IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 42, NO. 1. JANUARY 1994 121

Instrumental Variable Approach to Array Processing in Spatially Correlated Noise Fields

Petre Stoica, Mats Viberg, Member, IEEE, and Bjorn Ottersten, Member, IEEE

Abstract-High-performance signal parameter estimation from sensor array data is a problem which has received much atten- tion. A number of so-called eigenvector (EV) techniques such as MUSIC, ESPRIT, WSF, and MODE have been proposed in the literature. The EV techniques for array processing require knowledge of the spatial noise correlation matrix that constitutes a significant drawback. Herein, a novel instrumental 1,ariable (IV) approach to the sensor array problem i s proposed. The IV technique relies on the same basic geometric properties as the EV methods to obtain parameter estimates. However, by exploiting the temporal correlatedness of the source signals, no knowledge of the spatial noise covariance is required. The asymptotic properties of the IV estimator are examined and an optimal IV method i s derived. Computer simulations are presented to study the properties of the IV estimators in sam- ples of practical length. The proposed algorithm is also shown to perform better than MUSIC on a full-scale passive sonar experiment.

I. INTRODUCTION HE AREA OF direction-of-arrival (DOA) estimation T using radar and sonar systems has received considerable

attention in the recent signal processing literature. As a con- sequence of this research effort, several methods have been developed for DOA estimation applications. Most of these methods, such as the maximum likelihood (ML) technique and the eigendecomposition-based approaches (e.g., MUSIC), rely on the assumption that the additive sensor noise is spatially white (or, more exactly, that its covariance matrix is known to within a multiplicative scalar); see, e.g. [I]-[lo] and the references therein. However, the noise received by the array is usually a combination of multiple noise sources such as flow noise, traffic noise, or ambient sea noise, and is often correlated along the array; see [ 1 11-[ 131. The aforementioned parametric DOA estimation methods may perform very poorly in a situation where the noise is spatially correlated with unknown correlation structure. In particular, these techniques may produce highly biased or spurious estimates in such a

Manuscript received October 14, 1991; revised January 4, 1993. The associate editor coordinating the review of this paper and approving it for publication was Prof. S . Unnikrishna Pillai. This work was supported in part by the Swedish Research Council for Engineering Sciences.

P. Stoica is with the Department of Control and Computers, Bucharest Poly- technic Institute, Bucharest, Romania, on leave at the School of Engineering, Uppsala University, Uppsala, Sweden.

M. Viberg was on leave at the Information Systems Laboratory, Stanford University, Stanford, CA 94305. He is with the Department of Electrical Engineering, Linkoping University, Linkoping, Sweden.

B. Ottersten is with the School of Electrical Engineering, Royal Institute of Technology, Stockholm, Sweden.

IEEE Log Number 9213278.

case; see, e.g., [141. These observations motivated a rapidly increasing interest in developing DOA estimation methods which are insensitive to the spatial noise color.

A number of researchers proposed to employ a parametric model of the noise. In [15], [16], a spatial ARMA model is used to estimate the noise covariance matrix as a preliminary step. This estimate is then used to whiten the sensor noise, thus enabling application of standard DOA estimation methods. In [17], [18], the noise covariance is instead parametrized as a linear combination of a given number of known matrices. The noise and signal parameters are then estimated simultaneously using nonlinear regression techniques. Proceeding this way re- quires estimation of the noise parameters, which increases the computational burden significantly and, in addition, is often of minor interest per se. Moreover, misspecification of the noise model or inherent errors in estimating its parameters, even when the noise model is correctly specified, may adversely affect the DOA estimation performance.

Another approach to the problem of spatially correlated noise is based on the assumption that the correlation structure of the noise field is invariant to a rotation of the array, or a certain linear transformation of the sensor output vector [19]-[22]. These methods do not require the estimation of the noise correlation function, but they may be quite sensitive to deviations from the invariance assumption made, and they are not applicable when the signals also satisfy the invariance assumption.

A class of methods which appear promising in alleviating the drawbacks mentioned above is based on the instrumental variable (IV) approach. See [23], [24] for general results on this approach. The IV methods (IVM’s) usually make no assumption on the noise correlation structure (except for stationarity), and do not require estimation of the parameters in a noise model. Furthermore, they are computationally much less expensive than the eigendecomposition or ML-based techniques. In addition, the IVM’s are readily amenable to adaptive (i.e., on-line) implementation, a property which is not shared by the other methods.

The initial steps towards a reliable IV-based solution to the DOA finding problem have been taken by Moses and Beex [25]. However, the IV-based method introduced therein has a number of limitations and drawbacks.

‘For example, the method in 1191, [20] relies on the assumption that the noise covariance matrix is Toeplitz, and is not applicable to the common case of a uniform linear array illuminated by uncorrelated emitter signals since the signal covariance is then also Toeplitz.

1053-5878X/94$04.00 0 1994 IEEE

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 42, NO. I , JANUARY 1994

Its use is limited to uniform linear arrays (ULA). The DOA estimates obtained with this method may have poor statistical accuracy. No attempt was made in [25] to study this aspect and to propose means of improving the statistical properties, for example, by a suitable selection of the “instrumental variables.” The full-dimension model used in [25] for the array output leads to an ill-conditioned numerical problem. This drawback can be eliminated by using an SVD-based low-rank approximation procedure, but this will increase the computational burden significantly.

Other DOA methods based (although indirectly) on the IV principle have been proposed by Tewfik [26] and F’rasad and Chandna [27]. In addition to the limitations mentioned above, these methods have the drawback of being applicable only under certain conditions on the spatial noise correlation Etructure. For example, the method in [27] is usable only when the sensor noise has a banded covariance matrix, with a “bandwidth” sufficiently small compared to the number of sensors in the array.

The main purpose of this paper is to propose an IV-based methodology that eliminates the limitations and drawbacks of the existing techniques. Specifically, these include the following:

I ) The array can have an arbitrary geometry. 2 ) A general class of weighted IV estimates is introduced,

and their large sample accuracy properties are estab- lished. Furthermore, the minimum-variance estimate in this class is derived and its implementation discussed.

3) A reduced-dimension model of the array output is used to avoid the numerical difficulties associated with em- ploying the full-dimension model. The problem of op- timally selecting the reduced-dimension model is also addressed.

4) No constraint is imposed on the noise covariance matrix. The finite-sample performance of the proposed DOA esti-

mation method is studied by means of numerical examples and compared with the corresponding performance of the MUSIC algorithm. The practical application of the proposed technique i \ demonstrated on a full-scale underwater passive localization experiment.

