Method for array calibration in high-resolution sensor array processing

C.M.S. See, MSc

Indexing terms: Array processing, Calibration

Abstract: With few exceptions, most high- resolution sensor array algorithms for direction of arrival (DOA) estimation are sensitive to the errors in the nominal array manifold. Deviations from the true array manifold, typically resulting from mutual coupling, array channels mismatch and sensor positioning errors, can seriously degrade the algorithms’ performance. Several array calibration algorithms have been reported in the literature; however, only a handful have addressed the problem of calibrating the errors in the array manifold that result from the combined effects of mutual coupling, sensor positioning error and mismatch in the array channels. This problem is addressed in the paper. By extending the array parametrisation to describe the devi- ations of the nominal array manifold from the actual array manifold, the array calibration problem considered can be posed as that of esti- mating the calibration matrix and sensor posi- tions. The array calibration method proposed estimates the sensor positions and calibration matrix from a set of measured steering vectors. The usefulness and behaviour of the proposed calibration method are illustrated through simu- lated experiments.

List of symbols

(.)T = transpose (. )* = Hermitian transpose 11 I I F = Frobenius norm i = arg min (J(x)): i is the minimising argument

of J(x) diag ( I ) = diagonal matrix where the elements of I form

the diagonal T r ( . ) =trace vecd (A) = vector where the diagonal elements of A form

8 = Kronecker product 4) = Schur-Hadamard product X = an estimate of X E( .) = Expectation operator Z, = m x m identity matrix l,,, = m x n matrix of ones 1 0 = true value of the parameter x

the vector

0 IEE, 1995 Paper 1793F (E5), first received 19th April and in revised form 2nd December 1994 The author is with the Communications Research Division, Defence Science Organisation, 20 Science Park Drive, S(0511), Republic of Singapore

90

1 Introduction

The problem of estimating signal parameters from data collected by an array of spatially located sensors has received much attention. In recent decades, a variety of high-resolution algorithms for source localisation, e.g. References 1-3, have been reported in the literature. Although they have superior accuracy and resolution performance over conventional beamforming techniques, most high-resolution algorithms are sensitive to the errors in the nominal array manifold. In particular, devi- ations from the actual array manifold, resulting from mutual coupling, mismatch in the array channels and sensor positioning errors, can seriously degrade their per- formance [4-61. Hence, to achieve high-resoluton per- formance, array calibration is often necessary.

A common method of calibrating a sensor array is to store samples of the array manifold over the desired field of view. However, such a calibration procedure can be both tedious and time consuming. Alternatively, with suitable array parametrisation, array calibration can be posed as a parameter estimation problem. Recently, several researchers have adopted this approach and developed parametric array calibration algorithms that employ a substantially smaller number of sources. Exam- ples of such calibration algorithms using sources in known and unknown locations can be found in Reference 10-14 and their lists of references.

Although the parametric array calibration algorithms reported in the literature have demonstrated varying degrees of success, their applications are limited. In many situations, errors in the nominal array manifold usually result from the combined effects of mutual coupling, mis- match in the array channels and sensor positioning errors. However, few algorithms have been developed for this problem. Recently, robust DOA estimators that are insensitive to array perturbations have been reported in References 8 and 9. However, a drawback associated with these estimators is that they need to have a priori know- ledge of the array perturbation parameter distributions. This assumption is somewhat restrictive.

The approach developed here is based on earlier work by Pierre et al. 1181 and See [I91 and is motivated by the following observations. By extending the array para- metrisation to describe the deviations of the nominal array manifold from the actual array manifold, due to mutual coupling, sensor positioning errors and array channels mismatch, the array calibration problem can be

The author is grateful to Dr. W. Ser and Mr. B.C. Ng of the Defence Science Organisation of Singa- pore, and to the anonymous reviewers for their ! helpful comments and suggestions.

