IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-35, NO. 6, JUNE 1987 724

Array Shape

Part 11:

Calibration Using Sources in Unknown Locations-

Near-Field Sources and Estimator Implementation

YOSEF ROCKAH AND PETER M. SCHULTHEISS

Aktruct-Uncertainty concerning sensor locations can seriously de- grade the ability of an array to estimate the location of radiating sources. Array calibration then becomes an important issue. This pa- per deals with array calibration using spectrally disjoint near-field sources whose locations are not known a priori. .It calculates Cramer- Rao bounds on source location and array shape errors. It shows that, with mild restrictions on source-array geometry, the array shape er- rors can be made arbitrarily small by using a sufficient number of suf- ficiently strong calibrating sources. The required number of calibrat- ing sources is five for a four-sensor array, four for a five-sensor array, and three for an array of six or more sensors. Accurate calibration is not possible for a three-sensor array. While calibration can establish array shape with great accuracy, it cannot resolve a rotational uncer- tainty in array orientation. This uncertainty translates into a residual error in source bearings, but not in source ranges. Thus, incremental errors in target range due to sensor location uncertainty can be re- duced to any desired extent. The paper also proposes an actual cali- bration procedure and presents simulation results indicating that use- ful calibration can be obtained with calibrating sources of very mod- erate signal-to-noise ratio.

T I. INTRODUCTION

HE ability of an array to measure bearing and range to a source of radiation is seriously affected by un-

certainty concerning array shape and orientation. An ear- lier paper [ l ] dealt with the bearing estimation problem and arrived at the following conclusions.

1) Three noncollinear far-field sources can achieve ar- ray shape calibration. The bearings of these sources need not be known a priori. If the sources are spectrally dis- joint, the calibration errors tend to zero as the signal-to- noise ratio of the sources tends to infinity.

2) While array shape error tends to zero with increas- ing source strength, there remains a residual error in array orientation. This translates into a residual error in bearing for any target observed by the array. Only by providing a separate directional reference can one eliminate this error.

The present paper generalizes these results to accom-

Manuscript received April 8, 1986; revised December 16, 1986. This work was supported in part by the Office of Naval Research under Contract N00014-80-C-0092.

Y. Rockah is with Rafael, Haifa, Israel. P. M. Schultheiss is with the Department of Electrical Engineering, Yale

IEEE Log Number 8714126. University, New Haven, CT 06520-2157.

modate near-field calibration sources. It shows that array shape calibration of any desired accuracy can still be achieved with sources of sufficient strength, but that the number of required sources may exceed three (depending on the number of elements in the array). It further shows that target bearing estimates are subject to an irremovable uncertainty in rotation (as in the far-field case), but that there is no such residual error in the range estimate.

Since our discussion will draw heavily on material con- tained in [l], we summarize briefly the most relevant as- sumptions and observations in that paper.

A. Assumptions 1) The array and all sources are confined to a single

plane. All signal energy travels in this plane. 2) Within that plane, the array has an arbitrary, but

known, nominal geometry. x- and y-direction sensor dis- placements from their nominal locations are characterized as independent zero-mean Gaussian random variables with standard deviation u. Displacements of different sensors are statistically independent of each other.

3) The, displacement standard deviations u are small compared to all nominal intersensor distances as well as to the distances from all sources to all sensors. 4) Receiver noise is zero-mean Gaussian, with the same

spectrum at each sensor. Noise components at different sensors are uncorrelated.

5 ) All signals are zero-mean Gaussian random pro- cesses with known spectra, statistically independent of each other and of the receiver noise. The various source signals are spectrally (or temporally) disjoint.

B. Dejnitions and Relations 1) Potential performance is characterized by the Cra-

mer-Rao bound on the parameter vector 8. For N sources with bearings cyq, ranges rq, and M sensors with displace- ments (Axm, Ay,,,) from their nominal locations (xrn, ym), the parameter vector is

' e = (cyl, rl - - - QN, rN, Ax23 AY2

* - - AXM, AYM, Ax17 AYl)'. ( 1 )

0096-3518/87/0600-0724$01.00 O 1987 IEEE

ROCKAH AND SCHULTHEISS: ARRAY SHAPE CALIBRATION 125

Ax1, Ayl have been listed at the end because the first sen- sor will be used as a reference. Prior statistics are avail- able for the Axi and Ayi, but not for the ai and ri. We are therefore using a hybrid version of the Cramer-Rao in- equality [ l ] , which bounds variances of unbiased esti- mates of the (xi and ri, but mean square errors (MSE's) of the Axi and Ayi. Both quantities are averaged over the prior distributions of the Axi and Ayi.

2) Information concerning the components of 0 enters the data through the differential delays A$ = 71 - 7;

where 7% is the travel time of signal q to sensor m. Thus, the contribution of the qth source is described by a vector of differentials delays A,. The estimation error of this vector has the ( M - 1 )-dimensional Fisher matrix k,JA. k, depends on the signal spectrum S, ( w ), noise spectrum N ( w ), observation time T, and propagation velocity c through

JA is the constant matrix

J A = M I - l lT . (3)

1 is a vector all of whose elements are one. 3) Transformation from the Fisher matrix for differen-

tial delays to the Fisher matrix J for the parameter vector 0 requires the matrices of partial denvatives of the A$ with respect to the components of 0. We define

Rq = c

s, = c

3 0 a AX,

r N 1 - 1

L I + 0 2 j = c 1 kipj S q J A R ; ( ~ , ~ * ~ , T ) - '

Pj is defined by

pj = s~J~s,T - S~J*R;(R~J~R,T)-' R ~ J ~ S ~ T (7)

and P j / M is a projection matrix. The first term in (5) is the bound on bearing and range error in the absence of sensor displacements, designated here as CRLB ( aq, rq)o. The second term is the increment of the bound due to sen- sor location uncertainty, denoted by ACRLB ( a,, rq ) .

