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Robust Constant-beamwidth Beamforming Based on

Focusing Approach for Acoustic Imaging

Juan Wang* Qing Feng* Renbiao Wu* Zhigang Su* Jinfeng Yang* Jian Lit*Tianjin Key Lab for Advanced Signal Processing, Civil Aviation University of China, Tianjin 300300, P.R.China

Corresponding author: Renbiao Wu, E-mail: rbwu@cauc.edu.cntUniversity of Florida, Gainesville, FL, USA

Abstract-Focused beamformer is an efficient broadbandbeamforming method due to the advantages like ability tooperation in multipath environments and computationalefficiency. However, the robustness of focused beamformer isnot enough. To overcome the shortcoming, a novel robustbroadband beamforming method based on focusing approach ispresented in this paper to achieve constant beamwidth foracoustic imaging. The proposed method mainly combines theidea of Robust Capon beamforming (RCB) in the focusingapproach to get the more exact focusing transformation matrixand then reduce the focused data error. Thus it can greatlyimprove the robustness performance of focused beamformer.Simulation results show the feasibility of the proposed method

I. INTRODUCTION

Wideband array signal processing is applied in many fieldssuch as underwater acoustics, acoustic imaging,communications, and speech acquisition. The aboveenvironments require the array can receive the signals withoutdistortion over the entire bandwidth. Especially in theapplication of acoustic imaging, the beamformer ought toachieve consistent sound pressure level (SPL) estimationwhich means that for an acoustic wideband monopole sourcethe acoustic image of each frequency bin stays the same.However, the beamwidth of a beamformer have relations withthe signal frequency, the number of the array elements and theinterelement spacing. So, for a fixed array, the beamwidth isdifferent for different frequency and it decreases as thefrequency increases. This beamwidth variation as a functionof frequency will subject the signals incident on the outerportions of the main beam to lowpass filtering and lead todistorted signal spectra or inaccurate SOI (the signal ofinterest) power estimation. Hence, the design of thewideband constant-beamwidth beamformer, whosebeamwidth remains approximately constant over all thefrequency band of interest, is desirable.

In the past years, many approaches have been proposed forconstant-beamwidth beamformers, which are mainly based onthe the following three kinds of methods. Linear combinationof subarrays [1]: this kind of methods designs J subarrayswith similar structure for the corresponding frequency, andthe outputs in the other frequencies are the linearcombinations of the J subarrays by different weights.Unfortunately, these methods need too many sensors and arenot suitable for wide continuous frequency band. The second

kind of methods [2] [3] only use one array and all the sensorswork at the same time. They mainly weight per sensor withdifferent set of coefficients for different frequency bin toobtain the similar beamwidth. But most of them depend onsome mathematical technique and have relatively largecomputation complexity. The third kind methods are basedon wideband focusing approach which can be called focusedbeamformer [4]. These methods transform the signal spacesat different frequencies to the common one. Compared withthe two kind of methods mentioned above, they all have someadvantages such as requiring only one array, computationalefficiency and the ability to operate in multipath environments.However, the approaches have a higher requirement of theknowledge of signal arrival angles, or else the performance ofthis method will degrade badly.To improve the robustness of focused beamformer, a novel

robust constant-beamwidth beamformer which combines theidea ofRCB with focusing approach is proposed in this paper.Here, RCB not only plays the part of a good robustbeamformer but also effectively alleviates the estimationdistortion by modifying the transformation matrix in thefocusing approach when the steering vector error exists. Wecarry out the experiments in the similar circumstances asReference [5]. By comparison, our method proposed here,needs only one set of array and lower computation load due tosingle frequency weights. The simulation results demonstratethe efficiency of the proposed method.

The remainder of this paper is organized as follows.Problem formulation is given in Section 2. Our robustconstant-beamwidth beamforming method is proposed inSection 3. Section 4 includes the simulated and experimentalexamples. Finally, Section 5 contains the conclusions.

