IEEE TRANSACTIONS ON SIGNAL PROCESSING. VOL. 43, NO. 7. JULY 1995 1557

Direction Finding Using Noise Covariance Modeling Benjamin Friedlander, Fellow, IEEE, and Anthony J. Weiss, Senior Member, IEEE

Abstract-We consider the problem of direction finding in the presence of colored noise whose covariance matrix is unknown. We show that the ambient noise covariance matrix can be mod- eled by a sum of Hermitian matrices known up to a multiplicative scalar. Using this model, we estimate jointly the directions of arrival of the signals and the noise model parameters. We show that under certain conditions, it is possible to obtain unbiased and efficient estimates of the signal direction. The Cramer-Rao bound is used as the principal analysis tool. Computer simulations using the maximum likelihood estimator provide a validation of the analytical results.

I. INTRODUCTION RRAY signal processing is used in such diverse areas as A radar, sonar, communications, and seismic exploration.

Usually the parameters of interest are the direction of arrival (DOA) of the observed signals and the signal waveforms. Many high resolution methods for estimating these parameters have been proposed and analyzed. Most of these methods are variations of the MUSIC algorithm [3], which requires that the noise covariance matrix be known up to a multiplicative factor. In many radio frequency (RF) systems the dominant noise is the thermal noise that is approximately equal in all the channels. In these cases, the correct noise covariance is a scaled identity matrix (or a diagonal matrix, if the channels are unequal), and the assumption of known noise covariance is justified.

In this paper, we focus on the case where the dominant noise is external (ambient) noise. Ambient noise is the dominant noise source in RF systems operating in the HF and VHF frequency bands, and in most sonar systems [ 2 ] , and its presence introduces correlation between the noise processes of the different sensors.

When the correlation function of the noise is known, the direction-finding algorithm can be usually modified in a straightforward manner to take it into account. In practice it often happens that the noise environment is unknown, and is in fact changing slowly with time. The question then arises how to solve the estimation problem without the complete statistical characterization of the noise. Simply ignoring the presence of possible noise correlation may introduce significant bias into the direction estimates, causing unacceptable errors. This

Manuscript received October 6. 1993; revised November 21, 1994. This work was supported by the United States Army Research Office under Contract DAAL03-91 -C-0022. sponsored by U.S. Army Communications Electronics Command. Center for Signals Warfare. The associate editor coordinating the review of this paper and approving it for publication was Prof. Isabel Lourtie.

B. Friedlander is with the Deptartment of Electrical & Computer Engi- neering, University of California, Davis, CA 95616 USA.

A. J. Weiss is with the Electrical Engineering Dept., Tel-Aviv University, Tel-Aviv, 69978, Israel.

IEEE Log Number 941 1992.

problem has led to the development of various methods that attempt to take into account the presence of unknown noise correlation.

One such method is the covariance differencing technique proposed in [4]. In this method, two measurements of the array covariance are required, where the unknown noise covariance remains invariant, while the signal component undergoes some change between the two measurements. Then, the unknown noise covariance is eliminated by subtraction. However, as stated in [9], in many practical situations, only one measure- ment of the covariance is available.

Recently, a MAP approach was presented in [9], [IO]. This method assumes that the noise covariance matrix is completely unknown, except for the fact that it is a Hermitian positive definite matrix. This method is reported to perform better than MUSIC and maximum likelihood (ML) that are applied using the (wrong) assumption that the noise covariance is a scaled identity matrix. In [lo], it is shown that the MAP estimator proposed in [9] is in general inconsistent, except in special cases. A similar formulation that assumes completely unknown noise covariance was used in [SI. Wax advocated a technique based on Rissanen’s MDL principle for detection and localization of signals in noise with completely unknown noise covariance. However, this technique is asymptotically biased, as is the case in [9].

The authors of [7] and [ 121 proposed a parameterization of the noise covariance based on an AR model. The method proposed in [7] and the technique suggested in [12] for DOA estimation are both limited to linear uniform arrays. A formulation that is close in spirit to the formulation developed in this work is presented in [ 5 ] . While [5] concentrates on numerical techniques for DOA estimation, we focus on the CramCr-Rao bound (CRB) and the best performance that can be achieved.

In this work, we show that in the presence of ambient noise, the noise covariance matrix can be represented as sum of known matrices that are dependent on the array configuration. These matrices are multiplied by a an unknown constant whose size is determined by the intensity and the spatial distribution of the noise. Using the CRB, we examine the number of unknown noise parameters that can be estimated, the accuracy of the DOA estimates as a function of the number of noise parameters, and the effect of unknown noise covariance on the DOA estimates. We also evaluate by computer simulations the performance of a maximum likelihood estimator (MLE), and show that its performance is very close to the CRB.

We note that an approach that is similar in spirit to the one in this paper, was presented by LeCadre [13]. In that paper, it was proposed to parameterize the noise by an autoregressive (AR)

1053-587)3/95$04.00 0 1995 IEEE

1558 EEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 43, NO. 7, JULY 1995

model. The parameters of this model are jointly estimated with the signal parameter using a ML method. The parameterization

processes, and that the noise and the signals are uncorrelated, the data covariance matrix is - -

used in the present paper follows from a Fourier series representation of the spatial spectrum of the noise.

The structure of the paper is as follows. In Section II, we formulate the problem and describe the noise model on which our work is based. Section I1 summarizes the CRB for the joint estimation of the DOA's and the noise model parameters. A specific estimation algorithm based on the ML approach is described in Section IV. Section V contains numerical examples involving the CRB, and results of Monte Carlo runs using the estimation algorithm. Section VI has some concluding remarks.