11. DAT.~ MODEL AND PROBLEM STATEMENT

Consider a situation in which ri narrow-band signals im- pinge on an array composed of 111 sensors. The sensor output t’ector can be represented by (see, e.g., [ 11-43])

where y ( t ) the data measured on the array at time instant t (an

m x 1 vector) e ( t ) a background noise term ( r n x 1) . r . k ( t ) the kth (baseband) signal waveform, as received at

the first sensor

a ( H k ) the transfer vector between z k ( t ) and y(t) (also called the ktli direction or propagation vector);

the parameter(s) of the lcth signal. a(&) = [ l ; p2 (&)e j~ ,T’ (~ ’ ) . ’ . ’ , p n t ( 0 I , ) e ~ W J m ( ~ ~ ) 1 Hk

In the expression for a ( H k ) above, w, is the center frequency of the signals, T ~ ( Q ~ ) is the time required by a signal with parameter(s) HI, to travel from the first sensor (the reference) to the pth sensor, and p,(Bk) represents the gain and phase adjustments by the pth sensor. In general, 01, can contain several unknown parameters, such as azimuth and elevation angle, polarization, etc. Although the results to be presented can easily be extended to include this more general case, we will for notational convenience assume that 01, is a real scalar, referred to as the ktli direction-of-arrival (DOA).

The model ( I ) can be rewritten in the following more compact form:

( 2 ) ?At) = A ( e ) 4 t ) + 4 t ) where

e = is,. . . . e,]’ A ( 6 ) = [.(HI). . . . . a( e,)] z( t ) = [ X I ( t ) . ’ ’ . .2,, ( t ) ]

(3 1 (4) ( 5 )

The dependence of A on B will often be suppressed for notational convenience.

Next. we introduce some assumptions on the data model ( 2 ) . A l : The number of sensors is greater than the number of

signals (m > n) . and A ( 0 ) has full rank 71 for all distinct Hk E 8; where 8 represents the parameter set of interest.

A2: The noise vectors { e ( t ) } form a sequence of tempo- rally independent random vectors with zero means and identical (unknown) covariance matrix

T .

Q = E[e(t)e*(t)]. (6)

Furthermore, e ( t ) is assumed to be circularly symmet- ric, i.e.,

E [ e ( t ) e T ( t ) ] = 0. (7)

and e( / , ) and z(.) are assumed uncorrelated for all t and s . Here, the superscript (.)* denotes complex conjugate transpose and ( .)T means ordinary matrix transpose.

A3: The signal waveforms are arbitrary quasi-stationary processes. i.e., the limit

(8) 1

,V-X N pk. = lirn - ~ [ z ( t - / ~ ) z * ( t ) ] t=l

exists for k 2 0 1281. The signals are assumed to exhibit a “sufficient” temporal correlation so that the matrix

[ P’ ] (9) p31

has full rank for some M 2 1.

123 STOICA er al.: INSTRUMENTAL VARIABLE APPROACH TO ARRAY PROCESSING

Assumption AI is commonly imposed in DOA estimation problems to ensure uniqueness of the estimates. In what concems Assumption A2 made on the noise term, most of the available methods for estimating the parameters in ( 2 ) , for example, the ML and the eigenvector-based methods (EVM), assume that e ( t ) is both temporarlly and spatially white, and ML assumes in addition that e ( t ) is complex-Gaussian distributed. The assumptions of Gaussianity and spatial uncor- relatedness, which often may not hold in practical applications, were relaxed in A2. The assumption of temporal independence, which is more likely to be satisfied, was retained. However, this assumption is not critical to the use of IVM’s (as will be explained later). Note that the EVM’s, such as MUSIC and subspace rotation methods (ESPRIT and TAM), can also be used in situations where the noise is temporally correlated. The relaxation of the temporal independence assumption, however, complicates the analysis of the statistical properties of both IVM’s and EVM’s (see Remark 2 in Section IV for some additional comments on this aspect).

Regarding Assumption A3, this assumption along with A2 reveal the distinguishing characteristic that enables separating the signals from the noise: the signals are assumed to be temporally correlated, whereas the noise is white. In some applications where one does not have accurate a priori in- formation of the signal bandwidth, or when the transmitted signal has a “truly narrow-band” spectrum, the bandwidth of the receiver may be chosen (much) larger than that of the signal. Provided the noise has a flat spectrum over this band, Assumption A2 and A3 are then satisfied. A real-world example of such a case is presented in Section VII. In other applications, one tries to match the receiver bandwidth to the signal in order to maximize the signal-to-noise ratio. In such a case, the signal and noise have similar temporal correlation properties, and Assumptions A2 and A3 do not hold as stated. However, if the correlation time of the signal is longer than that of the noise, the “extended” IVM (see Remark 2 ) can still be applied. Otherwise, the IV approach presented herein should not be used.

The problem to be treated in this paper is the estimation of the DOA’s, {e,}, from a batch of N measurements, {y(l) , . . . y ( N ) } . The number n of signals is assumed to be given. When n is unknown, it can be estimated from the data by the techniques of [15], [16], [27], [29] (see also the discussion following the description of the DOA optimal IVM in Section V).

111. GENERAL INSTRUMENTAL VARIABLE METHOD

Let y(t) denote the signals measured on a subarray com- posed of (n + 1) sensors of the array under consideration. The model for y(t) is obtained by selecting the appropriate rows of (2), and is denoted

where

is an (n + 1) x n matrix. The following assumption is made on the reduced-dimension model (lo).

A4: The last n rows of A are linearly independent. Fur- thermore, for any (hypothetical) 8 # B k , k = l, ‘ . . , n, the vectors ii( el), . . . , ii( e,), ii( 0) are linearly indepen- dent.

For some array configurations, it may not be a priori known how to select a subarray satisfying Assumption A4. In such cases, a trial-and-error procedure is required: select a subarray, check if A4 is satisfied (see Remarks 4 and 5 in Section V for details), and if not, try another choice of subarray. In other cases, several subarrays satisfying A4 may be known to exist. For example, for ULA’s, any selection of (n + 1) consecutive sensors leads to a reduced-dimension model that satisfies A4. In such cases, it is possible to pose a problem of subarray selection so as to maximize the DOA estimation accuracy. However, it tums out that the solutions to this problem depends on the noise covariance matrix and, therefore, cannot be implemented without knowledge of this matrix (details for ULA’s can be found in Section V).

It follows from Assumption A4 that there exists a unique n-dimensional vector b satisfying

or, equivalently,

[l b*].(&) = 0, i = 1,. ’ ’ , n. (13)

It also follows from A4 that { 81, . . . , e,} are the only solutions to (13). Indeed, if 8 # 0 k , k = l , . . . , n is another solution, then the (n + 1)-dimensional vector [l bTIT is orthogonal to n + 1 linearly independent (by A4) vectors, which is impossible. The conclusion of the previous discussion is that the DOA’s can be uniquely determined from the locations of the n nulls of the function

f ( 0 ) = I[1 b*]ii(8)12, 8 E 0. (14)

Thus, under the assumptions made, the problem of estimating { 0 i } can be reduced to the estimation of b.

The vector b can be shown to be the unique solution of the following overdetermined system of equations:

Rb= -r (15)

where

and where { r k } are m x 1-vectors and {Rk} m x n-matrices defined by

I24 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 42, NO. 1, JANUARY 1994

To see this, note from (12), (17), and Assumption A2 that b satisfies

Collecting (1 8) for k = 1. ' . ' , Ad. we obtain

A [r R] [:] = [ ... "1 [ p l ] A * [ i ] = O . (19)

0 A PAi

Since the matrices A and [PT . . . . .P l f lT have full rank, it follows that (12) and (19) are equivalent. Thus, the only solution to ( 1 9) (or 15) is given by b.