II. PROBLEM FOMULATION

We focus herein on forming acoustic images using amicrophone array [5], which are obtained by determining thesound pressure estimates corresponding to the sphericalcoordinate system of a grid of locations. The signal at eachgrid location of interest is referred as the SOI.Next we introduce a wideband data model. Assume that a

wideband SOI impinges on an array with M elements whosecentre is located at the origin point of the coordinate system.The unite location vector of the SOI can be denoted by

0-7803-9740-X/06/$20.00 C 2006 IEEE F IA-2 ISCIT 200(6

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r = (sinOcosqv,sinOsinqv,cos 0), (1)

where A=0,[Q;z/2] is the angle of pitch, yec [Q2n2] is the azimuthangle. The propagation time delay between the source andthe m-th sensor is

m =rPm/C. (2)

Here C is sound speed, Pm = (p cOsg,8,, p sinx,8m O) is theunite location vector of the m-th sensor, where p is thediameter of the array circle and,/m = 2zrm /M is the anglebetween the m-th sensor and the positive horizontal axis. Letthe incident signal be denoted by s(t) , then the receivedsignal at the m-th sensor can be written as

Ym(t) =s(t-rm) + em (t), (3)

where em (t) is the additive white noise.We divide the output of each sensor into N non-

overlapping blocks with each block consisting of L samples.Apply an L-point FFT to each block to obtain L narrowbandfrequency bins. The data vector y(fk), for the k-th frequencybin of all the sensors can be written as

y(fk ) =ak (6R,0 O)S(fk ) + e(fk ), k=,**L1 (4)

where a k (60,V0) = [e j27T ef 2 fT2 ... e 27 ]T is the SOIsteering vector which depends on the k-th frequency bin andthe arrival angle(0 ,W0), and (.)T denotes the transpose,

and Y (ffk ) [Y1 (fk Y2(fk) YM (fk) ]Te(fk) = [el(fk) e2(fk) ..eM(fk) ]T

The covariance matrix for the k-th frequency bin can bedenoted by

Rk = E[y(fk)yH (fk)], k = 0,-* L-1 (5)

where E[.] is the expectation operator, and (.)H denotes theconjugate transpose.

In aeroacoustic measurement using arrays, the soundpressure response is normally shown. The intensity of thesound pressure response is measured on a logarithmic scale[6], which at k-th frequency bin is defined by

SPL = 20 log10 (Prms / Prej )dB (6)

where P s s ( ) /L2 denotes the root-mean-squared

pressure in Pa and Pref stands for the reference pressure

which is 20 Pa for air. Thus, we obtain the SPL estimationdependent on the power estimation for every frequency bin,

add them together over the entire analysis bandwidth and thencomplete acoustic imaging.

III. THE NOVEL METHOD FOR ACOUSTIC IMAGING

A. The Focusing Approach for Broadband Constant-Beamwidth Beamforming

The broadband constant-beamwidth beamforming based onfocusing approach generally involves a preprocessor thatfocuses the signal spaces at different frequencies to a commonone and a narrowband beamformer following the preprocessor.

Different focusing approaches mainly differ in the focusingtransformation matrices which in principle satisfy

Tk(6, )ak(6, y) aO (6,y), (7)

where Tk(0, w) is the transformation matrix for the fkfrequency bin according to the arrival angle (0, l).The rotate signal-subspace (RSS) [7] method is one of the

focusing approaches which will be used in this paper. It canbe formulated as

mi1 aO (0, t) -Tk (0, )ak (0, q)IIFTk ( t6T=)subject to Tk (0, y)Tk (0,Vq)=l

where the subscript |||F represents Frobenius matrix norm, Iis an identity matrix, Tk (0, ) can be given byTk (0, t) = V(fk)UH(fk ) ,where the columns of V(fk) andU(fk)are respectively the left and right singular vectors of

ak (O, )all(O, )-

We can see that the essential of the RSS method is totransform frequency dependent direction vectors into singlefrequency without altering the signal content. Consequently,the signal data vector at the frequency bins fk is written as

Y(fk) =Tk (O, V)Y(fk)

Tk(O, V)ak (do IV)s(fk)+Tk(O,V )e(fk) (9)