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posed as that of estimating the sensor positions and cali- bration matrix. The calibration matrix is an array param- eter that describes the combined effects of mutual coupling and array channels mismatch. Being unstruc- tured in general, the data from an array covariance are insufficient to obtain meaningful estimates of the cali- bration matrix. As observed in Reference 19, the principal eigenvector of the array covariance associated with a single source is asymptotically related to the true steering vector by a complex multiplicative contant. Hence, with a sufficient number of measured steering vectors, an array calibration method can be formulated in a least squares framework, where the sensor positions, calibration matrix and multiplicative constants are simultaneously identi- fiable from the data. However, for an m sensor array with n calibration sources available, a brute force estimation of all the parameters simultaneously would result in a (2m2 + 2m + 2n - 2)-dimensional optimisation problem! Fortunately, as this is a separable nonlinear least squares problem and taking advantage of the fact that the entries in the calibration matrix are the linear parameters, the search space can be reduced considerably. Similar ideas of estimating the calibration matrix from a set of mea- sured steering vectors have appeared in References 16 and 17.

This paper describes a two-step algorithm that esti- mates the sensor positions, calibration matrix and the associated multiplicative constants {ai};= , from n mea- sured steering vectors. This algorithm is referred to as the method for array calibration (MAC). The first step involves a (2m + 2n - 2)-dimensional search for m - 1 sensor positions and 2n complex parameters in {a,]:=,. Based on the estimated sensor positions and {a,}:=, obtained from the first step, the calibration matrix is estimated directly in the second step from a set of linear equations. As the first step is a multidimensional opti- misation problem, an iterative minimisation algorithm for determining the sensor positions and {a):= is intro- duced here as well.

2 Array data model

Consider an array of m isotropic sensors of arbitrary geometry impinged by a signal from a far field source located at 0. The signal waveforms are assumed to be narrowband of known centre frequency. Using one of the sensors as reference, the complex envelope of the array's output may be written as

r ( t ) = 4 0 , S)s(t) + n(t) (1)

where

a(0, $1 = [ 1, exp ( - j Inn T2), . . ., exp ( - j T,,,)IT

T, = [xi, yi][sin (e), cos (e)]= and i is the wavelength of the signal. n(t) and s ( t ) are the observation noise and signal waveform, respectively. x, and yi are the co-ordinates of the ith sensor with resoect to the reference sensor. The implicit assumptions made in this array model are that the array's sensors acted inde- pendently from each other and the array channels are identical. However, these assumptions are rarely satisfied and in many situations the signal observed at the output of each sensor is the sum of the sensor output when in isolation and the contributions from the signals reflected

I E E Proc.-Radar, Sonar Nauig., Vol. 142, No. 3. June 1995

from the neighbouring sensors. Hence, mutual coupling results, and the degree of coupling is particularly significant when the intersensor spacings are relatively small. To account for these deviations from the nominal array model, the array's output can be written as

(2)

(3)

r(t) = T w o , $)s(t) + n(t)

= Ca(0, $)s(t) + n(t)

where the m x m complex matrix T describes the coup- ling effects among the sensors, and the r is an m x m complex diagonal matrix whose iith entries are the unknown DOA independent gain and phase distortion of the ith sensor and its associated array channel. C is the calibration matrix and, in general, does not exhibit specific structure. As array perturbations considered herein are due to unknown mutual coupling among sensors, DOA independent sensor gain and phase mis- match and sensor positioning errors, the calibration matrix defined in eqn. 3 can be modelled without DOA dependence*.

It suffices to note that the degree of coupling among sensors and the distortion in array channel gain and phase responses vary with frequency. Throughout this paper, we assume that the observation noise is both tem- porally and spatially white and statistically independent from the signals.

3 Method for array calibration

The array covariance R associated with a single source located at 0 is given by

R = U: a,(e, +)axe, $1 + UP (4) where a,(& $) = Ca(0, $), uf = E(s(t)s*(t)), and U: I = E(n(t)n*(t)). The signal and noise power are denoted, respectively, by uf and U:. For the case of finite data, the array covariance can be estimated from

4 L

where L is the number of snapshots. It is clear from eqn. 4 that the principal eigenvector of

R is related to a,(& $) by a complex scaling constant. Denoting a,((?, $) as the normalised principal eigenvector of R, we have a,(0) = aa,(0, $). Owing to the presence of mutual coupling and array channels mismatch, the length of the true steering vector a,(@ is no longer a constant for all 0 [20]. Consequently, a is a complex scalar that varies with 0. When only finite data are available, consistent estimates of a.(& $) can be obtained from d. A robust approach for estimating U,(& $) in unknown coloured noise fields is presented in Reference 21.