Let R and 3 denote consistent estimates of R and r obtained from the available data, for example,

. N

[3 k] = $ O(t)Y*(t) t=AJ+l

where

Y ( t - 1)

Y ( t - (21)

UsiFg R . 3 . and observing (15), define the following esti-

(22)

where J@ is a positive definite weigh$ng matrix (which may be data dependent), and lIzl\*- = z*Wz. Thus, & is obtained

W as the least-squares solution to the following overdetermined system of equations:

mate b of the parameter vector b:

i = arg rnin llR/~ + ill2- W

1 / 2 ..- - 1/2 w Rbx-W 3 (23)

& = - ( R * W R ) - l f i * W 3 . (24)

- 112 where = k * / * W . It is well known that 5 is given by

(The inverse matrix in (24) exists, at least for sufficiently large N : see Remark 1 in Section IV.) The explicit expression (24) for b will be convenient for the theoretic:l analysis that follows. However, the actual computation of b should not be based on (24). Numerical procedures which are less sensitive to roundoff errors, such as the QR algorithm, are available for determining the L S solution of (23); see, e.g., [30].

The solution b given by (24) is a general instrumental sariable estimate. The reason for general is that (24) defines a whole class of estimates-a member of the class correspynds to a specific choice of W and M . The reason for calling b an

instrumental variable estimate is explained as follows. From (10) and (12), the following equation is obtained:

Making use of the temporal uncorrelatedness of e ( t ) , one can estimate the vector b in (25) by employing a-weighted overdetermined IV method with weighting matrix W and IV vector d ( t ) . This approach leads exactly to the estimate (24). The reader is referred to [23], [24], and [28] for general discussions of IVM's.

IV. STATISTICAL ANALYSIS

The IV estimate & introducedjn the previous section de- pends on the weighting matrix W and the number of block equations M in (15). It also depends implicitly on the subarray selected to define the reduced-dimension model (10). These three items should be chosen by the user. To set a conceptual framework for their selection, it is most useful to establish the statistical properties of b and the DOA estimates:

Theorem 1 below proves that & is a consistent estimate of-b. An immediate consequence of this is the consistency of { e , } as estimates of { e , } . Theorems 2 and 3 establish the asymptotic distributions of the normalized estimation errors f i ( b - b) and n( 8, - Ok). The selection of @. M , and the subarray so as to minimize the estimation error variance is then discussed in Section V.

Theorem I (Consistency): Assume that the mamx W has a limit with probability 1 (w.p.l), as N tends to

infinity. This limit, which is denoted W , is assumed to be positive definite. In adpition, assume that conditions Al-A4 hold. Then, liiriN+m b = b (w.p.1).

Proof: Under the assumptions above, the output signal is second-order ergodic:

lirri [3 k] = [r R] (w.p.1). (27) .v-x

Furthermore, it follows from (18) and A4 that the matrix R has full column rank. Thus, the matrix R*WR is nonsingular (W > 0 by assumption). These observations, (15), and standard results on stochastic convergence (see, e.g., [3 11) imply that

lini = -(R*WR)-lR*Wr

= (R*WR)-lR*WR b 1%' -t x

= b (w.p. l ) , (28)

0 Remark 1 : It follows from the proof above that R W R > 0

0

which establishes the assertion. r * 1

for sufficiently large values of N .

STOICA et al.: INSTRUMENTAL VARIABLE APPROACH TO ARRAY PROCESSING 125

Remark 2 : In Assumption A2, the noise was required to be temporally white. If, instead, e ( t ) is temporally correlated, but only for a finite number of time lags, i.e.,

E[e(t)e*(t - IC) ] = 0 fork > L , (29)

then it is easy to see that the previous discussion on the pa- rameter vector b and its estimate b remains valid, provided that the first covariance lag in the matrix [r R] is [rL+1 R L + ~ ] . The IVM obtained by this simple modification can be used to determine consistent DOA estimates in the presence of both spatially and temporally correlated noise fields. However, we

U Theorem 2 (Asymptotic Distribution of b) : Under the as-

sumpLions of Theorem 1, the normalized estimation error f i ( b - b) has a limiting zero-mean complex Gaussian distribution with second-order moments:

will not pursue this modification in detail,

where

s = E[d)(t)d)*(t)l (32) O = [ I b*]Q[1 b*]* (33)

and where Q is the covariance matrix of the subvector E ( t ) of e ( t ) :

Q = E[E(t)E*(t)]. (34)

Proof: Using the notation

E ( t ) = E * ( t ) [;I (35)

we can write

Since 4(t) and ~ ( t ) are independent, E[XN] = 0. The covariance matrix of X ~ T is

N

+ E[~(t)~*(IC)E(f)lE[.*(k)l

+ E[4(t)4*(t)lE[E(t).*(t)l

k=t+l

(40)

I N - M

N 0E[4(t)4*(t)l. - -

Application of Lemma B3 of [24] shows that X,v has a limiting zero-mean Gaussian distribution with covariance OS. Inter alia, it follows that X,y is of order2 Op(l) . This implies that R and in (38) can be replaced by their limiting values R and W , respectively, without affecting the asymptotic properties of f i ( b - b ) . Hence, the asymptotic distribution of f i ( b - b) is also zero-mean Gaussian with covariance

c = a ( ~ * ~ ~ ) - l ~ * ~ ~ ~ ~ ( ~ * ~ ~ ) - l (41)

which is identical to the expression (30). Using (7), it is straightforward to verify that E[XAvXxT] = 0, which readily

The parameters of major interest are the DOA estimates which are obtained from b. The following result provides the asymptotic properties of these quantites.

Theorem 3 (Asymptotic Distribution of {e,}):: Let e,. i = 1.. . . , n, be the ith Zmallest (separated) minimum of the function f ( 0 ) = I[1 b ]ii(0)12. Then, a(e, - 0,) converges in distribution to a zero-mean Gaussian random variable with variance

leads to (31). 0

(42) a* (0, )&(4 )

2l[l b*]?i'(&)l'

where .'(e,) = aii(O,)/aO, and E(H,) is composed of the n last elements of i i ( 0 , ) .

Proof: See Appendix A. 0 The explicit variance formulas derived above can be used,

N for example, to determine confidence intervals for the esti- d)( t )c*( t ) } [i] (37) mated DOA parameters. More explicitly, let q be a Gaussian

random variable with zero mean and unit variance, and define Pa by Prob(Iq1 5 Pa) = cy for some given confidence level a . Then, using Theorem 3, one obtains an a-confidence

. { fi t=Al+l

= - ( R * @ R ) - q j * @ N

(38) interval for the ith DOA

where the last equality follows from (25) and (35). Let where Var(8,) is a consistent estimate of the theoretical variance of e,, as given by (30) and (42). Calculation of

deterministic notatlon, see [32, sect 2.91.

l N 4(1)4t) . (39) 'The symbols O p ( ) and o p ( ) denote "in-probabillty" versions of the X N = -

fi t=M+1

~

126 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 42, NO. I , JANUARY 1994

Var (e,) entails estimation of 8,. R, S . b. and 0. which can be done subsidiary to IVM DOA estimation. Note that 0 can be consistently estimated by [see (25)]

(44) 1 IV

6 = [1 6'1% CY(t)Y*(t)[l &*I*. t=l

V. OPTIMAL INSTRUMENTAL VARIABLE METHOD

In this section, the formal choice of the quantities which are at the user's disposal is considered. It is clear from (42) that optimization of DOA estimation accuracy is equivalent to minimization of C with respect to W and M . This problem is similar in nature to that of determining optimal instrumental variables for estimating the parameters of an ARMA model (see [24], [33]). We can therefore make use of the tools developed for analysis of the latter problem.

