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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 10, OCTOBER 2003 2511

Design of Broadband Beamformers Robust AgainstGain and Phase Errors in the Microphone

Array CharacteristicsSimon Doclo, Student Member, IEEE,and Marc Moonen, Member, IEEE

Abstract—Fixed broadband beamformers using small-size mi-crophone arrays are known to be highly sensitive to errors in themicrophone array characteristics. This paper describes two designprocedures for designing broadband beamformers with an arbi-trary spatial directivity pattern, which are robust against gain andphase errors in the microphone array characteristics. The first de-sign procedure optimizes the mean performance of the broadbandbeamformer and requires knowledge of the gain and the phaseprobability density functions, whereas the second design proce-dure optimizes the worst-case performance by using a minimaxcriterion. Simulations with a small-size microphone array show theperformance improvement that can be obtained by using a robustbroadband beamformer design procedure.

Index Terms—Broadband beamformer, microphone character-istics, minimax, probability density function, robustness.

I. INTRODUCTION

I N MANY speech communication applications, such ashands-free mobile telephony, hearing aids, and voice-con-

trolled systems, the recorded microphone signals are corruptedby acoustic background noise and reverberation [1]–[3]. Back-ground noise and reverberation cause a signal degradation,which can lead to total unintelligibility of the speech andwhich decreases the performance of speech recognition andspeech coding systems. Therefore, efficient signal enhancementalgorithms are required.

Well-known multimicrophone signal enhancement tech-niques are fixed and adaptive beamforming [4]. Adaptivebeamforming techniques, such as the generalized sidelobecanceller (GSC) and its variants [5]–[8], generally have a better

Manuscript received October 1, 2002; revised March 12, 2003. This workwas carried out at the ESAT laboratory of the Katholieke Universiteit Leuvenand was supported in part by the F.W.O. Research Project G.0233.01 (Signalprocessing and automatic patient fitting for advanced auditory prostheses),the I.W.T. Project 020540 (Performance improvement of cochlear implants byinnovative speech processing algorithms), the I.W.T. Project 020476 [SoundManagement System for Public Address Systems (SMS4PA)], the ConcertedResearch Action Mathematical Engineering Techniques for Information andCommunication Systems (GOA-MEFISTO-666) of the Flemish Government,the Interuniversity Attraction Pole IUAP P5-22 (2002–2007), DynamicalSystems and Control: Computation, Identification and Modeling, initiatedby the Belgian State, Prime Minister’s Office—Federal Office for Scientific,Technical, and Cultural Affairs, and in part by Cochlear. The associate editorcoordinating the review of this paper and approving it for publication wasProf. Xiaodong Wang.

The authors are with the Katholieke Universiteit Leuven, Department ofElectrical Engineering (ESAT - SISTA), B-3001 Heverlee, Belgium (e-mail:[email protected]; [email protected]).

Digital Object Identifier 10.1109/TSP.2003.816885

noise reduction performance than fixed beamforming tech-niques and are able to adapt to changing acoustic environments.However, fixed beamforming techniques (with a fixed spatialdirectivity pattern) are sometimes preferred because they do notrequire a control algorithm in order to avoid signal distortionand signal cancellation and because of their easy implemen-tation and low computational complexity. Fixed beamformersare frequently used for creating the speech and noise referencesignal in a GSC, for creating multiple beams [9], [10], inapplications where the position of the speech source is assumedto be (approximately) known, such as hearing aid applications[11]–[13], and in highly reverberant acoustic environments.

In this paper, we are interested in designingrobust broadbandbeamformerswith a givenarbitrary spatial directivity patternfor a givenarbitrary microphone array configuration, using anFIR filter-and-sumstructure. Using traditional types of fixedbeamformers, such as delay-and-sum beamforming, differentialmicrophone arrays [14], superdirective microphone arrays [12],[15], [16], and frequency-invariant beamforming [17], it is gen-erally not possible to design arbitrary spatial directivity patternsfor arbitrary microphone array configurations. However, in [18]and [19], several procedures are described for designing broad-band beamformers with an arbitrary spatial directivity pattern.The design consists of calculating the filter coefficients such thatthe spatial directivity pattern optimally fits the desired spatialdirectivity pattern with respect to some cost function. Differenttechniques can be used, e.g., weighted least-squares filter de-sign, nonlinear optimization techniques [20]–[23], a maximumenergy array [24] or eigenfilters [19], [25]. Many such broad-band beamformer design procedures perform the design individ-ually for separate frequencies and/or approximate the (double)integrals that arise in the design by a finite sum over a grid offrequencies and angles. In this paper, we will calculate full inte-grals over the frequency-angle plane and, hence, perform a truebroadband design.