5 ) With the definition kmin = mini kj, (6) generates the inequality

CRLB(Ax2 9 * * AxM, Ay2 . * AyM, Ax1, A+) N \ -1

It follows that

lim (4 + kmin 5 Pj)-' = 0 kmin+ m 0 j = l

if and only if Cy= Pj is nonsingular. Under that condition, therefore, the accuracy of possible array calibration is limited only by the signal-to-noise ratio (SNR) of the cal- ibrating sources. Under the same condition, it follows that the right-hand side of (6a) becomes independent of u for sufficiently large kmin.

From this point on, the procedures for the far- and near- field cases diverge. The far-field assumption is equivalent

(4) to the assertion that the range is known to be infinite, so that R becomes a row vector and ( 5 ) represents a scalar. This can affect the rank of the Pj and can therefore change the calibration problem very substantially.

11. CALIBRATION WITH NEAR-FIELD SOURCES: CRAMER- RAO BOUNDS

A. Exact Location of One Sensor and Direction to a Second Sensor Known

As in the far-field case [l] , we start with the somewhat artificial assumptions that the location of the reference sensor is known and that one also knows the direction of a second sensor relative to the reference. The former

4) Using these definitions, [l] developed the following merely implies that the origin of coordinates is placed at general expressions for the Cramer-Rao bounds on source the reference and is therefore fairly innocuous. The sec- location errors and on the array geometry errors for any ond implies availability of a directional reference and is q , l l q s N therefore quite restrictive. Both will be relaxed later.

126 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-35, NO. 6, JUNE 1987

We begin by rewriting (7) in the form

This equation defines the matrix

Since J A is positive definite, the existence of its square root is assured. We observed earlier that ( 1 / M ) Pj is a projection matrix. It is a simple matter to demonstrate that Qj is also a projection matrix [of dimension ( M - 1 ) ] , projecting onto the orthogonal complement of the space spanned by the columns of & Rf. Rf consists of two linearly independent columns, and 4 is nonsingular. It follows that Qj has rank ( M - 3 ).

If we designate the sensor in a known location as sensor 1 , Ax1 = Ayl = 0. Using index 2 for the sensor whose direction relative to sensor 1 is known, we can, without loss of generality, place the origin of coordinates at sensor 1 with the x axis passing through sensor 2. Then

x1 = y1 = AX^ = Ay1 0

y2 = Ay2 = 0. (10)

Since Axl, by,, and Ay2 are known, the rows of (4) cor- responding to the derivatives with respect to these vari- ables are deleted. From the fact that Qj has rank ( M - 3 ) , it follows immediately that

rank Pj 5 N ( M - 3 ) . [ j r l 1 (11)

Since the dimension of Cy= Pj is ( 2 M - 3 ), a neces- sary condition for the nonsingularity of that matrix is

2M - 3 M - 3 ‘

N 1

This equation cannot be satisfied for M = 3. When there are four sensors, it requires a minimum of five sources, with five sensors a minimum of four sources, and with six or more sensors a minimum of three sources. Combining these observations with the discussion following (6a), one concludes that unless (12) is satisfied, one cannot cali- brate array geometry with arbitrary accuracy (or make the calibration error independent of the initial uncertainty de- scribed by o) by simply increasing the strength of the cal- ibrating sources or the length of the observation interval. This does not imply, of course, that no error reduction can be achieved when (12) is violated. One can show [ 11 that calibration useful for source localization is not pos- sible whenN = 1 . For 1 < N < ( 2 M - 3 ) / ( M - 3 ) , one would have to compute (5 ) for the particular geometry of interest to determine whether significant error reduc- tions can be obtained. Our concern here will be primarily with conditions that make Cyzl Pj nonsingular. We de- scribe source-array geometries with that property as per- mitting “accurate calibration.” In that language, (12) is a necessary condition for “accurate calibration. ”

In the far-field case Qj has rank M - 2 , so that the equivalent of (12) is

2 M - 3 M - 2 ’

N I

This is satisfied by N = 3 for all M 1 3 , and [l] shows that calibration is indeed possible with three sources. It is possible to prove [15] that accurate near-field calibration of a four-sensor array is possible with five sources (so that (12) is sufficient as well as necessary), when the calibra- tion source ranges are large compared to the array dimen- sions. Numerical computations suggest that (12), together with a requirement that neither sensor nor source geom- etries be collinear, is sufficient as well as necessary, but no formal proof is available at this time. Selected samples of the computational results are discussed later in this sec- tion.