Given that the data collected from n non-collinear calibration sources of known location are disjoint in time, we have

(6) J = CA - A , A

* A more general array verturbation model can be found in Reference 17, where I? is replaced- by a local Taylor expansion of the array pertur- bation around 0, by C = C, + C,(O - OJ + . . . . Although able to account for DOA dewndent model mismatch. the Derturhation model

I .

is only valid locally around 0,. Hence, for a large field of view and an array manifold that is highly nonlinear (possibly for large aperture array), this approach may need a substantial number of calibration sources to achieve the desired interpolation accuracy. However, in situ- ations where the sensors introduce 0 dependent gain and phase errors, the array data model in Reference 17 may be more appropriate.

91

where A = C4f91, $1, ..., 4 4 9 $)I 2. = C W I , $,I, . . ., Q" 1 $,)I

p = [a l , . . . , a,Jr

A = diag (p)

$o denotes the true sensor positions. It follows from eqn. 6 that the calibration matrix &', sensor positions $ and the scaling constants A can be obtained from the mini- mising arguments of

(7)

Observe that eqn. 7 is a nonlinear least squares problem with (2m2 + 2m - 2 + 2n)-dimensional parameter space. The elements in C and A are the linear parameters, and the nonlinear parameters are found in $. Clearly, direct minimisation of eqn. 7 is computationally expensive, par- ticularly for a large number of sensors m. A different form of the minimisation problem that is more suitable for our purpose is therefore needed. As eqn. 7 can be considered as a separable (nonlinear) least squares problem, whereby C is separable from # and A, a concentrated form J can be obtained. With $ and A held fixed, and minimising \ \ J l l F with respect to C, the least squares estimate of C is given by [19]

{e, $, A} = arg min llJllF C. *. A

d = AAJ*(AA*)-' (8)

J,,,(A, $) = Tr (A*A :̂ 2,AP')

Substituting eqn. 8 into eqn. 7, we have the following concentrated version of the l l J l l F [19]

(9) where

Pi = I - A*(AA*)-'A

In this paper, J,,,(A, $) is termed the MAC criterion. As all the calibrating sources are of finite power, p = vecd (A) is unconstrained except for p = 0. Thus, the estimates of pb. and A can be taken to be the minimising argument of

{$, A} = arg min JMAc(A, $) subject to p*p = K (10)

where the constant K is chosen to be K > 0. It can be seen from eqns. 8 and 10 that the estimation of C is separated from {$, A}. The advantage of this approach is obvious as it only requires a (2m + 2n - 2)-dimensional search for {$, A}, followed by a direct estimation of C from a set of linear equations.

To summarise, the proposed method of array cali- bration (MAC) consists of

(a) select (b) using the estimated $ and A, estimate C from

A. J

and A according to eqn, 10

eqn. 10.

3.1 Uniqueness Assuming that the array is unambiguous, then A, is a set of normalised eigenvectors representing 2mn - 2n inde- pendent measurements. The parameters to be estimated are C, Jr and A, which are described by 2mZ, 2m - 2 and 2n - 2 parameters, respectively. It follows that a neces- sary condition for the MAC approach to yield unique estimates of {C, $, A) is

where [K] denotes the largest integer smaller than K.

92

3.2 Minimisation algorithm Next, we introduce an algorithm for minimising J,,,(A, $). The basic idea is to decouple the minimisation of the MAC criterion such that each iteration only involves minimising .I,,&, $) with respect to $ or A, while the other is held fixed. As the criterion function is minimised at each step, it will converge to a minimum. However, depending on the initial estimates, the proposed algo- rithm can either converge to a local or a global minimum.

We minimise the cost function J,,dA, $) with respect to 9. by applying a Newton-type optimisation method. In this paper, we use the damped Newton method, where the estimate is calculated iteratively from

(12) $"+ 1) = $(I' - fi($'")- 1 q $ W ) Note that p, = pp and & ( I ) denote the estimates at the Ith iteration. The step length y, can be determined adaptively by selecting p-to-be the smallest integer such that .IMAc(& $ ( I ) ) > J%Ac(A, $(r+ll). H(*(l)) and F'($(')) are the Hessian and gradient of JMAC(A, $ ( I ) ) , respectively. The gradient and Hessian (approximate) can be expressed compactly by (vecd (PA'QPA, T)}

vecd (PrAtQP'A,P) V($) = -2 Re

H($) = 2 Re {BTBB}

respectively, where Q = A*J:A^,A

j2n A, = -AA,,, I

j2n A , = -AA,,, 1 d = [A: At]'