It is well known that fixed and adaptive beamformers arehighly sensitive to errors in the microphone array characteris-tics(gain, phase, microphone position), especially for small-sizemicrophone arrays. In many applications, the microphone arraycharacteristics are not exactly known and can even change overtime [26]. For superdirective beamformers, robustness againstrandom errors can be improved by limiting the white noise gain(WNG) of the beamformer, i.e., imposing a norm constraint or ageneral quadratic constraint on the filter coefficients [12], [15],[16], [27]. Limiting the WNG has also been applied in order toenhance the robustness of adaptive beamformers [28]. Another

1053-587X/03$17.00 © 2003 IEEE

2512 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 51, NO. 10, OCTOBER 2003

Fig. 1. Linear microphone array configuration (far-field assumption).

possibility is to perform a measurement or a calibration proce-dure for the used microphone array, which will, however, onlylimit the error sensitivity for the specific microphone array used[29], [30].

This paper discusses the design of broadband beamformerswith an arbitrary spatial directivity pattern, which are robustagainst unknown gain and phase errors in the microphonearray characteristics. In Section II, the far-field broadbandbeamforming problem is introduced, and some definitionsand notational conventions are given. Section III discussesseveral cost functions that can be used for designing broadbandbeamformers: the weighted least-squares cost function, theeigenfilter cost function based on a total least-squares errorcriterion, and a nonlinear cost function. For all consideredcost functions, we first discuss the general design procedurefor an arbitrary spatial directivity pattern and for frequency-and angle-dependent microphone characteristics. Next, themicrophone characteristics are assumed to be independentof frequency and angle, and we focus on the specific designcase of a broadband beamformer having a passband and astopband region. Using the considered cost functions, it ispossible to design broadband beamformers when the micro-phone characteristics are exactly known. However, in manyapplications, the microphone characteristics are not known andcan even change over time. Section IV describes two proce-dures for designing broadband beamformers that are robustagainst (unknown) gain and phase errors in the microphonearray characteristics. The first design procedure optimizesthe mean performance of the broadband beamformer for allfeasible microphone characteristics, whereas the second designprocedure optimizes the worst-case performance. Both designprocedures can be used with the discussed—and other—costfunctions. In Section V, simulation results for the differentdesign procedures and cost functions are presented. It is shownthat robust broadband beamformer design leads to a significantperformance improvement when gain and phase errors occur.

II. FAR-FIELD BROADBAND BEAMFORMING: CONFIGURATION

Consider the linear microphone array depicted in Fig. 1, withmicrophones and as the distance between theth micro-

phone and the center of the microphone array. Thespatial di-rectivity pattern for a source with normalizedfrequency at an angle from the array is defined as

(1)

where is the received signal at the center of the mi-crophone array, and is the frequency response of thereal-valued -dimensional FIR filter

(2)

where

......

(3)

When the signal source is far enough from the microphone array,the far-field assumptions are valid [31], i.e., the wavefronts canbe assumed to be planar, and all microphone signals can be as-sumed to be equally attenuated.1 Since the microphones are notnecessarily omni-directional microphones with a flat frequencyresponse, the microphone characteristics have to be taken intoaccount. The microphone characteristics of theth microphoneare described by the function

(4)

1Since we consider small-size microphone arrays in this paper, the far-fieldassumption will generally be valid. However, all expressions can be easily ex-tended to the near-field case [18], [19].

DOCLO AND MOONEN: DESIGN OF BROADBAND BEAMFORMERS ROBUST AGAINST GAIN AND PHASE ERRORS 2513

where both the gain and the phase canbe frequency-and angle-dependent. The microphone signals

, phase-shifted and filtered versionsof the signal

(5)

with the delay in number of samples equal to

(6)

where is the speed of sound propagation [340 (m/s)], andis the sampling frequency. Combining (1) and (5), thespatial

directivity pattern can be written as

(7)with the real-valued -dimensional filter vector , with

, and the steering vector equal to

...

...(8)

The steering vector can be written as

(9)

where is an -dimensional diagonal matrix con-sisting of the microphone characteristics, and hence,is the steering vector assuming omni-directional microphoneswith a flat frequency response equal to 1, i.e., ,

...