Theorem 4 (Optimal Choice of W ) : The matrix C (30) sat- isfies

c 2 CO = a(R"S-lR)-' v w > 0 (45)

(the inequality A 2 B . where A and B are Hermitian matrices, means that the difference ( A - B ) is a nonnegative definite matrix). Furthermore, the equality C = CO holds if W is chosen as

Proof:

S =

A straightforward calculation gives

(47)

APoA* APTA* ' . . AP;I-,A* APIA* APoA*

AP.II-IA* . . .

Thus, S is positive definite if, for example, Q is so (which is a weak condition indeed).

Next, note the following readily verified equality:

c - CO = n[(R*WR)-IR*W - (R*S-IR)-lR*S-l] x S[(R*WR)-lR*W - (R'SS-lR)-lR*S-l]*. (48)

Since S is positive definite, the inequality (45) follows im- mediately from (48). It also follows from (48) that the choice (46) of W makes C equal to its lower bound in (45).

0 Note that the optimal weighting matrix (46) depends on

the (unknown) signal and noise covariances. Fortunately, a consistent estimate of (46) can be obtained from the available sample as

where

S =

rnr-1 " ' r, J

and where

r, = 1 '-v y ( t - lk)y*(t). t = k + l

Note from Theorem 2 that use of the weight (49) instead of (46) in the IV estimator (24) does not affect the asymptotic accuracy. Thus, the choice W = 3-l of the weighting matrix is still asymptotically optimal.

Next, consider the minimization of estimation error variance with respect to M . It might be expected intuitively that the estimation accuracy increases with increasing M. For the IV estimator with an arbitrary choice of W, this property does not necessarily hold (see Section VI). For the IV estimator with the optimal choice (46) or (49) of W, however, the estimates become more accurate with increasing M , as shown in the next theorem.

Theorem 5 (Optimal Choice of M ) : Consider the covari- ance matrix C o ( M ) defined in (45) (the dependence of CO on hl is stressed by notation, for the covenience of exposition). The matrices C o ( M ) for M = 1.2, . . . . form a nonincreasing sequence i.e.,

(52) C"(1) 2 C"(2) 2 . " . Proofi First, note the following nested structures of the

matrices R ( M + 1) and S ( M + 1):

(53)

(54)

The exact expressions for the blocks X , Y , and Z are not important for this proof.

Next, make use of a standard formula for the inverse of a partitioned matrix (see e.g., Appendix A of [24]) to write

R * ( M + l )S- l (M + 1)R(M + 1) =

= R*(M)S-l(nl)R(M) + [X - z*s- ' (M)R(M)]* x [Y - z*s-'(M)Z]-1[X - z * s - 1 ( M ) R ( M ) ] . ( 5 5 )

Since S ( M ) > 0 for all M (by assumption), it follows that (see [24, Appendix A])

Y - Z*S- ' (M)Z > 0 forM = 1,2:.. . (56)

Thus, from (55) .

R*(hl+ l ) S - l ( M + 1)R(J4 + 1) 2 R * ( M ) S - ' ( M ) R ( M ) . (57)

0 In theory, M should thus be chosen as large as possible.

In practice. however. the value of M should be limited for

The stated inequalities, (52), follow from (57).

STOICA et al.. INSTRUMENTAL VARIABLE APPROACH TO ARRAY PROCESSING 127

at least two reasons. First, the developed asymptotic theory, which asserts that the accuracy should increase with increasing M , implicitly assumes that M is much smaller than N . Note, for instance, that the inverse matrix S-' may not even exist, unless M < N / m . Second, the computational burden corresponding to the IV estimator (24) increases linearly with M , and might become prohibitive if M is chosen too large. General rules for selecting M are not available. The value of this parameter should be chosen by a compromise between increasing the statistical accuracy and decreasing the computational burden, on an application-dependent basis.

Remark 3: Using the same technique as above, it is possible to prove that CO monotonically decreases also with increasing number of sensors m. provided the subarray remains fixed. The accuracy improvement for the optimal IVM may, however, often be only marginal. The reason for this is intuitively clear: increasing m (or M ) results in a larger dimension of the IV vector $( t ) used in the IVM. This should improve the accuracy only marginally if M (or m) is appropriately chosen, so that the dimension of $ ( t ) i s already "sufficient." For the IVM using nonoptimal weighting, the corresponding improvement may be more significant; see Example 6.2. It should also be mentioned that the variance of the DOA estimates may be substantially lowered by increasing m if the subarray aperture is also increased. However, this type of improvement is not

Finally, we consider the problem of selecting the sensors used to form the subarray output vector $( t ) . For an array with arbitrary geometry, it appears difficult to derive any general result on subarray selection. In fact, in such a case, there may exist only one subarray which satisfies Assumption A4, and thus, the problem of subarray selection does not appear. For ULA's, however, a result on the optimal subarray selection so as to maximize the estimation accuracy can be proved for a special class of subarrays. In the case of ULA's, any subarray composed of (71. + 1) consecutive sensors satisfies Assumption A4. There are (711 - 7 ~ ) possible subarrays which can be chosen in this fashion. The vectors b corresponding to all these subarrays are identical, a fact which will be used in what follows. To see this, note first that the matrix A associated with a ULA is given by

A = I "U'

guaranteed to be monotone. 0

1 . . . 1 ] (58) p J I' 71

e3 (177 - 1 ) P 1 (771 - 1) P n

where pk is a simple function of 6'1;. the distance d between two adjacent sensors and the kth signal wavelength Ak. viz.

(59)

Thus, if we let A, correspond to the subarray composed of the sensors k to k + 71. then

A, = AlA"l (60)

where

A = diag [&PI. . . . . e J P 7 ~ ] . (61)

-* -* It follows from (60) that the matrices A, , A,, . . . have the same (one-dimensional) null space, which establishes the uniqueness of b.

The problem of subarray selection to optimize the DOA estimation accuracy is the topic of the next theorem. Theorem 6 (Optimal Subarray Section for ULA's): Consider a uniform linear array, and let the kth subarray consist of the sensors k , IC + 1,. . . , k + n. The subarray within this class which minimizes the DOA variance (42) is given by

where Q, denotes the Q matrix corresponding to the kth subarray.

Proof: See Appendix B. 0 If the covariance matrix of the noise Q is Toeplitz, then

Q, does not depend on k . In such a case, it follows from the above theorem that one should not be concemed with subarray selection since none of the subarrays under consideration outperforms the others in terms of estimation accuracy. When the covariance matrix Q is not Toeplitz, a qualitative use of Theorem 6 leads to the intuitive recommendation of selecting the sensors with smallest noise variances to form the reduced- dimension model. A quantitative use of Theorem 6 would require detailed knowledge on the noise covariance matrix Q , which in general is unavailable in practice.