A' = (AA*)-'A

Asla = diag ([sin (el), . . . , sin (e.)]) A,,, = diag ([cos (el), . . ., cos (e,)])

P= CO 11,- B = I , @ i B = ( A P ~ A * ) 0 (I,,, 8 (A'QA'*))~

We will defer the proof to the Appendix.

identity From Reference 22, we have the following matrix

Tr (@*A@B) = d*(A 0 Br)+ (15) where 4 = vecd (e). Using eqn. 15, J,,,(A, $) can be written as

(16) As p is subject to the constraint p*p = K, JMAC(A, $) can be minimised with respect to p when p is taken to be the (minimum) eigenvector associated with the minimum eigenvalue of (AZAJ 0 (P')'.

The computational procedure of the proposed mini- misation algorithm can be summarised as

0 Set k = 0

J,,c(A, $) = p*((AtA,) 0 (P')r)P

0 REPEAT - Select $('+') that minimises J,,,(A('), 9"))

according to the Newton-type algorithm in eqn. 12.

of eqn. 16. - Select vecd (A('+')) to be minimum eigenvector

- Set k = k + 1.

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0 UNTIL lJMAc(A(k' , $('I) - JMAc(Alk+'), $(*+l))l < E

0 END and b show the algorithm's typical convergence behav- iour for L = 100 snapshots. As suggested in Section 3.2,

behaviour. Also-note from eqn. 10 that estimates-of C and A will be obtained up to a scaling constant, whereas estimates of I) will be determined up to a translational constant when no sensor positions are known a priori. In far field, this is unimportant.

0 -

__ p ~ ~ t ~ that ( . ) (XI is used to denote the estimates at the /& the cost function JMiC(A, $1 and x a d $ ) decrease mono-

iteration. As will be discussed in Section 4, the proposed minimisation algorithm exhibits favourable convergence

tonically to a minimum with the number of iterations.

m

x o

4 Numerical study and discussion

This Section presents the results from computer simula- tions of the proposed MAC approach. The purpose of this study is to investigate the convergence behaviour of the proposed minimisation algorithm and examine the capability of MAC in calibrating the errors in the array manifold due to the combined effects of mutual coupling, array channels mismatch and sensor positioning errors.

In all our simulations, we use a nominal eight-element uniform circular array of half-wavelength intersensor spacing. The beamwidth of this array is 44". The data are generated according to the model described in Section 2. The calibration matrix C is obtained from the following expression

(17) c = K I , + w where W is any arbitrary general complex matrix with 11 WII, = 1. For this study, the entries of P are generated from a complex Gaussian distribution, and K is fixed at 2.4. Signal-to-noise ratio (SNR) is defined as uf/u;. Unless otherwise stated, 13 calibration sources, located equally spaced over the field of calibration (FOC) of [ - x , n), will be used. The SNR of the calibration signals is fixed at 10 dB. The FOC is a section in the field of view where the calibration sources can be located. The signals from the calibration sources are assumed to be generated from a zero mean Gaussian process with variance uf. The sensor-positioning error per sensor as a percentage of 4 2 , xa,,($) is defined by

where

Computational experience has shown us that accurate estimates of the sensor positions by MAC using a finite number of measured steering vectors, in general, will result in accurate estimates of A and C. Therefore it is reasonable to use x,,,($) as a figure of merit for the proposed MAC.

The nominal and true sensor positions used in this study are shown in Fig. 1. Without loss of generality, the reference sensor (located at 0 ,O) is assumed to be known. If this assumption is not satisfied, the sensor positions will be estimated up to a translational constant. The sensor positioning error per sensor x,,&~) is 26% of 4'2, where @" denotes the nominal sensor positions. The factor for determining the step length is fixed at 0.5. For this study, the proposed minimisation algorithm uses the nominal sensor positions $. and Z. as the initial estimates of & and A, respectively. Extensive simulations have been conducted to investigate the convergence behaviour of the proposed calibration algorithm. Figs. 2a