...(10)

where is the -dimensional identity matrix. Theth el-ement of is equal to

(11)

where mod and , wheredenotes the largest integer smaller than or equal to

, and mod is the remainder of the division. Thesteering vector can be decomposed into a real and animaginary part . The real part

is equal to

(12)

where and are the real and the imaginaryparts of , and and are the real andthe imaginary parts of .

Using (7), thespatial directivity spectrum canbe written as

(13)

where , which can be written,using (9), as

(14)

where . The th element ofis equal to

(16)

where mod , mod ,, and . The matrix

can be decomposed into a real and an imaginary part. Since the imaginary part

is anti-symmetric, i.e., , thespatial directivity spectrum is equal to

(17)

The real part is equal to

(18)

where and are the real and the imaginaryparts of .

III. B ROADBAND BEAMFORMING COST FUNCTIONS

In this section, we discuss the design of broadband beam-formers when the microphone characteristics are ex-actly known. The design of a broadband beamformer consistsof calculating the filter , such that optimally fits thedesired spatial directivity pattern , where is anarbitrary two-dimensional (2-D) function in and . Severaldesign procedures exist, depending on the specific cost functionwhich is optimized. In this section, three different cost functionsare considered:


• a weighted least-squares (LS) cost function , mini-mizing the weighted LS error between the spatial direc-tivity pattern and the desired spatial directivitypattern [this cost function can be written as aquadratic function (cf. Section III-A)];

• the total least-squares (TLS) eigenfilter cost function, minimizing the TLS error between the spatial

directivity pattern and the desired spatial di-rectivity pattern [this cost function leads to ageneralized eigenvalue problem (cf. Section III-B)];

• a nonlinear cost function , minimizing the errorbetween the amplitudes of the spatial directivity pat-tern and the desired spatial directivity pattern

, not taking into account the phase of the spatialdirectivity patterns [minimizing this cost function leadsto a nonlinear optimization problem (cf. Section III-C)].

Obviously, it is also possible to use various other cost functions,which are, e.g., based on the “conventional” eigenfilter [19],[25], a maximum energy array [24], or a (nonlinear) minimaxcriterion [21]–[23].

We will consider the design of broadband beamformers overthe total frequency-angle plane of interest, i.e., we will not splitup the fullband problem into separate smallband problems fordistinct frequencies. Furthermore, we will not approximate thedouble integrals that arise in the design by a finite sum over agrid of frequencies and angles, as, e.g., has been done in [20] forthe nonlinear cost function . In [19], the three consideredcost functions have been discussed in more detail for omni-di-rectional microphones with a flat frequency response. Althoughthe nonlinear cost function leads to the best performance, thecomputational complexity for computing the filter coefficientscan be quite large, since an iterative optimization procedure isrequired. In [19], it has been shown that the TLS eigenfilter de-sign procedure is the preferred noniterative design procedure,since it leads to a better performance than the weighted least-squares, the “conventional” eigenfilter and the maximum energyarray design procedures.

For all considered cost functions, we will first discuss thegen-eral design procedurefor an arbitrary desired spatial directivitypattern and for frequency- and angle-dependent micro-phone characteristics . Next, the microphone charac-teristics will be assumed to be independent of frequency andangle, i.e., omni-directional, frequency-flat microphones. Evenif this assumption is not exactly satisfied in practice, it is gen-erally possible to split up the complete considered frequency-angle region into several smaller frequency-angle regions wherethis assumption holds. We will then focus on thespecific de-sign caseof a broadband beamformer having a desired response

in the stopband region andin the passband region . For the specific design case,we will consider a weighting function in the pass-band and in the stopband.

A. Weighted Least-Squares (LS) Cost Function

1) General Design Procedure:Considering the least-squares (LS) error , the weighted LS

cost function (e.g., used in [32] for the design of FIR filters) isdefined as

(19)

where both the phase and the amplitude of are takeninto account. is a positive real weighting function, as-signing more or less importance to certain frequencies or angles.This cost function can be written as

Re (20)

Using (17) and the fact that

Re

(21)

this cost function can be rewritten as the quadratic function

(22)

where

(23)

(24)

(25)

The filter , minimizing the weighted LS cost function, is given by

(26)

2) Omni-Directional, Frequency-Flat Microphones:Whenthe microphone characteristics are independent of frequencyand angle, the diagonal matrices containing the microphonecharacteristics are and .Using (12) and (18), the vectorand the matrix are nowequal to [assuming to be real-valued]

(27)

where

(28)


(29)

The th element of and the th element of are equalto

(30)

(31)

where , and .3) Specific Design Case:For the specific design case where

, in the passband and ,in the stopband, (28)–(31) become

(32)

(33)

(34)

The th element of and the th element of , i.e., or, can be computed as

(35)

(36)

where mod , mod , ,and . Similarly, the th element of and the

th element of can be computed by replacing within the integrals of (35) and (36). All these integrals can

be considered to be special cases of the integral

(37)

of which the computation is discussed in Appendix A.