To summarize, we propose the following. Optimal IVM for DOA Estimation

Step 1: Choose M and the subarray used to define the reduced-dimension data vector (some guidelines for this choice can be found above):

Step 2; Compute r k (for k = 0 . . . . , M - 1) using (S I ) and [f R] using (20). (The results of the former computation may be profitably used to simplify the latter significantly). Also, compute the inverse Cholesky factor S of the matrix S made of { f h } (since S is block Toeplitz, S can be efficiently computed using a Levinson-type algorithm; see, e.g., (241.)

Step 3: Solve the following overdetermined system of (IV) equations in an LS sense:

--1/2

---1/2

Step 4: Using b provided by the previous step, estimate { 6'i}

as the locations of the n smallest local minima of the function3

f ( H ) = I[1 b * ] i i ( 6 ' ) 1 2 6' E 8. (64)

Remark 4: The matrix R appearing in (63) may be nearly

The sources are closely spaced (inf+k(6'; - 6',1 is small compared to 1 / 7 n ) or the matrix (9) is nearly rank deficient. These cases correspond to near violation of assumptions A 1 and A3, respectively.

rank deficient in the following situations.

'In the case of a uniform linear subarray, 4, could altematively be determined by-a root-finding technique applied to the polynomial 1 + [ z . z 2 . . . . . :"]b . This will not alter the asymptotic properties of the estimate, but may improve the finite sample accuracy and reduce the computational burden.

128 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 42, NO. 1. JANUARY 1994

The rows of A included in A are (nearly) linearly depen- dent (violation of the first part of assumption A4). The number of sources n has been overestimated.

The rank of k can be tested when solving (63) by the QR algorithm. A "practical" rank less than n indicates that one of the above situations has occurred. 0

Remark5: If no problem is encountered in solving (63) , but the function (64) has more than n separated local minima, one can infer that part two of Assumption A4 was violated.0

Remark 6: Concerning Step 4 of the procedure, it might be tempting to consider in this step a generalized function of the form

ml(6') (65)

where g(6') > 0 for all 6' values, and try to choose g(6') so as to optimize the corresponding DOA estimation accuracy. However, it can be shown as in [5] that any loss function of the form (65) leads to DOA estimates with the same asymptotic accuracy. Consequently, there is no function g(6') which could outperform (in large samples) the simplest choice y(6') = 1.0

Finally, we point out a possible modification of the IV approach to DOA estimation described previously. This mod- ification is obtained by considering the full-dimension model instead of the reduced-dimension model [i.e., taking $(t) = y ( t ) , A = A. and E ( t ) = e( t ) ] . The (modified) matrix R has dimensions hfm x m, but its rank is still equal to 71.

The IV system of equations (63) should then be solved using rank approximation followed by pseudoinversion; see [ 341. Failure to recognize this fact may lead to a numerically ill- conditioned estimation problem. The requirement of low-rank approximation and/or pseudoinversion of a matrix of relatively large dimensions leads to an increase in the computational burden. This possible extension of the IVM technique is not pursued in any further detail herein.

VI. NUMERICAL EXAMPLES In this section, the performance of the proposed IVM is

investigated through numerical evaluation of the theoretical estimation error variance. Furthermore, the applicability of the asymptotic results in finite samples is tested by computer simulations. In all experiments, the scenario is as follows.

Array: A uniform linear array (ULA) with half-wavelength spacing is employed. The manifold vectors are normalized to have length fi and first element 1; see (58).

Signals: Two equipowered point sources are located at 0" and 5" with respect to array broadside. The baseband signal waveforms are generated as correlated complex AR( 1) processes:

where (-1 = 0 . 9 ~ ' . ~ ' ~ . a 2 = 0.9eo.3JT. and where [ ( t ) = [E1 ( t ) E2 (t)]' is a sequence of circularly symmetric complex Gaussian vectors with covariance matrix

It is easy to verify that the signal covariances for this case are given by PI, = P0Dk where

r l 0.9 1

Noise: The noise consists of a spatially correlated back- ground noise with exponentially decaying correlation among the elements. The noise covariance matrix has ilcth element

(69) Q i k = ff 2 a I i- k I ( X P ) (i- k )

where ( Y = 0.9 and where a2 is adjusted to give the desired signal-to-noise ratio (SNR).

IVM-DOA: The first three elements of the array form the subarray from which g ( t ) is obtained. The DOA estimates are determined by a root-finding technique as described in Step 4 in Section V. Two different schemes are considered: IVM-LS applies an identity weighting W = I to the IV equations (24), and IVM-OPT uses the optimal weighting W = 3-l (50). The number of sources is fixed at 71, = 2 .

MUSIC: For comparison, the MUSIC algorithm is applied to the scenario of Example 1 below. No spatial prewhitening of the data is performed since the noise covariance is assumed unknown. The signal subspace dimension is chosen equal to the known number of sources, n = 2.

The noise correlation model (69) corresponds to a first- order spatial autoregressive model. Note that the same noise covariance is used in [35] and [36].

The sample statistics in the examples below are based on 200 independent trails, each consisting of N = 2000 snapshots. The empirical root mean-square (rms) error of 6'1

is calculated and displayed with the symbols x and 0 . The theoretical standard deviations, as obtained from (42), are shown with solid and dashed lines.

Example 6.1-Varying M : In the first example, the number of sensors is fixed at vi = 4 and the element SNR (signal- to-noise ratio), defined as 1010g,~ ({P~}ll/o*), is 10 dB. The IVM-LS and IVM-OPT techniques are applied with different numbers of covariance lags M ranging from 1 to 8. In Fig. l(a), the IVM sample statistics and theoretical standard deviations for the source at 0' are depicted. The empirical rms error is well predicted by the theory for the IVM-LS method, whereas for IVM-OPT, this is the case only for small values of A f . This is due to an increasing bias of the estimates that apparently is induced by the weighting for large M. The empirical standard deviation of the estimates does indeed follow the theoretical curve quite closely. For reference, the MUSIC cost function is displayed in Fig. l(b), the dotted vertical lines indicating the true locations of the sources. Clearly, MUSIC cannot resolve the sources in this scenario, and in addition, there is a spurious peak near 26O.O

Example 6.2-Varying m: In this example, the number of covariance lags is fixed at M = '2. and the number of sensors is varied from 7r i = 3 to r r i = 12. In all cases, the IVM subarray is composed of the first three sensors. Fig. 2 shows the sample statistics and theoretical standard deviations for the source at 0". Comparing Figs. l(a) and 2, one sees that the

STOICA et al.: INSTRUMENTAL VARIABLE APPROACH TO ARRAY PROCESSlNG

0.6

1

09

08

07

06

0 5

0 4

0 3

0 2

0 1

'10 5 0 5 10 15 20 25 30 35

1 , , , , - , , ,

MUSIC I

5 10 15 20 25 30 35

DOA (deg)

(b)

Fig. 1. (a) Theoretical and empirical rms error of IVM-DOA estimate in degrees versus the number of covariance lags; (b) five realizations of thi: MUSIC spatial pseudospectrum. True DOA's indicated by dotted vertical lines.