B. Exact Location of One Sensor Known; No Directional Reference

The introduction of one additional unknown ( Ay2) adds one row to the matrix Sj [(4)], but leaves Qj [(9)] un- changed. The dimension of Pj [(S)] increases by one, and if Sj has full column rank (as it would for nonpathological array geometries), Pj acquires one additional eigenvalue of zero. What is not clear algebraically is that Cy= Pi can now no longer be made nonsingular by increasing the value of N . From a geometrical point of view, this con- clusion is, of course, not at all surprising: rotation of the array by a fixed angle is indistinguishable from rotation of all sources by the same angle. Thus, one can hope to calibrate array shape, but not angular orientation. This geometrical insight provides the clue for the analytical treatment. One transforms from 2 M variables specifying sensor locations to 2 M - 1 variables specifying distances between sensors (hence array shape) and one variable specifying angular orientation. Refer to Fig. 1 , and let

4 = tan-’ (%) for Ax2, Ay2 << x2. (14)

Since the displacements Ax,,,, Ay,,,, m = 2 , * * - , M are very small compared to the dimensions of the array, one can use a first-order Taylor expansion around the

ROCKAH AND SCHULTHEISS: ARRAY SHAPE CALIBRATION 121

(X,.Y,) X

Fig. 1 . Array shape described by intersensor distances.

nominal locations. In matrix notation,

L Ay2 1 TA is the corresponding matrix for the case of known ref- erence direction.

Algebraic manipulations outlined in Appendix A yield the following. If good calibration is possible with a direc- tional reference,

lim CRLB(AL, - - AL2M-3, 4 )

and

L o ( l

cy and r specify target location. k is the SNR factor [(2)] associated with the target and can therefore be small even when all calibrating sources are strong.

Since the Li completely determine array shape, (16) can be read as asserting that “accurate calibration” can be obtained for array shape, but not for array orientation. The equivalence of the expressions in the lower-right-hand comer of (16) and the upper-left-hand corner of the sec- ond term in (17) shows that uncertainty in sensor loca- tions causes an incremental error in target bearing pre- cisely equal to the residual uncertainty in array orientation. The zero in the lower-right-hand comer of the second term in (17) indicates that the uncertainty in sensor locations causes negligible incremental error in target range when the calibrating sources are strong enough (or the observation time is long enough). The physical expla- nation is that the only remaining uncertainty concerning array geometry is one of angular orientation, which does not affect the range estimate. For calibrating sources of finite SNR, the incremental error is, of course, finite, but the numerical examples of Section 11-D indicate that they can become quite small for SNR’s well within the prac- tical range.

C. Uncertainty in All Sensor Locations Results have been obtained only for ranges large com-

pared to the array dimensions. With that restriction, one finds the asymptotic rotational error by a straightforward extension of the far-field argument [ 11 :

where rij is the nominal distance between sensors i.and j . The same procedure yields the following asymptotic

bound on translational error:

lim CRLB(AX: + AY:) = a2 kmin -t m C C ri.

(19) If all rii have the same order of magnitude, both terms in (19) vary as M-’. For large M , therefore, the translational rms error tends to zero with M- /2.

For the bearing and range errors, one finds, in complete analogy with Section 11-B,

Thus, the incremental error in bearing tends to q5cu, while the incremental error in range tends to zero.

D. Numerical Examples For the geometry described in Fig. 2 ( M = 6), numer-

ical computations were performed, based on (5) and (6), using parameter values d = 1, X. = (d/10) , Tw = 10, u2 = 0.0025. All calibration sources had the same ( SNR),, zero outside of a band W centered at wo and con-

728 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASP-35, NO. 6, JUNE 1987

stant within that band.’ X. is the wavelength at frequency a, = n/3 WO.

following: (SNR), = 0.5

signals, j = 1, 2, 3; C R L B ( ~ , ) ~ = 5.7 - 1 0 - ~ CRLB(r,), = 7.1

signal; and ACRLB (a,)No = 4.5 * lod4 ACRLB (rt)No = 25.5.

signals, respectively.

Computations were carried out for various values of the rl = rz = r 3 = rt 10

1) aj, rj-direction of arrival and range of calibration

2) a,, r,-direction of arrival and range of the target

3) (SNR),, (SNR),-SNR of the target and calibration

The Largest Diagonal Elements of CRLB (Ax , AY 1c.1

(SNR), Largest 2nd Largest 3rd Largest 4CRLB ( al)cal ACRLB ( r, leal

1 8.0 . 10-4 6.8 . 6.8 . 8.6 * lo-’ 2.8 2 6.2 . 10-4 6.2 . 1 0 - ~ 6.1 . 1 0 - ~ 5 .3 . 1 0 - ~ I .4

10 5 .0 . 10-4 5.0 . 1 0 - ~ 4.0 . 1 0 - ~ 2.1 . io-’ 0.3 100 1.7 . 10-4 1.7 . 1 0 - ~ 1.3 . 1 0 - ~ 6.1 . 0.03

Nomenclature:

CRLB(a,)o, CRLB(r,)o bounds without sensor location uncertainty

ACRLB ( )NO, incremental emirs due ACRLB ( rt )No to sensor location un-

certainty without cal- ibration

CRLB(AL)cal, CRLB(at)cal, CRLB ( rt >tal bounds after calibra-

tion.