B. TLS Eigenfilter Cost Function

Eigenfilters have been introduced for designing 1-Dlinear-phase FIR filters [33] and 2-D FIR filters [34], [35].In [19] and [25], two noniterative broadband beamformerdesign procedures based on eigenfilters have been discussed.The “conventional” eigenfilter technique minimizes theerror between the spatial directivity patterns and

and requires a reference fre-quency-angle point . The TLS eigenfilter minimizes thetotal least-squares (TLS) error between the spatial directivitypattern and the desired spatial directivity pattern

and does not require a reference point. In [19], ithas been shown that the performance of the TLS eigenfilteralways exceeds the performance of the weighted LS and the“conventional” eigenfilter cost functions and therefore is thepreferred noniterative design procedure.

The TLS eigenfilter cost function is defined as

(38)which can be written as

(39)where is defined as

(40)

For computing the TLS error, the expression is usedin the denominator of (39) instead of the usual expressionsince represents the total area under the spatial direc-tivity spectrum . The TLS eigenfilter cost function in(39) can be rewritten as the Rayleigh-quotient

(41)

where

(42)

with , , and defined in Section III-A. The filterminimizing in (41) is the generalized eigenvectorof and , corresponding to the smallest generalizedeigenvalue. After scaling the last element of to 1, theactual solution is obtained as the first elements of

.In the case of omni-directional, frequency-flat microphones,

and for the specific design case, we can use similar expressionsas derived in Sections III-A2 and 3.


C. Nonlinear Criterion

1) General Design Procedure:Instead of minimizingthe LS error or the TLS error,one can also minimize the error between the amplitudes

because in general, the phase of the spa-tial directivity pattern is not relevant. This problem formulationleads to the cost function

(43)and gives rise to a nonlinear optimization problem, which hasto be solved using iterative optimization techniques. Theseiterative optimization techniques generally involve severalevaluations of in each iteration step. A complexityproblem now arises because the filter coefficients cannot be extracted from the double integral because of thesquare root in the term

[19]. Hence, for everyintermediate , the double integrals have to be recomputednumerically, which is a computationally very demandingprocedure. However, by slightly changing the nonlinear costfunction in (43), it is possible to overcome this computationalproblem: Instead of minimizing the error between the ampli-tudes and , it is also feasible to minimizethe error between the square of the amplitudes and

and define the cost function

(44)

which is again independent of the phase of the spatial directivitypatterns. Using (13) and (17), the cost function can bewritten as

(46)

with

(47)

(48)

(49)

The cost function can be minimized using iterative op-timization techniques, which are discussed in Section III-C3. Aswill be shown in Section III-C2, the filter coefficients can beextracted from the double integral in (47), such that these doubleintegrals only need to be computed once.

2) Omni-Directional, Frequency-Flat Microphones:Whenthe microphone characteristics are independent of frequencyand angle, the matrix can be computed similarly asin (27) as

(50)

where

(51)

(52)

Using (16), the expression , arising in the computa-tion of can be written as

(53)

(54)

(55)

where

mod mod

mod mod

(56)

Since is real, only the real part of the exponentialfunction in (55) has to be considered, i.e.,

(57)

Hence, can be written as

(58)


where

(59)

(60)

The double integrals in (59) and (60) only need to be computedonce (since and are independent of ). Therefore,the function , and, hence, also the total cost function

, can be evaluated without having to calculate doubleintegrals for every . This result also remains true when themicrophone characteristics are frequency- and angle-dependent.

3) Nonlinear Optimization Technique:Minimizingrequires an iterative nonlinear optimization technique, for whichwe have used both a medium-scale quasi-Newton method withcubic polynomial line search and a large-scale subspace trustregion method [36], [37]. In order to improve the numerical ro-bustness and the convergence speed, both the gradient and theHessian

(61)

(62)

can be supplied analytically. In [18] and [19], it has been shownthat can be calculated as

(63)

with the th element of equal to

(64)

and the th element of equal to

(65)Hence, stationary points , i.e., filter coefficients for whichthe gradient is 0, satisfy

(66)

In addition, it can be shown that the quadratic formin a stationary point is equal to

(67)

Since, in general, the matrix , defined in (49), is positive-definite, the quadratic form is strictly positivein all stationary points, except for , where it is equal tozero. Therefore,all stationary points are either local minima orsaddle points(except for , where the Hessian is nega-tive-definite, such that it is the only local maximum). Simula-tions have indicated that for each design problem, a number oflocal minima exist, which are generally related to the symmetry

present in the considered problem. However, the cost functionin all local minima seems to be approximately equal, such thatany of these local minima can be used as the final solution forthe broadband beamformer.