dependence on m and M is similar. A large value of T I L results in a bias in the IVM-OPT estimates, which in tum increases

Example 6.3-Varying SNR: In the last example, the values M = 2 and m = 4 are used, and the SNR is varied from 4 to 20 dB. The results are presented in Fig. 3. There is a close agreement between the theory and the empirical results in this case, which is expected since the quotient m M / N is small. It is seen that the rms error decreases approximately as l /SNRdB for both IVM-LS and IVM-OPT. U

As seen in the previous examples, the asymptotic ex- pressions predict the performance of the estimators with a reasonable accuracy in these scenarios. However, for "large" values of the quotient m M / N , an unpredicted bias is intro-

the empirical rms error. 0

129

SNR (dB)

Fig. 3. m s error of first DOA estimate versus signal-to-noise ratio.

duced in the estimates which results in a larger rms error than predicted by the estimate. As indicated in Fig. l(a) and 2, this effect appears to be more prominent for the IVM-OPT method. Indeed, the IVA-LS method can give more accurate estimates for large rri and/or M . Note, though, that the appropriate weighting can significantly decrease the rms error for smaller values of rnM/N.

The MUSIC method clearly fails in the scenario of Fig. I(b). The MUSIC technique requires that the number of sensors is increased to ni 2 9 or, for 7n = 4. that the SNR is 220 dB for resolving the signals. The comparison with the MUSIC method is, of course, not entirely fair since MUSIC is not designed to work in spatially correlated noise fields. Other algorithms have been proposed to alleviate the effects of correlated noise; see Section I . Unlike the technique presented herein, none of these exploits the temporal correlation of the signals. Thus, a comparison with these methods is also not realistic.

The theoretical analysis showed that the performance of the estimators can be improved by appropriate weighting and increasing M . It is therefore interesting to examine how this effect depends on the scenario. It is our experience that the presence of closely spaced DOA's (compared to the array aperture) emphasizes the dependence on the weighting and on 711 and M. Placing :he poles of the signals closer to the unit circle has the opposite effect. For pure sinusoidal signals (the limiting case), the IVM with identity weighting and M = 1 performs, in general, close to the optimal IVM with M 2 1. If, on the other hand, the signal spectra are more flat, a large value of M is recommended. It should also be noted that the convergence rate towards the asymptotic ( N >> 1) estimation error variance tends to be slower for signals with narrow spectra.

VII. AN UNDERWATER PASSIVE LOCALIZATION APPLICATION

This section presents some results of an application of the proposed technique to data collected during a full-scale underwater experiment in the Baltic sea. A uniform linear array (ULA) of seven hydrophones, spaced 0.9 m apart, is used to record the wavefield generated by a source located at a known azimuthal angle equal to 16.8' relative the array broadside. The array is not calibrated and the spatial correlation structure of the background noise is unknown. When applying the estimation methods, the array is assumed to be a perfect ULA

130 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 42, NO. 1, JANUARY 1994

0 9 ~

a

104 , , , , , . , , I

d B I I I IVMOPT --

MUSIC / "

I 1 0 3

102

10'

100

10 1

10 2

50 100 150 200 250 300 350 400 450 500 1034 ' ' ' ' I

Hz

ia)

d B 1 O 3 T ' A ' ' ' ' ' '

10-5 5 10 1 5 20 25 30 35 40 45 i o Hz

ib)

lines; (b) spectrum of filtered and resampled signal. Fig. 4. (a) Spectrum of received signal; passband indicated by dotted vertical

with omnidirectional sensors, and for the MUSIC method, the background noise is assumed spatially white. It should be remarked that using an isotropic noise model, see [13], does not improve the behavior of the MUSIC estimates in this experiment.

The raw data consist of 1000 real-valued snapshots in each sensor, and the original sampling frequency is 1 kHz. A typical FFT spectrum of the received signal is depicted in Fig. 4(a). A dominant peak at the frequency 280 Hz is present in all sensors. Since the speed of propagation is approximately 1500 m/s, this frequency corresponds to a wavelength of about 5.2 m. The total array length is thus only one wavelength, giving a 3 dB beamwidth of about 50".

In order to apply the proposed technique, the raw data are first filtered through a bandpass filter, with passband 25G300 Hz. An additional high-pass filtering is also applied in order to eliminate strong low-frequency components of the signal. The last 800 samples (to allow for filter initialization) of the filtered signal are then Hilbert transformed and resampled by a factor of 10, thus providing 80 samples of complex data, sampled at the frequency 100 Hz. An FFT of the resulting signal is shown in Fig. 4(b). Note that the passband of the original signal is shifted down to the range 50-0 Hz (i.e., it is mirrored) by the resampling. As seen in the figure, the resulting spectrum is fairly Aat with a dominant peak which hopefully originates from the emitter signal. Hence, it is believed that the assumptions A2 and A3 are essentially satisfied. Fig. 5(a) shows the normalized spatial pseudospectra corresponding to MUSIC and the optimally weighted IVM with M = 1. TO use as much array aperture as possible, a two-step procedure is used when applying the IVM. A rough estimate of the

-7

DOA (deg)

ib)

Fig. 5. (a) IVM and MUSIC spatial pseudospectra. One batch of N = 80 samples. True location indicated by a dotted vertical line; (b) as above, but data divided into five batches of S = 16 samples each.

source location is first performed by using a subarray formed from sensors 1 and 2. A local search (to avoid the ambiguity) around this preliminary estimate is then obtained by using a subarray formed from sensors 1 and 7. To get an impression of the statistical variability of the DOA estimates, and also of their behavior in samples of smaller length, the available data are also split into five contiguous sets of 16 samples each. The corresponding results are depicted in Fig. 5(b). Clearly, the proposed IVM provides significantly less biased DOA estimates than those obtained via MUSIC in the studied case, whereas the standard deviation of the estimates is similar for the methods.

The previous experiment has been repeated with 16 other emitter positions. The obtained results are similar to those presented above. Although IVM did not outperform MUSIC uniformly, it did indeed give better results in most scenarios. It should be noted that other important sources of errors, apart from the unknown noise covariance, are expected in the present data set. These include imperfect sensor positions and characteristics, as well as deviations from the narrow-band plane wave model. The effect of the latter cannot be mitigated by the proposed approach.

It should be stressed that the chosen preprocessing of the data is dictated by the desire to eliminate the need for knowledge of the spatial noise color. If the noise covariance is known, other techniques can be used, which might give better results. Let us also mention that MUSIC and traditional beamforming give essentially identical results on these data. This should not be surprising since only one emitter is present in the experiment.

STOICA et al.: INSTRUMENTAL VARIABLE APPROACH TO ARRAY PROCESSING 131

VIII. CONCLUSION

This paper presents an algorithm for narrow-band DOA estimation in the presence of ,spatially correlated noise with unknown correlation structure. The signal waveforms are assumed to exhibit a “sufficient” temporal correlation, whereas the noise is assumed to be temporally white. The IV principle is invoked to obtain an estimate of a vector in the noise subspace. The estimated noise subspace vector is then used in a “traditional” manner, such as in MUSIC, to obtain the final DOA estimates. A reduced-order model is employed for the noise subspace to reduce the computational requirements and the risk of spurious DOA estimates.