Three auxiliary sources were used, which is the mini-

Case a-Exact Location of One Sensor and Direction

Location set 1 (short range):

mal number for M = 6.

to a Second Sensor Known:

a1 = 0 a2 = n/4 a 3 = a / 2

a, = a/3

r1 = r2 = r3 = r, = 2

(SNR), = 0.5

CRLB(~,), =2.3 CRLB(~,),= 3.9

ACRLB(a,), = 1.25 ACRLB(rf)No = 2.06 -

Comparison of CRLB (Ax, AY),,~ with the precalibra- tion uncertainty of o2 = 0.0025 shows that significant calibration is achieved even at the lowest SNR’s consid- ered here. Improved knowledge concerning array geom- etry translates very directly into smaller incremental er- rors for target bearing and range. The effect is particularly dramatic for the geometry-sensitive range measurement at long ranges. Here the total range error is dominated by the effects of sensor location uncertainty in the absence of calibration, but by basic fluctuation errors when calibra- tion is performed (even at the lowest (SNR), considered here).

Case b--Exact Location of First Sensor Is Known; No Directional Reference: Case b is for the same two loca- tion sets as in Case a.

Location set I :

short-range (SNR), = 0.5

a1 = 0 a2 = a/4 ay3 = n/2

ff, = n/3

rl = r2 = r, = r, = 2

CRLB(a,>, = 2.3 - CRLB(r,),= 3.9 - ACRLB ( = 2.7 * ACRLB(T,),~ = 2.35 -

The Largest Diagonal Elements of CRLB ( A *, AY L I

(SNR), Largest 2nd Largest 3rd Largest ACRLB ( ACRLB

1 1.4 1 0 - ~ 8.9 . 1 0 - ~ 6.2 . 1 0 - ~ 5.3 . IO-’ 2 .9 . 1 0 - ~ 2 1.3 . 1 0 - ~ 7.2 . 1 0 - ~ 4.5 . 1 0 - ~ 3.6 . lo-’ 2 .4 . 1 0 - ~

10 9.2 . 4.5 * 2.4 . 1.4 . lo-’ 1.6 . 1 0 - ~ 100 2.5 . loWq 1 . 1 * 5.7 - IO-’ 3.0 . 4.7 . 1 0 - ~

Location set 2 (long range): (SNR), A C R L ~ ( ~ , ) , , I ACRLB (rrlcal

a1 = 0 a2 = 4 4 a 3 = a/2 1 2

‘Disjointness of sources would then have to be achieved in the time lo

2.14 . 1 0 - ~ 2.14 . w 4

3 .1 . 10-4 2.6 . 10-4

2.12 . 1 0 - ~ 1.8 . 10-4 100 2.12 . 1 0 - ~

domain. 4 .8 . IO-’

ROCKAH AND SCHULTHEISS: ARRAY SHAPE CALIBRATION 729

(401

Fig. 2. Regular six-element array.

Location set 2:

long-range (SNR), = 0.5

CRLB(~,), = 5.7 . 10-~ CRLB( r,)o = 7.1

ACRLB(a,), = 6.75 * ACRLB(r,), = 32.8

1 2

10 100

2.6 . 1 0 - ~ 2.5 2.3 10-4 2.1 1 0 - ~

3 1.5 0.3 0.03

Here the improvement in bearing accuracy through cal- ibration is sharply limited by the residual rotational un- certainty characterized by 9. The value of 2.1 X to which the incremental mean square bearing error con- verges for high SNR's is simply the asymptotic MSE of 4. Much more important from a practical point of view is the impact of calibration on the incremental range' error, particularly at long ranges. As anticipated from the dis- cussion in Section 11-B, the incremental error declines steadily with increasing (SNR),. What is added by the numerical computations is the insight that transition from a total range error dominated by location uncertainty to one dominated by fluctuation errors occurs at values of (SNR), well within the practical range.

111. IMPLEMENTATION A. General Considerations

We have demonstrated that auxiliary sources of suffi- cient strength and numbers provide the potential for use- ful array shape calibration. We now propose a procedure for actually achieving such gains. We demonstrate that this procedure perforins useful array shape calibration un- der certain conditions and will show by example that it can approach the performance specified by the Cramer- Rao bound for moderate SNR's and observation times.

We consider an array of M sensors and use N calibrating sources. The j th source yields a set of ( M - 1 ) linearly independent delays described by the delay vector 8j. Con- catenating these vectors, one obtains the N ( M - 1) di- mensional vector of observed differential delays:

Using 8j and the nominal sensor locations (xo, yo), it is a simple matter to compute first estimates of source bear- ings ajo (and ranges rj0 in near-field calibration). Their concatenation with the nominal sensor locations will be designated as uo, the initial estimate of the vector of un- known parameters. For the far-field case,

We propose to improve upon the estimate uo by a simple iterative procedure. In the interest of compactness, we will pursue the argument only for far-field sources where the required N is 3. An analogous procedure can be used in the near-field case when the proper number of sources are available. The underlying idea is the following.

Geometry specifies a unique relation between 8 and the vector u of unknown parameters:

8 = f ( u ) . (23 1 If one disallows the usual pathological geometries (line arrays, coincident sources) and sets Axl = Ayl = 9 = 0 (know directional reference and reference sensor loca- tion), (23) becomes, at least locally, a l : l relation. We shall postulate (Ax, Ay ) and ( a - q,) sufficiently small so that successive approximations to u remain within the same region of 1 : 1 mapping.

Substituting (22) into (23), one obtains a vector ii0,

which will, in general, not coincide with the measured 8. Two factors contribute to the discrepancy.