4) Specific Design Case:For the specific design case con-sidered in Section III-A3, the matrices and and thescalars , and are equal to

(68)

(69)

(70)

where , , and have been defined in Section III-A.The integrals in (69) and (70) can be considered to be specialcases of the integral (37), of which the computation is discussedin Appendix A.

IV. ROBUST BROADBAND BEAMFORMING

Using the cost functions presented in Section III, it is pos-sible to design broadband beamformers with an arbitrary spatialdirectivity pattern , when the microphone characteris-tics are exactly known (and fixed). However, thesebeamformers are known to be highly sensitive to errors in themicrophone array characteristics (gain, phase, and microphoneposition) [15], [26], [29]. Small deviations from the assumedmicrophone array characteristics can lead to large deviationsfrom the desired spatial directivity pattern, especially whenusing small-size microphone arrays, e.g., in hearing aids andcochlear implants (cf. Section V). Since, in practice, it isdifficult to manufacture microphones having exactly the samecharacteristics, it is practically impossible to exactly knowthe microphone array characteristics without a measurementor a calibration procedure. Obviously, the cost of such acalibration procedure for every individual microphone arrayis objectionable. Moreover, after calibration, the microphonecharacteristics can still drift over time [26].

Instead of measuring or calibrating every individual micro-phone array, it is better to consider all feasible microphone char-acteristics (in this paper, we only consider gain and phase2) andto either optimize one of the following criteria.

• The mean performance, i.e., the weighted sum of thecost functions for all feasible microphone characteristics,using the probability of the microphone characteristicsas weights (cf. Section IV-A). This procedure requiresknowledge of the gain and the phase probability densityfunctions (pdf), which can, e.g., be obtained from themicrophone manufacturers. It will be shown that for gainerrors only the moments of the gain pdf are required,

2Robustness against microphone position errors can actually be considered aspecial case of robustness against phase errors since a position error correspondsto a frequency- and angle-dependent phase error [38].


whereas in general, for phase errors complete knowledgeof the phase pdf is required. We will apply this meanperformance criterion to the weighted LS and the non-linear cost function, whereas it is not straightforward toapply this criterion to the TLS eigenfilter cost function.When optimizing this mean performance criterion, it is,however, still possible that for some specific gain/phasecombination (typically with a low probability), the costfunction is quite high.

• Theworst-case performance, i.e., the maximum cost func-tion for all feasible microphone characteristics, leading toa minimax criterion (cf. Section IV-B). This is a strongercriterion, since the cost for the worst-case scenario is min-imized. We will apply this criterion to all considered costfunctions.

The same problem of gain and phase errors has been studied in[27]. However, in [27], only the narrowband case for a specificdirectivity pattern, with a uniform pdf and a LS cost function,has been considered. The approach presented here is more gen-eral in the sense that we consider broadband beamformers withan arbitrary spatial directivity pattern, arbitrary probability den-sity functions, and several cost functions.

A. Weighted Sum Using Probability Density Functions

The total cost function is defined as the weightedsum of the cost functions for all feasible microphone character-istics, using the probability of the microphone characteristics asweights, i.e.,

(71)

where is the cost function for a specificmicrophone characteristic , and is theprobability density function (pdf) of the stochastic variable

, i.e., the joint pdf of the stochastic variables(gain) and(phase), . We assume that is in-

dependent of frequency and angle or that is availablefor different frequency-angle regions, such that the problem caneasily be split up. Without loss of generality, we also assumethat all microphone characteristics aredescribed by the same pdf . Furthermore, we assume that

and are independent stochastic variables such that the jointpdf is separable, i.e., , where is thepdf of the gain , and is the pdf of the phase. For thesepdfs, the relation

(72)

holds. We will consider two cost functions from Section III: theweighted LS and the nonlinear cost function (it is not straight-forward to apply this criterion to the TLS eigenfilter cost func-tion). Remarkably, the same design procedures as for the nonro-bust design in Section III can be used, and we only require someadditional parameters, which can be easily calculated from thegain and the phase pdf.