Note that the transformation matrix depends on thedirection of the SOI arrival (DOA) and the focused datadepend on the transformation matrix. Therefore, we can

obtain the desired Tk(0,Vf) and signal data providing thatthere are no steering vector errors. Then the costant-beamwith beamformer can be obtained by combining RSSmethod with any narrowband beamforming such as

conventional delay-and-sum (DAS) method. However, inpractice the knowledge of the SOI direction can not beestimated precisely due to differences between the assumedsignal arrival angle and the true arrival angle or between theassumed array response and the true array response (arraycalibration errors). Whenever this happens, the inaccurate

(8)

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transformation matrix brings on the error of the focused signaldata. In this situation, any narrowband beamformer followingthe focusing preprocessor will all cause the array outputdistorted. As a result, the robustness performance of focusingbeamformer is drastically reduced. To overcome thisshortcoming, we present a constant beamwidth beamformingmethod combining the idea of RCB with focusing approachRSS for example in the next section.

B. The Robust Constant-beamwidth Beamforming Based onFocusing Approach

The RCB method is first proposed in Reference [8] as arobust narrowband beamforming. It is turned out to belong tothe extended class of diagonal loading approach, andfortunately, the corresponding value of diagonal loading canbe calculated precisely based on the uncertainty of thesteering vector. In a word, the RCB algorithm improves theperformance of the beamformer by getting the estimation ofthe steering vector most close to the real one. Here theproposed method uses the RCB as narrowband beamformerfollowing the focusing preprocessor. In addition, theemphasis is that it applies the result of RCB to the focusingpreprocessor. That's why our method can improve therobustness of the focused beamformer. In this section, wewill describe the proposed method in detail. We refer to thismethod as robust constant-beamwidth RCB (R-CBRCB).

The RCB at the reference frequency f0 bin can beformulated as

For the beamformning at the frequency fk, we focus thesignal data to the reference frequency bin with thetransformation matrix i (9, V) which is obtained by thecorrected steering vector OandAk

Tk(,Vf) =V(fk)U (fk) (13)

So the data vector is transformed to the referencefrequencyfJ like (12). And in the following narrowbandbeamformer, let the weight vector w k at the frequencyfk bethe same as w0 from RCB, i.e.

Wk =Wo (14)The above is the description of the proposed robust

constant beamwidth beamforming R-CBRCB. Then weanalyze its beamwidth and powerwidth.The beampatterns for the frequency bins J0 and fk is

written as

(15)Bo (O, f) = |wHOaO(0a f)O

Bk (O, W) = WJO,Vf)ak (O,,)

Because of

w HT(O,y,)ak (O, W) = |W Oao (0, v)I

(16)

max ao) subject to Ro - j2a0aoa 2-aO

|a_-aO| < c2~o _

where Ro and ao are respectively the covariance matrixsteering vector for the frequency with center at f0, ao i;assumed steering vector, and £ is a user parameterregard to the error of steering vector. The robust Cibeamforming gives the array weights vector w0 forreference frequency bin Jo

w = 0oaa HRH0 0 0'

(17)

(10) when ik(6, Vf) and ak(0,Vf) vary to cover each direction ofinterest, we can achieve constant beamwidth beamforming.

In the same way, the power estimations for the fo and fkand frequency bins can be expressed respectively as

s thewith 60 =w0 RoWoaponthe = &02wOa (0, tV)aH(a tw)w0 +o'Wo[W0,

07k = WoR k W

= WoTk(O,q)RkTk (O,q/)WO= Uwo aO(0,V)aOH (0,)wo + c3nwo wO,

(1 1)

where a is the calibrated steering vector which can beobtained by

ao = ao - (I +AR)l a', (12)

where A is the diagonal loading factor.Due to the fact that the arrival angle of different

frequencies of the wideband SOI is same, the steering vectorsof different frequencies only differ in the frequency. Thesteering vector ak for k-th frequency bin fk can be obtainedwhen a0for the f0 frequency bin is gained by (12).

(18)

(19)

where U0 and Uk are respectively the real power of the signalat f,and fk' and Rk =E[y(fk)yH(fk)]. In this way, thepower estimation for each frequency bin can be obtained. It iseasily to understand from (18) and (19) when w0 varies tocover each direction of interest, the powerpattern is similar forall the frequency bins. So the constant beamwidth andpowerwidth are achieved, and the acoustic imaging is fulfilled.

