 2π/λ. Thu, for a 1D array with element uniformly paced by a ditance d, the entire vector become: jkdu jk 2du jk ( N 1) du T [ 1 e e e a( u) = ] (3) Implementation of a CBF beamformer at a ingle bearing (or beam) imply involve rotating the phae of each hypotheized ignal (i.e., a plane wave at arrival angle pecified by u) by the appropriate amount to caue the ignal at each element to combine coherently. Thi procedure amount to imply pre-multiplying the ignal by a H, where () H denote the conjugate tranpoe operation. By extenion, we can repreent the entire CBF operation, where we potulate a large number of arrival direction, a a linear filter with fixed coefficient. Recall that N i the number of element, and defining N θ a the number of hypotheized arrival angle (beam), a N N θ teering matrix to perform the entire beamforming operation (all N θ beam at a ingle frequency), where z i a N 1 vector repreenting the received data at each element. Here, the output b i a N θ 1 vector repreenting all output beam. Figure 2 (blue line) how a imulated CBF profile for two ideal (noie-free environment) target: one at u=-0.1 and the other at u=0.1. A i evident from the figure, the target have different SNR value. There are 51 enor element and CBF form 201 beam, at the array deign frequency (i.e., the element pacing i λ/2 at the plane wave frequency). The figure clearly indicate peak at the arrival angle correponding to the two target, a well a a good deal of idelobe tructure. The red line how a more realitic beam profile, where additive noie corrupt the arriving ignal at each element. Figure 2: Standard CBF Proceing (2 Target). C. Broadband A powerful technique for improving target SNR and localization i to take advantage of the broad frequency band afforded by many paive enor. Simple incoherent combination of beam profile (like thoe in Figure 2) over multiple frequencie reult in improved performance. Figure 3 provide a imple example. Here, beam profile for the ame cenario a Figure 2 appear for different frequencie. The blue line indicate the profile for the wavelength correponding to the element pacing, and the green line how the profile for the wavelength that correpond to one-fourth the element pacing. Note that operating the array at thi frequency caue patial aliaing (or grating lobe ) artifact. The red line indicate imple incoherent addition of beam profile over a band of frequencie from DC to four time the array deign frequency (i.e., element pacing of 2λ at that frequency). Note that the red line poee narrower mainbeam width than many of the low-frequency bin, with idelobe tructure a well a grating lobe almot eliminated by the averaging proce. 473

3 repreent direction with no ignal preent. Many author have produced tool to olve matrix equation ubject to the contraint that the olution be pare. Some have demontrated that, under certain condition [3][5], olving Figure 3: Broadband CBF. III. BASIS PURSUIT CBF proceing i a very effective technique and it i employed in many real cenario. However, it doe not exploit phyical characteritic of the environment that may lead to improved performance. Thi ection decribe a plane-wave model for beamforming, and develop an alternate formulation of the beamforming problem baed on the Bai Puruit ( BP ) method. A. Plane-wave ignal We have already written a ingle plane-wave ignal arriving from arrival angle θ (where u co(θ ) ) via Equation (3) above, and alo written the expreion for a collection of unit-amplitude plane-wave ignal a A in Equation (4). Given thi notation, a imple expreion decribe a et of ignal arriving at variou angle θ m : arg min i equivalent to olving 1 ubject to A z 2 σ arg min 0 ubject to A z σ 2 (8) i.e., the one-norm minimization generate the maximally pare olution to the matrix equation (Equation (6)). Returning to the imulation of Figure 2 and Figure 3, we compute the BP olution to the ame input data. The particular BP implementation i publicly available [4] from the author of [3]. The reult appear in Figure 4, where the blue line reproduce the noie-free CBF olution from Figure 2, and the red line how the pare BP olution. We have picked enible value for the optimization parameter required by the algorithm and error criterion. The BP approach i able to iolate the two plane-wave target without any fale target in the idelobe tructure. (7) z = A + n (6) Here, i the et of complex ource ignal (N θ 1); A i the teering matrix, which map ignal to received data (a enor model); n i an (N θ 1) noie vector at each of the potential ource location; and z i the data received at the enor element (N 1) The crucial aumption here i that there i a potential ource of energy (target) at each of the N θ arrival angle, and all entrie in the vector correponding to angle where no target i preent are zero. In other word, mot of the N θ entrie in the complex ignal amplitude vector are zero and i pare. Figure 4: BP v. CBF for two Plane Wave Signal. A more realitic cenario i one with corrupting noie, a in Figure 2. We added Gauian noie to the target ignal, and the reult appear in Figure 5. Here, the noie floor puhe the far-out, noie-free, 40dB CBF idelobe (with repect to maximum ignal) up by about 10dB to approximately 30dB. The BP olution, however, i virtually unaffected by the noie, and till produce two ignal peak at the correct arrival angle with no fale ignal. B. Bai Puruit Approach to Beamforming Our application of BP to the beamforming problem relie on two baic aumption. One i narrowband ignal arrive at the array a plane-wave. Thi i a common aumption made in mot beamforming algorithm. The econd aumption i that there are a mall number of ignal compared to the number of available enor element. For example, the cenario that gave rie to Figure 2 and Figure 3 ha two ignal. In thi cae, mot beam 474