1) Weighted LS Cost Function:The mean performanceweighted LS cost function can be written as

(73)

The cost function for a specific mi-crophone characteristic is equal to (22), i.e.,

(74)

By combining (73) and (74), the mean performance weightedLS cost function can be written as

(75)

(76)

which has the same form as (22). Using (30), theth element ofis equal to

(77)

(78)

(79)

(80)

where

(81)

such that

(82)

Using (31), the th element of is equal to

(83)


(84)

If , is equal to

(85)

where is the variance of the gain pdf, i.e.,

(86)

whereas, if , is equal to

(87)

where is the mean of the gain pdf, and

(88)

(89)

(90)

(91)

such that

(92)

The matrix can now be easily computed as

......

...

(93)where is an -dimensional matrix with all elementsequal to 1 and denoting element-wise multiplication. As canbe seen, we only need the mean and the variance of the gain pdf

, whereas in general, complete knowledge of the phase pdfis required.

Frequently used probability density functions are a uniformdistribution (with boundary values and )

(94)

and a Gaussian distribution (with mean and standard devia-tion )

(95)

For a uniform distribution, the different gain and phase param-eters are equal to

For a Gaussian distribution with meanand standard deviation, the variance is equal to , whereas the phase

parameters , , and have to be calculated numerically.2) Nonlinear Cost Function:The mean performance non-

linear cost function can be written as

(96)

The cost function for a specific mi-crophone characteristic is equal to (46), i.e.,

(97)

By combining (96) and (97), the mean performance nonlinearcost function can be written as

(98)

(99)


Similarly to (93), the matrix is equal to

......

...

(100)Using (58), can be written as

(101)

(102)

where

(103)

(104)

(105)

where and are defined in (56). The expressionin (102) has the same form as (58), such that

the same nonlinear optimization techniques as described inSection III-C3 can be used for minimizing . The calcu-lation of the parameters , and is discussedin Appendix B. For the calculation of , we only requirethe (higher order) moments of the gain pdf , whereasfor the calculation of and , in general, completeknowledge of the phase pdf is required. In Appendix B,it is also shown that for a symmetric phase pdf, ,such that

(106)

B. Minimax Criterion

For the minimax criterion, which optimizes the worst-caseperformance, we first have to define a (finite) set of microphonecharacteristics ( gain values and phase values)

(107)

as an approximation for the continuum of feasible microphonecharacteristics and use this set to construct the -di-mensional vector

...(108)

which consists of the used cost function (weighted LS, TLSeigenfilter, nonlinear, or any other cost function, e.g., definedin [19]–[23]) at each possible combination of gain and phasevalues. The goal then is to minimize the -norm of ,i.e., the maximum value of the elements

(109)

which can, e.g., be done using a sequential quadratic program-ming (SQP) method [36], [37]. In order to improve the numer-ical robustness and the convergence speed, the gradient

(110)

which is an -dimensional matrix, can be sup-plied analytically. As can easily be seen, the larger the valuesand , the denser the grid of feasible microphone character-istics, and the higher the computational complexity for solvingthe minimax problem. However, when only considering gain er-rors and using the weighted LS cost function, the number of gridpoints can be drastically reduced.

Theorem 1: When considering onlygain errors and usingtheweighted LS cost function, the maximum value of ,for any , occurs on a boundary point (of an-dimensionalhypercube), i.e., or , .This implies that suffices, and only consistsof elements. This is not necessarily the case for the TLSeigenfilter and the nonlinear cost function.

Proof: When considering only gain errors, the weightedLS cost function in (74) can be written as

(111)

where and , and

...(112)


The expression can be rewritten as

(113)

where is an -dimensional submatrix of ,i.e.,

(114)

If we substitute into , thenin (113) can be rewritten as

(115)

where is an -dimensional vector consisting of the micro-phone gains

(116)

Similarly, if we define as , where is an-dimensional subvector of , , and

, then the weighted LS cost function can bewritten as

(117)

Since is a positive-(semi)definite matrix, ,such that

(118)

and is a positive-(semi)definite matrix for every.Therefore, the weighted LS cost function is a quadraticfunction (with a single minimum), such that the maximum valueof for all points inside an -dimensional hypercube,defined by , , occurs on oneof the boundary points of the hypercube.