1) The delay measurements are noisy. There is no rea- son to expect noisy delay measurements to lie in the range space off ( u ), so that 8 need not satisfy (23).

2) There are sensor displacements. Since the compu- tation of 8o assumes zero sensor displacements, one would expect to differ from 8. However, in the absence of delay measurement noise, 8 lies in the range space off ( u ) and therefore on the surface specified by (23).

Since we cannot be sure that 8 satisfies (23), we choose as our estimate # the value of u that minimizes 11 8 - f ( u ) 1 1 2 , i.e., the point on the surfacef( u ) closest to 8. This leads to the usual least squares procedure of succes- sive approximations

uk+l = uk + Auk (24)

where Auk is the least squares solution of

Equation (25) is, of course, simply a linear Taylor expan- sion of (23) about u = uk. Iff ( u ) is linear over the region

730 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-35, NO. 6, JUNE 1987

between uo and the true value of u, the point closest to 8 is found in a single step. Otherwise, it is approached with increasing k .

The least squares solution of (25) is

The required nonsingularity of A;Ak is assured by the assumption that (23) is (locally) a 1 : 1 mapping, which results in a matrix Ak of full column rank. A more direct demonstration of that property is given in Appendix B.

We mustnow reexamine the assumption Axl = Ayl = 4 = 0. For the far-field case, small translational errors ( Axl , Ayl ) are clearly unimportant, and we need to deal only with the issue of unknown angular orientation 4. It is tempting to argue from the bounds computed earlier that the data contain no information concerning 4 and that no loss is therefore incurred by choosing 4 = 0. How- ever, since we are dealing with an ad hoc procedure of parameter estimation, it is not clear that the estimated ar- ray shape is independent of the postulated 4. We will therefore generalize the technique developed in the pre- ceding paragraphs by introducing the additional parame- ter 4 and show that the iterative procedure leads to pro- gressively improved estimates of array shape, but does not alter the initially postulated 40. Section 111-B ad- dresses that problem.

When this has been accomplished, there will still re- main the selection of an appropriate value of 4. The data provide no direct clue, but the choice is not completely arbitrary. What one must exploit at this stage is the prior knowledge that the sensors are near their nominal loca- tions. Given information concerning array shape gathered from the data, a computationally attractive (although by no means unique) choice of 4 is the value that minimizes the sum of the squared displacements of all sensors from their nominal locations. That minimization is carried out in Section 111-C.

B. Calibration Without Directional Reference As in the previous section, we confine discussion to the

far-field problem, noting that the near-field problem can be treated similarly and yields similar results.

To handle the case of known reference sensor location but unknown direction to the second sensor, we begin with the case of known reference direction and introduce the same transformation as in Section 11-B. Let LA have com- ponents 1/2a:, the ai being distances, between sensors. Then the vector of increments A LA is related to ( Ax, Ay ) by the equation

T A is a nonsingular matrix defined in (15). As in the ear- lier discussion, A x = ( Ax2, + * * , AxM) and Ay = ( Ay3, * * , AyM ) T , generating a total vector of length ( 2 M - 3 ) in (27). Define

then

where

using this transformation (25) becomes

where

A, = Ak(

Since Ak has full column rank, i i k also has full column rank. (For any vector x # 0, ( Til),x = y # 0, and Ak y # 0). Now define the vector of unknown parameters to include Ayz and hence allow for unknown reference di- rection.

A U = ( A a l , AcY~, A a 3 , AX,, . , AX,, Ay3 ,

* - , AYM, A y 2 ) '. (33) A series of algebraic manipulations outlined in Appen-

dix C convert (23) to the form

= c ( 8 -

where A Pj = Aaj + 4. The left-hand side of (34) is independent of 4, confirm-

ing that the delay measurements contain no information concerning the rotation angle 4. The equation remains un- changed if one sets 4 = 0 or Ax2 = 0, and one then has the case of known reference direction (since APi = Aai ). As far as the estimation of array shape is concerned, one therefore incurs no loss of performance by postulating 4 = 0 and proceeding as if the direction from sensor 1 to sensor 2 were known.

We note that the proposed procedure is an approxima-

ROCKAH AND SCHULTHEISS: ARRAY SHAPE CALIBRATION 73 1

tion of Newton's method for minimizing Hence, (41) becomes

€ = (6 - f ( u ) ) ' ( 8 - ( u ) ) . (35) This can be shown by recognizing A i ( 8 - f ( u ) ) as ( - a€ / au ) k and ( A ~ A ~ ) as an approximation to ( a€ '/ au ) (see for example [ 151). Indeed, numerical simulation in- dicates that the convergence is almost quadratic.

C. Choice of 4 As suggested earlier, we choose 4 to minimize the sum

of squares of sensor displacements ( Axm, Aym) from their nominal locations for an array with estimated shape spec- ified by 8 L . The relation between sensor displacements on one side and array orientation and estimated shape on the other is

[iyx] = T-'[ 71 where Tis given by (A28). The objective is

Ax

AY 6 = [AX', Ay'] [ 1.