Y - l t lx

0 X

X O

-zl , , , ~ 0,

x o

-3 1 0 1

x

Fig. 1 x True 0 Nominal

True and nominal sensor positions

The MAC criterion J,,(A, @), in general, is multi- modal. Hence, the proposed algorithm may converge to a local minimum, particularly for large errors in the initial estimates. It is therefore of interest to examine the pro- posed algorithm's probability of converging to the true sensor positions for different degrees of perturbaeon. Here, we ignore the finite sample effects, whereby A , is taken to be A , . The graph illustrated in Fig. 3 is the probability of convergence of the proposed algorithm as a function of xOde($") for n = 12 and n = 13. The average error of the nominal sensor position xave(+m) varies from 5% to 40%. Each point on the graph is calculated from 100 independent trials. A different realisation of $, obtained from a Gaussian distribution, is used for each trial. For this study, the proposed algorithm is said to haveA converged to the true sensor positions when xaue($) < 0.1%. Although only asymptotic results are pre- sented here, they indicate that the proposed algorithm is capable of calibrating arrays with large sensor posi- tioning errors. Also, the results indicate that the probabil- ity of convergence improves as the number of calibration sources increases. It suffices to remark that, when the algorithm converges to a local minimum, the DOA esti- mates may be biased, and the resolution and accuracy performance will be degraded. The extent of the degrada- tion depends on how 'far' the estimated C and @ associ- ated with the local minimum deviate from their actual values. In many situations, the sensor positioning errors are relatively small, and therefore the proposed cali- bration method can converge to the global solution with high probability.

In the examples that follow, the proposed mini- misation algorithm is terminated when one of the follow- ing conditions is satisfied

(b) after lo00 iterations (a) I/ xaUe($(' + I ) ) - x . . ~ ( $ ( ~ ) ) II < 0.1

(c) lIJMAC(A(k+l~, $'"I)) - J,,,(A'k', $q < ( E = 10-8) I

To show the accurate estimates of $ that can be achieved by the proposed MAC, Fig. 4 shows the estimated sensor

I E E Proc.-Radar, Sonar Navig., Vol. 142, No. 3, June 1995 93

positions for L = 100 snapshots from 20 independent trials. Note that, in all the trials, the sensor positions are accurately estimated.

a t

1000 2000 3000 number of iterations

a

\ 1000 2000 3000

number of iterations b

Fig. 2 a cost function J ~ ~ ~ ( A , $)against number of iterations b Sensor positioning error per sensor I.&) against numbet of iterations

Convergence curues ofproposed algorithm

Next, to demonstrate the accuracy and resolution improvement of the DOA estimates from arrays cali- brated by the proposed MAC, we consider the case of two uncorrelated signals impinging the array from - lo" and lo". The SNR of each signal is fixed at 10 dB. Fig. 5 displays the MUSIC angular responses of an uncalibrated array and an array calibrated by the pro- posed algorithm. The graphs display the MUSIC'S angular responses from 10 independent trials. For each trial, a,(O) is estimated using L = 100 snapshots, and 100 snapshots are used for DOA estimation. As suggested by the MUSIC'S angular responses, DOAs estimated using the calibrated array manifold (by MAC) result in better resolution and acccuracy.

The graphs from Fig. 6 illustrate the expected sensor positioning error per sensor of the estimated sensor posi- tions E{X.J+)} as a function of L for FOCs [ -n, n) and [ - tn, fn). The calibration sources are located equally spaced for each FOC. Each point in the graphs is calcu- lated from 100 independent trials. As seen from the graphs, the proposed algorithm resulted in better accu-

94

racy performance with wider FOC, and x.,&$) decay monotonically with L. In fact, for the asymptotic case, that is t$(O, $) -+o,(B, t) as L + m, exact estimates of $ (or xaUe(+) = 0) are obtained for both FOCs.