The asymptotic (for large amounts of data) variances of the DOA estimates are derived, and optimal choices of user- defined quantities are discussed. The behavior of the proposed IV-DOA estimation scheme is examined through computer simulations. The effect of optimally weighting the IV equa- tions is studied, and the applicability of the asymptotic vari- ance formula in finite samples is investigated. A full-scale underwater experiment is also included.

Finally, it should be pointed out that the IVM proposed in this paper represents only one of several ways to exploit temporal correlation of the signal waveforms. As already mentioned in the text following Remark 6, one could exploit the full-dimension model and noise subspace to estimate the DOA’s. Another possibility to further exploit the temporal correlation of the signal is to introduce explicitly a dynamical model of the signal waveforms, for instance, AR models. It would then be possible to combine traditional prediction-error methods (or IV methods) from system identification, see [24], [28], to parametric DOA estimation techniques. Naturally, this will increase the accuracy of the DOA estimates, at least when the specified signal model is correct. However, additional work is required to determine if this can motivate the inherently increased computational burden associated with such an approach.

APPENDIX A PROOF OF THEOREM 3

Since f (0) has a minimum point at 6’ = 8,. we have

For large values of N, 8, is close to the true DOA of the i th source (Theorem 1). Thus, it is possible to expand the left-hand side of (A. l ) in a Taylor series around 0i:

0 = f’(6’i) + f^”(Q,)(ei - 0;) + o(lf ’(6’;)l). (A.2)

Let f”(S;) denote the probability limit of ?(e;). Assuming f (Si) # 0, we have from (A.2) - / I

Some straightforward calculations give

?(e,) =2Re{[1 b*]iii’(Si)ii*(6’;)[l i*]*} (‘4.4)

=2Re{[l i*]ii’(Oi)ii*(6’i)[O (i-b)*]*} 64.5) =2Re{[1 b*]ii’(Oi)ii*(Oi)[O (i-b)*]*} + o( lb - bl)

(-4.6)

and - / I f (Si) = lim 2Re{[1 i*]ii’(6’i)ii’*(6‘i)[l &*I*

N-w

+ [l i*]ii”(6’i)ii*(ei)[l &*I*} (A.7) =2l[ l b*]ii’(Sz)l2. (A.8)

In the equalities (AS) and (A.8), we have used (13). Combin- ing (A.3), (A.6), and (A.8) gives

Re {[l b*]ii’(O;)iii*(6’i)[O (6 - b)*]*} ~ [ i b*]ii’(ei)12 JN(8, - 6’i) = -JN

+ O ( J N 1 6 - bl). (A.9)

Theorem 2 now shows that fl(8; - 0 i ) has a limiting zero-mean Gaussian distribution. To evaluate the asymptotic variance of the numerator in (A.9), note first that for any two scalars U and W.

(A.lO) 1 2

Re (U) Re (w) = -[Re (uw*) + Re (uv)].

Thus,

NE(Re {[I b’]ii’(S;)ii*(di)[O (i - l ~ ) * ] * } ) ~ = TI + 572 (A.11)

where

N TI = T [ [ l b*]ii’(Si)l2ii*(&)

.E([! b - b ] [? b - b ]*)ii(6’,) (A.12)

([l b*]ii’(0i))2ii*(Si) 2

. E ( [? b - b ] [! b - b ] T ) i i * T ( S i ) } . (A.13)

It is readily verified that

and from (31), it is clear that

(A.15) ”30 lim T2 = 0.

In conclusion, it follows fro,m (A.9), (A . l l ) , (A.14), and (A.15) that the variance of m(0; - 6’;) in the limiting distribution is given by

ii* ( 6 ’ ; ) (A.16)

2l[l b*]ii’(Sz)12 NE(& - oil2 =

which immediately gives (42).

I32 IEEE TRANSACTIONS ON SIGNAL PROCESSING. VOL. 42, NO. I , JANUARY 1994

APPENDIX B PROOF OF THEOREM 6

Let RI, and CI, denote the matrices R and C corresponding to the kth subarray. Also, let Ak denote the matrix composed of the last n rows of Ak. It follows from (18) and (60) that

AP1 AP1

APM APn1

= [ ; ] A; = [ ; ] A-(k-l)A; = UA-(” l )A* 1 ’

(B.1) The matrices W and S in the expression of Ck do not depend on k . Making using of this observation and of (B.l), one can write

where

Inserting (B.2) into the numerator of the variance formula (42), we obtain

Next, note from (60) that

i - i I , (H,) = L ? @ , ) e J P J - 1 ) .

Thus.

Using the fact that b does not depend on k (the subarray index), along with (BSt(B.7) above, we obtain the following expression for var (Qi) corresponding to the kth subarray:

where only f l k depends on k . The proof is thus concluded.

ACKNOWLEDGMENT We wish to thank the Swedish National Defense Research

Institute (FOA 2) for providing the data used in Section VII. We are also grateful to the reviewers for their constructive comments to an earlier version of this manuscript.