Substituting (36) in (37) and differentiating to 4, one obtains the stationary point

0 = [O; 11 (T')-'T-l[ AL 1 . Denote

( TT)-'T-' = [ 1; B a

a c

then (38) becomes

aTAL + c3 = 0

or

1 = --a'AL.

c

Denote

4 = h'K;' A L. (4-4) When a priori statistics are given for Ax, Ay, one can use them to obtain the best least square estimate. Assume that [ Ax'/ Ay'] ' is a zero-mean Gaussian vector with co- variance K, and assume, for the moment, that A L is known. Since (36) is a linear transformation, [ A LT, 41 ' is also a zero-mean Gaussian vector with covariance K L

= TK,TT. Partitioning KL into the components

one can write the conditional probability density of 4 given A L as

where from [15]

and

o: /AL = f2 - hlKT'h2. (48)

Since the received data do not give relevant information concerning 4, the best estimate is / .L+/AL, Therefore,

3 = hlKT'AL. (49)

When the elements of [ AX', Ay'] 'are independent with the same variance c r 2 , K, = a2 I , and

K L = u'TT', ( 5 0 )

so that h2 = h and K2 = K 1 . Thus, (44) and (49) are identical in form. At high SNR's, the estimate AL will be close to the true value AL, and its use in (49) may therefore be viewed as an approximation to the best a pos- teriori estimate of 4. It can also be shown that the mean square estimation error of 4 resulting from use of (49) equals the asymptotic (high SNR) value of CRLB (4).

Then, using the standard formula for the inversion of a D. Relation to Maximum Likelihood Estimation partitioned matrix,

The iterative procedure described in Sections III-A-III- 1

a' = (f- hTK;'h) - h'(Kl - hhT)-'

= (f - hTK; 'h) hT( f l l - hh')-'

= (f- hTK;'h) hT($K;' + f 2 ~ ) = h'K;'. (43 ) sources, Ks is given by

1 1 -- C is closely related to maximum likelihood estimation. At c f f combinations of SNR and observation time such that the

1 approximately Gaussian with covariance matrix Ks close

E ( 8 ) = f ( 4 (51)

differential time delay estimates 8 are reasonably accu- rate, one can expect the associated estimation errors to be

- K;'hhTKr' to CRLB( 8) . Then from (23),

1 - - hTK;'h where u is the vector of true parameter values. For three

732 IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-35, NO. 6, JUNE 1987

(SNR), MSE 0.5 CRLB

MSE CRLB

MSE CRLB

lo EiEB

3.18 . 6.85 . 1 0 - ~ 15.2 . 1 0 - ~ 14.4 . 10-~ 5 .74 . 1 0 - ~ 5.79 . 1 0 - ~ 3.72 . 1 0 - ~ 3.35 + 1 0 - ~ 4.78 . 1 0 - ~ 2.86 . 3.10 . 1 0 - ~ 6.79 1 0 - ~ 7.44 . 1 0 - ~ 3.57 . 1 0 - ~ 3.59 . 1 0 - ~ 2.57 . 1 0 - ~ 3.04 + 1 0 - ~ 4.08 1 0 - ~

1.36 . 1 0 - ~ 1.76 . 1 0 - ~ 4.53 . 1 0 - ~ 4.96 . 1 0 - ~ 2.10 . 1 0 - ~ 1.89 1 0 - ~ 1.30 . 10-~ 1.48 a 1 0 - ~ 2.08 . 1 0 - ~

6.80 . 1 0 - ~ 1.50 . 1 0 - ~ 3.62 . 1 0 - ~ 3.30 . 1 0 - ~ 1.39 . 1 0 - ~ 1.20 1 0 - ~ 9.96 . 1 0 - ~ 8.12 . 1 0 - ~ 1 . 1 1 . 1 0 - ~ 6.58 . IO-’ 6.58 I 1 0 - ~ 2.75 . 1 0 - ~ 3.00 . 1 0 - ~ 1.20 . 1 0 - ~ 9.91 . 1 0 - ~ 6.65 . 1 0 - ~ 7.20 . 1 0 - ~ 1.05 . 1 0 - ~

1.21 + lo-’ 2 .6 . 1 0 - ~ 5.76 . 1 0 - ~ 5.5 . 2.17 . 1 0 - ~ 2.18 . 1.41 . io-’ 1.23 - 1 0 - ~ 1.82 . 1 0 - ~

1.35 . 2.77 * lov4 6.66 . 6.60 . 2.77 . 2.55 . 1.83 . 1.70 + 2.27 *

1.27 . IO-’ 2.19 . 6.67 * 7.27 . 2.72 * IOw5 2.10 . IO-’ 1.38 3 1.40 . lo-’ 1.40 .

1 1 . -1

kl(MZ - llT) 0 0

K6 = 0 k2(MZ - l lT) 0

0 k3(MZ - l lT)

( 5 2 ) Since this is independent of u, the maximum likelihood

estimate is simply the value of u that minimizes [ 8 - f ( u ) ] TKS1 [ 8 - f ( u ) 1. The proposed iterative proce- dure minimizes [ 8 - f ( u ) ] T [ 8 - f ( u ) ] . Therefore, it differs from the maximum likelihood procedure only in replacing Ks by I , i.e., in ignoring differences in SNR associated with different sources (described by the kj) as well as dependences between the differential delay esti- mates caused by the fact that all sensor pairs use a com- mon reference [which generates the matrix MI - llT of (3)]. This omission can be corrected by replacing 6 and f ( u ) by w 8 and w f ( u ) throughout Sections III- A and 111-B. An example worked out in the next section suggests that for SNR’s above a very moderate level, the use of K6 = Z results in little performance degradation, at least when the three sources have similar power levels.