V. SIMULATIONS

This section discusses the simulation results of robustbroadband beamformer design for gain and phase errors in themicrophone characteristics. Since the effect of gain and phaseerrors is more profound for small-size microphone arrays, wehave performed simulations for a small-size linear nonuniformmicrophone array consisting of microphones at positions

m, corresponding to a typical configurationfor a next-generation multimicrophone BTE hearing aid. Thenominal gains and phases of the microphones are and

, . We have designed an end-fire broad-band beamformer for a sampling frequency 8 kHz withpassband specifications – Hz, 0 –60and stopband specifications – Hz,

TABLE IDIFFERENTCOST FUNCTIONS FORWEIGHTED LS, TLS EIGENFILTER, AND

NONLINEAR ROBUSTBEAMFORMERDESIGN (� = 1;N = 3; L = 20)

80 –180 , cf. Sections III-A3 and C4. For the TLS eigenfilter,the matrix is computed with frequency and angle specifi-cations – Hz, 0 –180 , cf. Section III-B.The used filter length , and the stopband weight .

We have designed several types of beamformers using theweighted LS cost function and the nonlinear criterion:

1) a nonrobust broadband beamformer (not taking into ac-count errors, i.e., assuming , );

2) a robust broadband beamformer using a uniform gain pdf( , );

3) a robust broadband beamformer using a uniform phasepdf ( , );

4) a robust broadband beamformer using a uniformgain/phase pdf ( , , ,

);5) a robust broadband beamformer using the minimax crite-

rion (only gain errors are taken into account, ,, ).

Using the TLS eigenfilter cost function, we have designed a non-robust beamformer and a robust beamformer using the minimaxcriterion. For all beamformer designs, we have computed thefollowing cost functions:

1) the cost function without phase and gain errors ( ,);

2) the mean cost function for the uniform gain pdf;3) the mean cost function for the uniform phase pdf;4) the mean cost function for the uniform gain/phase

pdf;5) the maximum cost function when the gain varies

between and .We will plot the spatial directivity pattern in the frequency-angleregion (300–3500 Hz, 0–180 ) and the angular pattern for thespecific frequencies (500, 1000, 1500, 2000, 2500, 3500) Hz.

Table I summarizes the different cost functions for theweighted LS, the nonlinear, and the TLS eigenfilter nonrobust


Fig. 2. Spatial directivity pattern of nonlinear nonrobust design for no gain and phase errors (� = 1,N = 3,L = 20).

Fig. 3. Spatial directivity pattern of nonlinear nonrobust design for gain and phase errors (� = 1,N = 3,L = 20).

Fig. 4. Spatial directivity pattern of nonlinear gain/phase-robust design for no gain and phase errors (� = 1,N = 3,L = 20).

and robust broadband beamformer design procedures. Obvi-ously, the design procedure optimizing a specific cost functionleads to the best value for this cost function (bold values). Thisimplies that when no gain and phase errors occur, the robustdesign procedures lead to a higher cost functionthan thenonrobust design procedure. However, the nonrobust designprocedure leads to very poor results whenever gain and/orphase errors occur (e.g., compare for the nonrobust andthe robust design procedures and see the figures). All robust

design procedures (using pdf and minimax criterion) yieldsatisfactory results when gain and/or phase errors occur.

Fig. 2 shows the spatial directivity pattern of the nonrobustbeamformer, designed with the nonlinear cost function, whenno gain and phase errors occur. Fig. 3 shows the spatial direc-tivity pattern for microphone gains and phases

, i.e., small deviations from the nominal gains andphases. As can be seen from this figure, the beamformer perfor-mance dramatically degrades, especially for the lower frequen-


Fig. 5. Spatial directivity pattern of nonlinear gain/phase-robust design for gain and phase errors (� = 1,N = 3,L = 20).

Fig. 6. Spatial directivity pattern of nonlinear minimax design for no gain and phase errors (� = 1,N = 3,L = 20).

Fig. 7. Spatial directivity pattern of nonlinear minimax design for gain and phase errors (� = 1,N = 3, L = 20).

cies, where the spatial directivity pattern is almost omni-direc-tional, and the amplification is very high.

Figs. 4 and 5 show the spatial directivity pattern of thegain/phase-robust beamformer, designed with the nonlinearcost function, when no errors occur and when gain and phaseerrors occur. As can be seen from Fig. 4, the performanceof this beamformer is worse than the performance of thenonrobust beamformer when no errors occur. However, as canbe clearly seen from Fig. 5, when gain and phase errors occur,the performance of the gain/phase-robust beamformer is muchbetter than the performance of the nonrobust beamformer.