REFERENCES

R. 0. Schmidt, “Multiple emitter location and signal parameter esti- mation,” in Proc. RADC Spectrum Estimation Workshop (Rome, NY), 1979, pp. 243-258. D. Johnson, “The application of spectral estimation methods to bearing estimation problems,” Proc. IEEE. vol. 70, pp. 1018-1028, Sept. 1982. M. Kaveh and A. J. Barabell, “The statistical performance of the MUSIC and the minimum-norm algorithms in resolving plane waves in noise,” IEEE Trans. Acoust. Speech Signal Processing. vol. ASSP-34, pp. 331-341, Apr. 1986. K. Sharman and T. S. Durrani, “A comparative study of modem eigen- structure methods for bearing estimation-A new high performance approach,” in Proc. 25th IEEE Conf. Decision Conrr. (Athens, Greece), Dec. 1986, pp. 1737-1742. P. Stoica and A. Nehorai, “MUSIC, maximum likelihood and Cram& Rao bound,” IEEE Trans. Acousr. Speech Signal Processing, vol. 37, pp. 72C741, May 1989. -, “MUSIC, maximum likelihood and Cram&-Rao bound: Further results and comparisons,” IEEE Trans. Acoust. Speech Signal Process- ing. vol. 38, pp 2140-2150, Dec. 1990. B. Ottersten and M. Viberg, “Analysis of subspace fitting based methods for sensor array processing,” in Proc ICASSP ‘89 (Glasgow, Scotland), May 1989, pp. 2807-2810. B. D. Rao and K. V. S. Hari, “Performance analysis of ESPRIT and TAM in determining the direction of arrival of plane waves in noise,” IEEE Trans. Acoust. Speech Signal Pi-messing, vol. 37, pp. 1990-1995, Dec. 1989. -, “Performance analysis of root-music,” IEEE Trans. Acoust. Speech Signal Processing, vol. 37. pp. 1939-1949, Dec. 1989. M. Viberg and B. Ottersten, “Sensor array processing based on subspace fitting,” IEEE Trans Signal Processing. vol. 39, pp. 11 10-1 121, May 1991. B. F. Cron and C. H. Sherman, “Spatial correlation functions for various noise models,” J . Acousr. Soc.. Ame r... vol. 34, pp. 1732-1736, Nov. 1962. E. M. Arase and T. Arase, “Correlations of ambient sea noise,” J . Acoust. Soc. Amer-., vol. 40, no. I , pp. 205-210, 1966. R. J. Thalham, “Noise correlation functions for an isotropic noise field,’’ J . Acoust. Soc. Amer.. vol. 69, pp. 213-215, 1981. M. Viberg, “ Sensitivity of parametric direction finding to colored noise tields and undermodeling,” Sigrial Proc.essing, vol. 34, no. 2, 1993. J. P. Le Cadre, “High resolution methods in presence of noise correlated sensor outputs,” preprint, 1988. -, “Parametric methods for spatial signal processing in the presence of unknown colored noise tields,” IEEE Trans. Acoust.. Speech, Signal Processing. vol. 37, pp. 965-983, July 1989. J . F. Bohme and D. Kraus, “On least squares methods for direction of arrival estimation in the presence of unknown noise fields,” in Proc. ICASSP’88 (New York. NY), 1988, pp. 2833-2836. D. Kraus and J. F. Bohme, “Asymptotic and empirical results on approximate maximum likelihood and least squares methods for sensor array processing,” in Proc,. ICASSP’90 (Albuquerque, NM), 1990, pp. 2795-2798. R. T. Williams, A. K . Mahalanabis, L. H. Sibul, and S. Prasad, “An efficient signal subspace algorithm for source localization in noise fields with unknown covariance,” in Proc. ICASSP’88 (New York, NY), 1988, pp. 2829-2832. S. Prasad. R. T. Williams. A. K . Mahalanabis, and L. H. Sibul, “A transform-based covariance differencing approach for some classes of parameter estimation problems,” IEEE Trans. Acoust. Speech Signal Processing. vol. 36. pp. 631441. May 1988. A. Paulraj and T. Kailath, “Eigenstructure methods for direction-of- anival estimation in the presence of unknown noise fields,” IEEE Trans. Acoust. Speech Sigtial Processing. vol. 33, pp. 806-81 I , Aug. 1985. F. Tuteur and Y. Rockah. “A new method for detection and estima- tion using the eigenstructure of the covariance difference,” in Pro(,. ICASSP’YY (Tokyo, Japan), 1986. pp. 281 1-2814. T. Soderstrom and P. Stoica, 1ristr.umental l/ariahle Methods for System Identifrc.ation. Berlin: Springer-Verlag, 1983. -, System Identification. R. L. Moses and A. A. (L.) Beex. “Instrumental variable adaptive array processing.” IEEE Trans. Aernsp. Elec.trrm. Syst.. vol. 24, pp. 192-202, Mar. 1988. A. H. Tewfik, “Robust direction finding in the presence of unknown colored noise,” preprint. 1988. S. Prasad and B. Chandna, “Signal subspace algorithms for direction of

London: Prentice-Hall, 1989.

STOICA et a / INSTRUMENTAL VARIABLE APPROACH TO ARRAY PROCESSING 133

amval estimation in the presence of a class of unknown noise fields,” in Proc. ICASSP’88 (New York, NY), 1988, pp. 2913-2916.

1281 L. Ljung, System Identifrcation: Theory for the User. Englewood Cliffs, NJ: Prentice-Hall, 1987.

1291 J.-J. Fuchs, “Estimation of the number of signals in the presence of unknown correlated sensor noise,” in Proc. ICASSP’RY (Glasgow, Scotland), May 1989, pp. 2684-2687.

[30] G. H. Golub and C. F. Van Loan, Matrix Computations, 2nd ed. Baltimore, MD: Johns Hopkins Univ. Press, 1989.

[31] W. F. Stout, Almost Sure Convergence. [32] K. V. Mardia, J. T. Kent, and J. M. Bibby, Multivariate analysis.

London: Academic, 1979. [33] P. Stoica, B. Friedlander, and T. Soderstrom, Instrumental variable meth-

ods for ARMA models,” in Control and Dynamic Systems-Advances in Theory and Applicarions. Vol. 25. (C. T. Leondes, Ed.). New York: Academic, 1987, pp. 79-150.

134) P. Stoica. T. Soderstrom, and F. -N. Ti, “Asymptotic properties of the high-order Yule-Walker estimates of sinusoidal frequencies,” IEEE Trans. Acoust. Speech Signal Processing, vol. 37, pp. 1721-1734, Nov. 1989.

[35] J . P. Reilly, K. M. Wong, and P. M. Reilly, “Direction of arrival estimation in the presence of noise with unknown, arbitrary covariance matrices,” in Proc. ICASSP’89 (Glasgow, Scotland), May 1989, pp. 2609-26 12.

[36] M. Wax, “Detection and localization of multiple sources in noise with unknown covariance,” IEEE Trans. Acoust. Speech Signal Processing. vol. 40, pp. 245-249, Jan. 1992.

New York: Academic, 1974.

Petre Stoica received the M.Sc. and D.Sc. degrees, both in automatic control, from the Bucharest Polytechnic Institute, Romania, in 1972 and 1979, respectively. In 1993 he was awarded an Honorary Doctorate by Uppsala University, Sweden.

Since 1972, he has been with the Department of Automatic Control and Computers at the Polytechnic Institute of Bucharest, Romania. where he currently holds the position of Professor of System Identification and Signal Processing.

Dr. Stoica is a Corresponding Member of the Romanian Academy.

Mats Viberg (S’87-M’90) was born in Linkoping, Sweden, on December 21, 1961 He received the M.S degree in applied mathemdtics in 1985, the Lic Eng. degree in 1987, and the Ph.D degree in electncal engineenng in 1989, all from Linkoping University, Sweden

He joined the Division of Automatic Control at the Department of Electrical Engineering, Linkoping University in 1984, and since November 1989 he has been a Research Associate From October 1988 to March 1989, he was on leave

at the Informations Systems Laboratory, Stanford University as a Visiting Scholar. From August 1992 to August 1993 he held a Fulbright-Hayeq grant scholarship as a Visiting Researcher at the Department of Electrical dnd Computer Engineering, Bngham Young University and at the Informations Systems Laboratory, Stanford University. His research interests have focused on statistical signal processing and its application to array signal processing. system identification, and communicatlon systems

Bjorn Ottersten (S’87-M’89) was bom in Stockholm, Sweden, in 1961. He received the M.S. degree in electrical engineering and applied physics from Linkoping University, Linkoping, Sweden, in 1986, and the Ph.D. degree in electrical engineering from Stanford University, Stanford, CA, in 1989.

From 1984 to 1985, he was employed with Linkoping University as a Research and Teaching Assistant in the Control Theory Group. He was a Visiting Researcher at the Department of Electrical Engineering, Linkoping University during 1988.

In 1990 he was a Postdoctoral Fellow associated with the Information Systems Laboratory, Stanford University, and from 1990 to 1991 he was a Research Associate in the Electrical Engineering Department at Linkoping University. In 1991 he was appointed Professor of Signal Processing at the Royal Institute of Technology (KTH), Stockholm, and he is currently head of the Department of Signals, Sensors & Systems at KTH. His research interests include stochastic signal processing, sensor array processing, system identification, time series analysis, and spread spectrum communications.