E. Example (Known Directional Reference) Monte Carlo simulations were carried out for the array

used in Section 11-D assuming the following conditions. 1) The array has six elements with nominal geometry

described by Fig. 2. d = 1, X,, = d/10, TW = 10. 2) There are three far-field calibration sources with the

same (SNR),, and bearings a1 = ( ~ / 3 ) , a z = T, a3 = ( 5 / 3 ) a.

3) The calibration sources are disjoint. 4) (SNR), is high enough that the estimation errors of

the differential delays 6; are approximately zero-mean Gaussians with covariance matrix CRLB( 8; ) =

5 ) (SNR), is zero outside of a frequency band W cen- tered at wo. It is constant within that band.

6) The Ax,,,, Aym are zero-mean i.i.d. Gaussian random variables each with variance a* = 2.5 X loF3.

Table I lists MSE’s obtained by averaging squared er- rors resulting from the iterative procedure over 300 in- dependent Monte Carlo trials. Cramer-Rao bounds for the

( l /k;) (MI - 1lT)- ’ .

Even at (SNR), = 0.5, the experimental results are above the Cramer-Rao bounds by a factor never greater than 2.5. At higher SNR’s, the discrepancy becomes so small that experimental error occasionally brings the MSE below the bound.

The iterative procedure used K6 = I . This is clearly not optimal. The principal conclusion to be drawn from the computation is that even a very simple algorithm can come quite close to the absolute lower bound on error. Some improvements could undoubtedly be achieved by using the full maximum likelihood procedure. For the range of SNR, considered here, it would be limited to less than 4 dB (less than 2 dB for SNR, > 1 ). Any such improve- ment would be bought at the expense of implementing power estimation procedures for the various calibration sources as well as somewhat greater computational com- plexity in the iterative algorithm.

APPENDIX A

From (15),

CRLB(AL1 * - * A L ~ M - ~ , 4 ) = TCRLB(Ax, Ay) TT N

;= 1 ]-ITT

+ (TT)-l ( kjSj j = 1

same quantities are listed for comparison. The matrix T-‘ can be written as

ROCKAH AND SCHULTHEISS: ARRAY SHAPE CALIBRATION 733

(A3) lim CRLB(a, r ) = - ( M A R T ) - ' 1

k m i n - + m k

where * S J A R ~ ( I W , R T ) - 1 ( A W

where R, S, and k are associated with the target and B is t = [last column of T - ' I

. . . & & I T (A4) the matrix on the right-hand side of (A10). a+ ' a4 Using (A6) and (A7),

From simple geometrical considerations [ 121 , STT- 'B(TT) - 'S =

U 2 rl rT. (A13)

tT. = [ -y2, -y3, ' ' 7 -yM, x39 * ' 3 x,, * (M)

(x : + Y : ) If one writes R, = [rl, r2] ,, it is easy to check that

Sj't = (r1), for any j .

1=2

Moreover, writing JA = 66 and noting that the vec- (A6) tor &rl is the first column of G R T ,

CRLB (A&, * * 9 AL2M-3, 4 ) The matrix in the squared brackets is a pseudoinverse of &RT. Substituting (A13) and (A14) into (A12),

lim CRLB(a, r )

+ [LTil)TETil j O]]-' (A8) him-

where = - k (MART)- ' + ; 0

1

From the results of Section 11-A,

lim (E + ( T ; ~ ) ~ T ; I ) - ' = o (A9)

where kmin = min, k,. Therefore, for N large enough for good calibration with known reference direction,

kmin 4

lim CRLB(AL1, * , AL2M-3 , 4 ) kmin -+

where

APPENDIX B RANK OF Ak IN (25)

As in earlier derivations, we deal only with the far-field case; the near-field case differs only in algebraic com- plexity. If we represent each set of arrays equivalent un- der translation and rotation by the member described by Axl = Ayl = 0 , 4 = 0, we need to consider only the case of known reference sensor location and known direction to a second sensor. In that situation, - 7

a a x a a y

as2 as2 a a x a A y

. (A16) 2 U = U k

Furthermore, the incremental error in a and r of the target satisfies

L

From (4)

as, as. 1 as, aaj a a x I a a y

c - = R,?, c [ - I-] = S T , (A17)

IEEE TRANSACTIONS ON ACOUSTICS, SPEECH, AND SIGNAL PROCESSING, VOL. ASSP-35, NO. 6, JUNE 1987 734

so that

Now suppose that Ak does not have full column rank. Then there exists a vector v such that

A k V = 0, V # 0. (A191

Since the RT are column vectors and the matrices ST have dimension (M - 1 ) X (2M - 3 ), we decompose v as foilows

v = (v , , 2/29 v 3 , v,') ( A20 ) T

where v, is a vector of dimension (2M - 3). From (A18) and (A19),

v,.R,T = S:U, j = 1, 2, 3. (A21)

Since the vj are scalars and the RT are not equal to zero, it follows that v, = 0 * vj = 0, j = 1 , 2, 3. Thus,'the vector v, cannot be a null vector. Multiplying both sides of (A21) by the nonsingular matrix a,

vj G R ~ = JS;[SFV, j = 1 , 2 , 3 . ( ~ 2 2 )

On the other hand, it was shown in Section I11 that the matrix

3

W = S , G Q j & S F (A23 1 is positive definite. The matrix Qj is a projection matrix onto the subspace orthogonal to the column space of C A R : . Therefore, since us # 0,

j= 1

v;wvs > 0 ( A24 1 or

3

vSTSjCQj&Sfvs > 0. ( M 5 ) j = l

But from (A22),

ej &s:us = vjej GR: = o j = 1 , 2 , 3.