Figs. 6 and 7 show the spatial directivity pattern of the min-imax beamformer, designed with the nonlinear cost function,when no errors occur and when gain and phase errors occur.Similar conclusions can be drawn for the minimax beamformeras for the gain/phase-robust beamformer.

VI. CONCLUSIONS

In this paper, two procedures have been presented for de-signing fixed broadband beamformers that are robust againstgain and phase errors in the microphone array characteristics.


The first design procedure optimizes the mean performance byminimizing a weighted sum using the gain and the phase proba-bility density functions. The second design procedure optimizesthe worst-case performance by minimizing the maximum costfunction over a finite set of feasible microphone characteris-tics. We have used the weighted LS, the TLS eigenfilter, anda nonlinear cost function for designing broadband beamformerswith an arbitrary spatial directivity pattern. Simulation resultsfor the different design procedures and cost functions show thatrobust broadband beamformer design for a small-size micro-phone array indeed leads to a significant performance improve-ment when gain and phase errors occur.

APPENDIX ACALCULATION OF DOUBLE INTEGRAL FORFAR-FIELD

The integral

(119)

is equal to

(120)

such that, in fact, we need to solve integrals of the type (120)

(121)

where

(122)

Normally, this integral can be solved numerically without anyproblem, but a special case occurs when because then,a singularity occurs in the denominator, with

(123)

such that numerically calculating the integral could leadto numerical problems when . By using the Taylor expan-sion of around , we can derive a function

(124)

which is a good approximation for around and whichis independent of . If we now define the function

, we can prove (by applying L’Hôpital’s ruletwice) that for any , is finite and is equal to

(125)

For details, see [18].

Hence, the function can be integrated numericallywith no problem. In fact, the total integralcan be written as

(126)

(127)

APPENDIX BCALCULATION OF , AND FOR ROBUST

NONLINEAR CRITERION

Depending on the values of , ,, and , different cases have to be considered:

• Four equal values:

(128)

• Three equal values and one different value:,

(129)

(130)

(131)

• Two equal values and two equal values:

(132)

(133)

(134)

where

(135)


(136)

• Two equal values and two different values:

(137)

(138)

(139)

(140)

all other cases

(141)

• Four different values: ,

(142)

(143)

(144)

For a symmetric phase pdf , i.e., a function for which, , for a certain , it can easily

be proved that since

(145)

(146)

(147)

such that for we obtain

(148)

ACKNOWLEDGMENT

The authors would like to thank the reviewers for their valu-able comments and suggestions.

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Simon Doclo (S’95) was born in Wilrijk, Belgium, in 1974. He received theM.Sc. degree in electrical engineering and the Ph.D. degree in applied sciencesfrom the Katholieke Universiteit Leuven, Leuven, Belgium, in 1997 and 2003,respectively.

Currently, he is a post-doctoral researcher with the Electrical Engineering De-partment, KU Leuven. His research interests are in microphone array processingfor acoustic noise reduction, dereverberation and sound localization, adaptivefiltering, speech enhancement, and hearing aid techology.

Dr. Doclo received the first prize “KVIV-Studentenprijzen” (with E. DeClippel) for his M.Sc. thesis in 1997, and in 2001, he received a Best StudentPaper Award at the IEEE International Workshop on Acoustic Echo and NoiseControl. He was secretary of the IEEE Benelux Signal Processing Chapterfrom 1997 to 2002.

Marc Moonen (M’94) received the electrical engineering degree and the Ph.D.degree in applied sciences from the Katholieke Universiteit Leuven, Leuven,Belgium, in 1986 and 1990, respectively.

Since 1994, he has been a Research Associate with the Belgian National Fundfor Scientific Research (NFWO). Since 2000, he has been an Associate Pro-fessor with the Electrical Engineering Department, KU Leuven. His researchactivities are in mathematical systems theory and signal processing, parallelcomputing, and digital communications.

Dr. Moonen is Editor-in-Chief forEURASIP Journal of Applied Signal Pro-cessing, Associate Editor for IEEE TRANSACTIONS ONCIRCUITS AND SYSTEMS

II, and is a member of the editorial boards ofEURASIP Journal on WirelessCommunications and Networkingand Integration, the VLSI Journal. He re-ceived the 1994 KU Leuven Research Council Award, the 1997 Alcatel Bell(Belgium) Award (with P. Vandaele), and was a 1997 “Laureate of the Bel-gium Royal Academy of Science.” He is secretary/treasurer of the EuropeanAssociation for Signal, Speech, and Image Processing (EURASIP), and he waschairman of the IEEE Benelux Signal Processing Chapter from 1997 to 2002.

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