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RCB can be readily combined with the focusing approachRSS to obtain a robust constant beamwidth for the desiredfrequency band, as explained above. Note in R-CBRCB, theRCB makes important contribution to focusing preprocessor

in the focused beamformer. It modifies the arrival angleinformation of the SOI, yields the more accurate focused datafrom the corrected transformation matrix, and then mitigatesthe estimation distortion of focused beamformer. Thus therobustness of the broadband beamformer is improved greatly.Moreover, R-CBRCB needs low computation load and lessarray sensors as well as inherits the strength of RCB in therobustness against finite sample size problems, highresolution, and excellent interference suppression capability.

IV. SIMULATION RESULTS

In the simulated examples below, we consider an array [9]which consists of 17 microphones arranged in two circles ofeight microphones each with the diameter respectively 3.89inches and 3.89/2 inches and one microphone at the array

center. A wideband monopole source with flat spectrum islocated at (0,0,60) in the array coordinate system and a

spatially white Gaussian noise with SNR equal to 20dB andthe SPL equal to 20dB for each frequency bin. We use an

(a)RCB

4..5

4

8192-point FFT on the non-overlapping blocks of simulateddata. The sampling rate is 142.857 kHz.

A. Comparison ofthe Signal Spectrum Results Obtained viaDifferent Methods

In this simulated example, we have a look at R-CBRCBfrom the point of view of output signal spectrum. The SOI iswith flat spectrum from 10 khz to 20 khz and the vector error

£=0.75 Without focusing approach, the output spectrumwith the frequency from the beamformer such as RCB isshown in Fig.1 (a). It can be seen that there is serious energy

loss when the frequency increase. Fig. 1 (b) shows the resultsfrom the conventional beamformer DAS following thefocusing approach RSS. It is obvious that the outputspectrum has big estimation error and is not steady. Fig. 1 (c)shows the results from the RCB following the focusingapproach RSS in which RCB is not applied to the focusingpreprocessor. The output of beamformer can achieveapproximately consistent. However, relatively big fluctuationof output spectrum exists due to the unrobustness offocusing preprocessor. By comparison, our R-CBRCB ismore robust and can achieve good estimation without energy

loss in high frequency. Fig. 1 (d) shows the efficiency.

(b) DAS following RSS

x 1Q

HI

0 0- 1 1-R 2 2-CFrMequency(Hiz) x,1 o4

(d) R-CBRCB

14

.2

(c) RCB following RSS

Fig. 1. Comparison of the signal spectrum results obtained via different methods with the vector error £=0.75

l1

A

1 1111 I 1. ..l II, . 111,11

825

B. Comparison ofthe Acoustic Imaging Results Obtainedvia Different Methods

Fig.2 compares the acoustic imaging results from threekinds of broadband beamforming method for the narrowbandfrequency bins at 10 kHz, 20 kHz and 40 kHz, with the vectorerror,c=0.75, where 10 kHz is viewed as reference frequency.The wideband source is with flat spectrum from 10 khz to50khz. Fig.2 (a) (b) (c) show the results from theconventional beamformer DAS following the focusingapproach RSS. It approximately maintains the same SPLestimations across the frequency bins. However, because thesteering error exists, the estimation images of 20 kHz and 40kHz have severe distortion. Furthermore, this method alsohas poor resolution and much higher sidelobes. Fig.2 (d) (e)(f) show the results from the RCB following the focusing

approach RSS. RCB is not applied to the focusingpreprocessor and only as a narrowband beamformer. So thiskind of method has much better resolution, but the estimationimages still have severe distortion as a result of the unrobustfocusing approach. Fig.2 (g) (h) (i) show our method R-CBRCB. The distortion of estimation images is effectivelymitigated as shown. It is due to the matter that R-CBRCBmodifies the transformation matrix of the focusing approachby the calibrated steering vector derived from RCB. Inaddition, R-CBRCB also has much better resolution andlower sidelobe and hence better interference rejectioncapability. Consequently, the consistent SPL estimation ofacoustic images is achieved across the frequency bins. It isobvious that R-CBRCB significantly outperforms the othermethods.