X O ' 10 20 30 40

sensor positioning error per sensor, "io

Probability of converging to the true sensor positions against Fig. 3

x = 13 calibration sources = 12 calibration sources

LS*J

Y t

t

!& & t

I - L 1 0 1

Estimated sensor positions by the proposed algorithmfrom 20 X

Fig. 4 independent trials + Estimated sensor positions x True sensor positions

In many situations, degradation in resolution and accuracy performance of most high-resolution source localisation algorithms is due to the combined effects of mutual coupling, array channels mismatch and sensor positioning errors. The MAC we propose can therefore be useful in many array applications. Although the inter- polation procedure by Schmidt [7] reduces the number of calibration sites required by the conventional array calibration method, a substantial number of them are still needed when the array is required to resolve closely spaced emitters. Consider the array in the simulation study as an example. To achieve an interpolation accu- racy of lo, the grid size of 3" is required for the case of two signals about 8" (0.2 beamwidth) apart. It follows

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that, for a field of view of [-n, n), a total of 45 cali- bration sources are required, whereas the proposed cali- bration algorithm only uses 13 calibration sources in this numerical study.

c F I X

0 4 - a

m L aJ m -

brating the errors in the (nominal) array manifold due to the combined effects of mutual coupling, mismatch in the array channels and sensor positioning errors. The pro- posed method can be applied to any unambiguous array,

L

I X E

X * I X X

x x x

degrees 0

degrees b

Fig. 5 M U S I C ’ S angular responses of an uncalibrated array and an array calibrated by the proposed algorithm from IO. independent trials (true DOAs indicated by ueriical dotted lines) ~1 Uncalxbrated b Calibrated

In general, the calibration matrix varies with fre- quency. Hence, the calibration matrices associated with the frequencies of interest need to be estimated to achieve high-resolution performance. Although sensor positions are not a function of frequency, computational experience has demonstrated that more accurate estimates of C and + and better probability of convergence can be achieved at higher frequency, i.e. larger effective aperture or smaller beamwidth. Hence, it may be useful to estimate C(f) and $ by MAC atf=fm, and estimate the remain- ing C(f) using the calibration method described in Reference 19 with the estimated &. Note that ‘(f) is to stress the dependence on frequency. Also note that this approach is computationally simpler, as the calibration matrices, other than C(f,,,), are estimated by non- iterative means.

5 Conclusions

We have presented an array calibration that we refer to as the MAC. The proposed MAC is developed for cali-

IEE Proc.-Radar, Sonar Navig., Vol. 142, No. 3, June 1995

16

4000 number of snapshots L

Fig. 6 estimated sensor positions EO&($)) against the number of snapshots L. x FOC-z .4

Expected auerage sensor positioning errors per sensor of the

FOC[-flr,fn)

and the number of calibration sources needed is deter- mined (almost linearly) by the number of sensors. The MAC approach is a two-step procedure that involves a (multidimensional) search for the sensor positions and {mi}:=l, followed by the solution of a set of linear equa- tions for the calibration matrix. The search space required by this approach is considerably smaller than a ‘brute force’ estimation of all the parameters simulta- neously. As the first step is a multidimensional opti- misation problem, we have also introduced a minimisation algorithm for determining the sensor posi- tions and {ai};= Results from simulation studies revealed that the proposed MAC can result in accurate estimates of the sensor position and the calibration matrix. Furthermore, the numerical examples have also demonstrated the potential accuracy and resolution improvement that can be achieved by arrays calibrated by the proposed MAC. Simulation studies conducted to examine the convergence behaviour of the proposed minimisation algorithm indicated that, although a modest number of iterations are required for con- vergence, a global solution may be obtained even for large sensor positioning errors.

6 References

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2 ZISKIND, I., and WAX, M.: ‘Maximum likelihood estimation via the alternating Droiection maximization algorithm’, IEEE Trans.. 1988, ASP-36, pp.-1553-1560

3 VIBERG, M., OTTERSTEN, B., and KAILATH, T.: ‘Detection and estimation in sensor arrays using weighted subspace fitting’, IEEE Trans., 1991, ASP-39, (ll), pp. 2436-2449

4 FRIEDLANDER, B.: ‘A sensitivity analysis of the MUSIC algo- rithm’, IEEE Trans., 1990, ASP-38, (lo), pp. 1740-1751

5 FRIEDLWDER, B.: ‘Sensitivity analysis of the maximum like- lihood direction-finding algorithm’, IEEE Trans., 1990, AES26, (6), pp. 953-968

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6 ZHU, J.X., and WANG, H.: ‘Effects of sensor position and pattern perturbations on CRLB for dwection finding of multiple narrow- band sources’. Proc. 4th ASSP Workshop on Spectral estimation and modelling, Minneapolis, Minesola, 1988, pp. 98-102