4 Figure 5: BP v. CBF for Plane Wave Signal in Noie. C. Application The BP approach produce olution with imulated data characterized by much more favorable idelobe profile than CBF. In addition, another application for BP method i operating a reduced number of enor element. Thi often occur on deployed array in the preence of failed hydrophone. Additionally, it may ait in the deign of future array which can deliver higher preciion beamforming then the conventional approach. The imulation hown in Figure 6 and Figure 7 illutrate the need for an effective approach. Figure 6 repeat the noiy CBF beam profile for the two target in blue. The red line indicate CBF performance with a random et of 20 hydrophone removed (of the original 51). In thi cae, the proceing i the ame a Equation (5) but with the A matrix reduced, each deleted column correponding to a miing hydrophone. It i very clear that performance ha declined ignificantly, with raied inner idelobe level and a 15 db increae in the noie floor region. Figure 7: CS-BP v. CS-CBF with Failed Hydrophone. IV. REAL DATA RESULTS A. Paive Sonar Diplay Modality Thi ection illutrate the performance of the BP approach uing real data collected during an at-ea experiment. The ingle time graph in Figure 3 (red line) i ueful becaue it provide information about where target are located in pace at a frozen time naphot. However, a more ueful diplay provide the bearing information preent in that curve over time. Figure 8 how an example of uch a diplay. Each horizontal lice how the equivalent of the red line in Figure 3 in graycale (i.e., a bearing profile, in coine-pace). White pixel repreent high-energy bearing-time event, and black pixel correpond to low-energy. The vertical axi repreent time. Thu object that move in the x-direction on thee diplay have bearing rate relative to the enor over time. Vertical object, like all the one in thi real-world cenario, do not have ignificant bearing change over time. Becaue of it format, thi diplay i known a a Bearing-Time Record ( BTR. ). Figure 6: CBF with Failed Hydrophone. The BP approach i able to produce nearly full-fidelity reult with the dramatically reduced number of enor. Figure 7 how the outcome in red, and the reduced-enor CBF reult appear for reference in blue. Comparing the red BP curve here with the full-aperture BP curve (Figure 5, red) indicate that the large number of miing hydrophone (40%) ha not ignificantly impacted target-finding performance. Figure 8: BTR Diplay: Full-Aperture CBF v. BP. Thi figure how that the CBF (left-panel) and BP (rght-panel) approache produce comparable reult for the full-aperture cae. The ame target are evident in both image. 475

5 B. BP in the Preence of Reduced Senor Element Reproducing the condition of Figure 6 (eliminating a ignificant number of the hydrophone) over all frequency bin of coure ha a negative impact on BTR quality. In the left panel of Figure 9, CBF proceing eem to have lot or almot lot two of the 5 clearly viible, non-endfire target. However, BP, operating on the ame exact input data, i able to maintain thee target on the diplay. Figure 11: Target-SNR curve: BP-CS v. CBF-CS. REFERENCES Figure 9: BTR Diplay: CS-Aperture CBF v. BP. The next two figure quantify the gain BP enjoy over CBF in thi reduced enor ituation. In Figure 10, color overlay indicate target bearing. A imple maximum amplitude earch over a manually pecified window (both for CBF and BP) determine thi truth data. The target truth bearing-time trajectorie in Figure 10 provide a method for etimating target SNR on the repective BTR diplay. On a per-target, per-time-lice bai, ignal i defined a the BTR amplitude in the ignal beam, and noie i defined a the median amplitude over a local window (centered at the target truth for each target). Figure 11 how the SNR for each target (correponding to the five target in Figure 10, numbered left to right) a a function of time for the entirety of the data that appear in Figure 8. With no exception, the BP-CS approach produce a higher SNR target that the CBF-CS proceing. [1] Emmanuel J. Cande and Michael B. Wakin, An Introduction to Compreive Sampling, IEEE Signal Proceing Magazine, vol. 25, no. 2, pp , March, [2] Harry L. Van Tree, Detection, Etimation, and Modulation Theory, Part IV, Optimum Array Proceing. New York: John Wiley & Son, [3] Ewout Van Den Berg and Michael P. Friedlander, Probing the Pareto Frontier for Bai Puruit Solution, Siam J. Sci. Computing, Vol. 31, No. 2: pp [4] [5] E. J. Cande, J. Romberg, and T. Tao, Stable ignal recovery from incomplete and inaccurate meaurement, Comm. Pure Appl. Math., 59 (2006), pp Figure 10: CS-Aperture BTR: With Target Overlay. 476