(a) DAS following RSS with fo=10 kHz

(d) RCB following RSS with Jo =10 kHz

(g) R-CBRCB with fo =OkHz

(b) DAS following RSS with f, =20 kHz

(e) RCB following RSS with fJ =20 kHz

(h) R-CBRCB with f = 20 kHz

(c) DAS following RSS with f=40 kHz

(f) RCB following RSS with f2 = 40 kHz

(i) R-CBRCB with f2 = 40 kHz

Fig. 2. Comparison of the acoustic imaging results obtained via different constant beamwidth beamforming algorithms for the frequency bins at fo=O kHz,

f1 =20 kHz, f2 =40 kHz with the vector error £=0.75

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C. Comparison ofthe Powerwidths for Different MethodsIn acoustic imaging application, the goal is to consistently

estimate the SPL which depends on the SOI power, so we

compare the 3-dB powerwidths of different methods as

function of the frequency. Note that the powerwidths of theRCB decrease drastically as the frequency increases. Themethods of conventional DAS and RCB following RSS can

achieve approximately constant powerwidths, but thepowerwidths have relatively big fluctuations. However, our

R-CBRCB not only can achieve constant powerwidths butalso has much smaller powerwidths and good steadypowerwidths level as can be seen from the figure 3.

the arrival angle information of the SOI and yields the more

accurate focused data from the corrected focusing matrix forthe focusing preprocessor. Thus the proposed method can

greatly improve the performance of focused beamformingwith the advantages such as fewer array sensors,

computational efficiency, much higher resolution, andexcellent interference suppression capability.

ACKNOWLEDGMENT

This work is supported in part by the National NaturalScience Foundation of China under grant 60325102 and60428101.

REFERENCES

6

1 115 2 2-5 3 3-5 4

FrequencyrHz) X 1 4

Fig. 3. Comparison of the powerwidths for different methods with thevector error 1

V. CONCLUSIONS

A robust broadband constant-beamwidth beamformingbased on focusing approach for acoustic imaging is proposedin this paper. It improves the robustness of focusedbeamformer by combining the idea of RCB. Here, RCB notonly plays the part of a good beamformer but also modifies

[1] R. Smith, "Constant beamwidth receiving arrays for broad bandsonar systems," Acoustica, 1970, 23.

[2] Ward. D. B, Kennedy R A, Williamson R C, "Theory anddesign of broadband sensor arrays with frequency invariant far-field beam patterns," J. Acoust. Soc. Amer, 1995, 97(2).

[3] Yinxin Yang, Chao Sun, "A new method of broadband constantbeamwidth beamforming for arbitrary geometry arrays," ActaAcoustica 2001, 26.

[4] S. Simanapalli, M. Kaveh, "Broadband focusing for partiallyadaptive beamforming," Aerospace and Electronic Systems,IEEE Transactions on Volume 30, Issue 1, Jan. 1994Page(s):68 - 80

[5] Z. Wang, J. Li, P. Stoica, "Constant-beamwidth and constant-powerwidth wideband robust Capon beamformers for acousticimaging," Journal of the Acoustical Society of America,2004.

[6] D.T. Blackstock, "Fundamentals of physical acoustics," JohnWiley Sons, New York, NY, 2000.

[7] Yongliang Wang. Hui Chen, Yingning Peng, "The theory andarithmetic of space spectrum estimation," Published by theUniversity of Springer, 2004.

[8] P. Stoica, Z. Wang, and J. Li, "Robust Capon beamforming,"IEEE Signal Processing Letters, 2003.

[9] W. M. Humphreys, "Design and use of microphone directionalarrays for aeroacoustic measurement", AIAA Paper98-0417, 36st Aerospace Science Meeting and Exhibit,Reno,NV,January 12-15, 1998.

f ~ ~ ~ ~~~ rgR-R- B- DASfflowing RSSRRBCR1CB5fblowing R4SS

S~~~~~~~~& X 'CBRCB

.~~~