7 SCHMIDT, R.O.: ‘Multilinear array manifold interpolation’, IEEE Trans., 1992, AssP-40, (4). pp. 857-866

8 WAHLBERG, B., OITERSTEN, B., and VIBERG, M.: ‘Robust signal parameter estimation in the presence of array perturbations’. Proc. IEEE Int. Conf. Acoust., S w h , Simal Process., Toronto, Canada, 1991, pp. 3277-3280

9 SWINDLEHURST, A.L., and KAILATH, T.: ‘A performance analvsis of subsoace-based methods in the oresence of model errors. Par; I: The m;sic algorithm’, IEEE Trans., 1992, AssP-40, (7); nn 175R-1774 rr.-.--

10 SEYMOUR, L., COWAN, C., and GRANT, P.: ‘Bearing estimation in the mesence of sensor wsitioning errors’. Proc. IEEE Int. Conf. Acousi, Speech, Signal Prbcess., 1987, pp. 2261-2267

11 NG, B.C., and SER, W.: ‘Array calibration using sources in known locations’. IEEE Int. Conf. on Communications Systems, Singapore, 1992, pp. 836-840

12 WEISS, A., and FRIEDLANDER, B.: ‘Array shape calibration using sources in unknown locations - a maximum likelihood aproach’, IEEE Trans., 1989, ASSP-22, (3), pp. 1958-1966

13 FRIEDLANDER, B., and WEISS, A.: ‘Eigenstructure methods for direction finding with sensor gain and phase uncertainties’. Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., 1988, pp. 2681- 2684

14 FRIEDLANDER, B., and WEISS, A.: ‘Direction finding in the pres- ence of mutual coupling’, IEEE Trans., 1991, AP-39, pp. 273-284

15 FUHRMANN, D.R.: ‘Estimation of sensor gain and phase’, IEEE Trans., 1994, AssP-42, (I), pp. 77-87

16 KOERBER, M.A., and FUHRMANN, D.R.: ‘Array calibration by Fourier series parameterization scaled principal components method‘. Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., 1993, IV, pp. 340-343

17 OlTERSTEN, B., VIBERG, M., and WAHLBERG, B.: ‘Robust source localization based on local array response modelling’. Proc. IEEE Int. Cod. Acoust., Speech, Signal Process., 1992, 2, pp. 4 4 - 444

18 PIERRE, I., and KAVEH, M.: ‘Experimental performance of Cali- bration and direction-finding algorithms’. Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., Toronto, 1991, pp. 1365-1368

19 SEE, C.M.S.: ‘Sensor array calibration in the presence of mutual coupling, gain and’ phase mismatch’, Electron. Lptt., 1994, 30, (5).

20 STEINBERG, B.D.: ‘Principles of aperture and array system design’ (John Wilcy & Sons, New York, 1976)

21 TALWAR, S., PAULRAJ, A., and GOLUB, G.H.: ‘A robust numerical approach for array calibration’. Proc. IEEE Int. Cod. Acoust., Speech, Signal Proms., Minnesota, 1993, IV, pp. 316-319

pp. 373-374

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7 Appendix

The gradient and an approximation of the Hessian of J,,,(A,. $) in eqns. 13 and 14 are derived in this Section. The denvative of the projection matrix P’ with respect to +i is C3l

(19) _- - -pLA:At - (P’A:At)* Wi

where Ai = dA/a+,. Using eqn. 19, the first derivative of J,,& $) with respect to Si can be written as

(20) ~- aJMAc(A’ ‘I - - 2 Re {Tr (A’QPIA,)} a+,

After some manipulations, the gradient of eqn. 12 follows. The second derivative of the projection matrix P’ is

c31

-- a2pL -n+n* aSi a+.,

where

II = PIAzA’AfA’ + A’*A, P’ArA’ - P A ; A‘

- P’AfA’At*A, P‘. + P’A:A’A:At

For small perturbations in the sensor locations, we can ignore terms involving P’Q and QP’. It follows that the ikth element of the Hessian can by approximated by

aZJMAc(A’ = 2 Re {Tr (AiP*A:A’QA’*)} (22)

After some algebraic manipulations, the compact expres- sion of the Hessian of eqn. 13 follows.

a+i +k

96 IEE Proc.-Radar, Sonar Nauig., Vol. 142, No. 3, June 1995