(A26) Hence,

0s wvs = 0, T

which is a contradiction. Q.E.D.

APPENDIX C Following the same pattern as in (14)-(15), we write

(ALl, - - * 7 A L 2 M - 3 , 4 ) T

= T ( A x 2 , * * , A x M , A y 3 , * , A y M , A Y ~ ) * (A27)

where 4, as usual, represents the rotation of the array due to sensor location error. T is the transformation matrix

* = LoA I l,xJ T t B

where TA is the same matrix as in (15). Now let P1, P2, p3 be the angles of the calibration

sources with respect to the perpendicular to the actual line between the first and the second sensor.

Then

where from (14)

A Y 2 + = - . x 2

Defining

Then

or

A B

A B = T B A u

1 0

1

0 1

0

0 0 0 -

I I I I I I I I I I I I I I I I I I I I I I 1

I

(A33

Returning to the set of linear equations (25), and using the symbol B k to denote the equivalent of A k ,

Then from (A33),

B k ( A B ) , = C ( 8 - 6 k ) (A35 1 where

ROCKAH AND SCHULTHEISS: ARRAY SHAPE CALIBRATION 735

Using the partitioning in (A33),

where tk is the last column of the matrix (T-’) k and is given in (A4). Therefore, the last column of ( T i 1 ) k will be

(Z), = ( - 1 , -1, -1, ti) . T (A381

Now, from (A6),

sjTt = Rj’. ( A39 1 Therefore, by inspection, from (A18)

Bk(T)k = O , ( A40 )

which says that the last column of B k = Bk ( ~i~ ) k is equal to zero. Bk differs from the Ak [defined in (A18)] only in the fact that the ST have an additional column, whose ele- ments are ( as, /’( a ~ y , 1.

Therefore, we can write

Bk = [Ak akl (A411

where the vector ak is

Therefore, from (A37), (A40), (A41)

r ! 01

REFERENCES [l] Y. Rockah and P. M. Schultheiss, “Array shape calibration using

sources in unknown locations, Part I: Far-field sources,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-35, pp. 286-299, Mar. 1987.

[2] M. J . Hinich and W. Rule, “Bearing estimation using a large towed array,” J . Acoust. SOC. Amer., vol. 58, pp. 1023-1029, 1975.

[3] M. J . Hinich, “Bearing estimation.using a perturbed linear array,” J . Acoust. SOC. Arner., vol. 61, pp. 1540-1544, 1977.

[4] G . C. Carter, “Passive ranging errors due to hydrophone position uncertainty,” U.S. Navy, J . Underwater Acoust., vol. 29, pp. 79- 89, 1979.

[5] P. M. Schultheiss and J . P. Ianniello, “Optimum range and bearing estimation with randomly perturbed arrays,” J . Acoust. SOC. Amer.,

[6] E. Ashok and P. M. Schultheiss, “The effect of an auxiliary source on the performance of a randomly perturbed array,” in Proc. ICASSP ’84, Apr. 1985, p. 40.1.

[7] C. N. Dorny, “A self-survey technique for self-cohering antenna sys- tems,” IEEE Trans. Antennas Propagat., vol. AP-26, pp. 877-881, 1978.

[8] C. N. Dorny and B. S . Meaghr, “Cohering of an experimental non- rigid array by self-survey,” IEEE Trans. Antennas Propagat., vol.

[9] H. L. Van Trees, Detection, Estimation and Modulation Theory. New York: Wiley, 1968, pt. I, ch. 2.

[lo] W. J. Bangs, “Array processing with generalized beamformers,” Ph.D. dissertation, Yale Univ., New Haven, CT, 1971.

[ l l ] P. M. Schultheiss and E. Weinstein, “Lower bounds on the locali- zation errors of a moving source observed by a passive array,” IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-29, pp. 600- 607, 1981.

[12] Y. Rockah and P. M. Schultheiss, “Array shape calibration with sources in. unknown locations,” Yale Univ. Cen. Syst. Sci. Rep., Yale Univ., New Haven, CT, July 1985.

[13] C. R. Rao, Linear Statistical Inference and Its Applications, 2nd ed. New York: Wiley, 1973.

[14] H. Cramer, Mathematical Methods of Statistics. Princeton, NJ: Princeton University Press, 1946.

[15] Y. Rockah, “Array processing in the presence of uncertainty,” Ph.D. dissertation, Yale Univ., New Haven, CT, 1986.

V O ~ . 68, pp. 167-173, 1980.

AP-28, pp. 902-904, 1980.

According to (32), the matrix Ak( T i 1 ) k = Ak has full COl- umn rank.

Substituting (A43) into (A34) and using (A33) to define ( A IC) k, one obtains

Yosef Rockah, for a photograph and biography, see p. 299 of the March 1987 issue of this TRANSACTIONS.

Peter M. Schultheiss, for a photograph and biography, see p. 299 of the March 1987 issue of this TRANSACTIONS.