11. PROBLEM FORMULATION In this section, we present the models for the data and the

noise, and define the set of parameters that is the subject of this study. We start by introducing some necessary notation.

A. Notation

We use lower case boldface letters to denote vectors and upper case boldface letters to denote matrices. In addition, we use the following conventions throughout the paper. (.)* conjugate

transpose Hermitian transpose

Re{.} real part Im{.} imaginary part E{ .} expected value tr{-} trace diag{x} I1 . II Euclidean norm X x Y X 8 Y Kronecker product I identity matrix Xn vecCX]

(.IT U H

diagonal matrix whose diagonal is the vector x

Hadamard (Schur) product

nth element of the vector x a concatenation of the columns of X.

B. The Data Model

Next, we describe the data model for the narrowband DOA estimation problem. To simplify the exposition our discussion is confined to azimuth-only systems, i.e., the sensors and signals are assumed to be co-planar.

We consider an M-element array of sensors and L nar- rowband far-field signal sources, and define the M x 1 vector a( 8) to be the complex array response for a source at direction 8. The array manifold is defined to be the continuum M = {.(e) : 8 E [ -T ,T ] } .

The outputs of the M array elements at the kth sample are arranged in an M x 1 vector

R e ~ { x ( k ) x ~ ( k ) ) = A P A ~ + E (3)

is the noise where P is the signal covariance matrix and covariance matrix.

111. MODELING OF THE NOISE COVARIANCE MATRIX

The noise in most receiving systems consists of internal noise and external noise. The internal noise is produced by the electronic equipment and includes thermal noise and weak versions of other signals in the system, such as clocks and local oscillators. The external noise is defined as an unwanted random signal that is intercepted by the sensor. If the system is designed well, so that there is no coupling between the receiving sensors, and the thermal noise is the main source of internal noise, then a good model for the internal noise covariance is a scaled identity matrix, PI. This model assumes that the thermal noise intensity is the same in all sensors and that there is no correlation between the noise at any two sensors.

Next, we define an appropriate model for the external noise. The noise intensity as a function of azimuth 8 at a given time instant k is a random function denoted by v(O(k). Therefore, the noise vector at the array output (assuming external noise only) is given by

n(k) = a(8)v(elk) de. (4) 1: We assume that v(8lk) satisfies the following relations:

E{w(elk)v*(olj)} = E(e)d(O - a ) d k j (5) E{.(elk).(.lj)} = 0 (6)

where S k j is the Kronecker delta, 6 ( ~ ) is Dirac's delta and & ( e ) is the spatial power density function of the noise. Using these relations we get for the external noise covariance

E{n(k)nH(k)} = E{ 1; a(e)v(elk) do/" a(a)v(alk) do --K

(7) J-li

Since E ( @ ) is a periodic function it can be represented by Fourier series as follows:

03

€ ( e ) = ce cos(ee) + se sin(t0) (8) e=0

where

Assuming that the signal vectors and the noise vectors are realizations of stationary, zero mean Gaussian random

In general, the Fourier series representation of &(e) will have an infinite number of terms. However, in most practical cases

FRIEDLANDER AND WEISS: DIRECTION FINDING USING NOISE COVARIANCE MODELING 1559

of interest the noise power spectral density function is smooth and varies slowly with direction. Thus, E ( # ) can be usually approximated quite well by a Fourier series with a small number of terms, say, L.

Substituting (8) in (7) and assuming that for 1 > L the coefficients of the Fourier series are zero, we get

L

E{n(k)nH(k)} cpzp + spgp ( 1 1)

(12) ~p = 1; a(s)aH(#) cos(^^)

(13) ~f = i_il a(s)aH(o) sin(P0) d#.

All cp and % p are Hermitian. It can be shown. that for any 2- D array with omnidirectional sensors, the following relations hold:

f = O

where

- A

- A

Im{%p} = Im{%f} = 0. P = 2n (14) Re{%p} = Re{%!} = 0. P = 2n + 1 (15)

where n = 0. 1 . 2 , . . . Also, for any 1-D array (linear array) with omnidirectional sensors, the following relations hold:

Im{Cc} = O. I = 2n + 1 (16) Re{%p} = 0, P = 2n. (17)

We therefore conclude that for linear array with omnidirec- tional sensors, the matrices cp for odd P, and the matrices % p

for even P, vanish. We note that in this paper we have considered the case

of a 2-D noise field. This model is appropriate, for example, in representing the underwater acoustic noise environment in shallow water. The approach presented here can be extended in a straightforward manner to the 3-D case.

Remark: Note that it is impossible to uniquely identify the coefficients c( and S F if the noise matrices are not linearly independent. In general, the noise matrices are confined to a M2-dimensional real linear space due to the fact that these matrices are Hermitian. However, for certain arrays this space may be considerably smaller. Consider, for example, a uniform linear array consisting of omnidirectional sensors. In this case, the matrices % p , % p are all Toeplitz, and therefore they are confined to a ( 2 M - 1)-dimensional real space. Thus, in the general case we cannot estimate more than M2 noise parameters and in the case of uniform linear array with omnidirectional sensors we are limited to 2 M - 1 noise parameters. Further restrictions on the number of noise parameters are presented later.

IV. THE CRAM~R-RAO BOUND

In this section, we present the CRB for estimating the directions of arrival, the entries of the signal covariance matrix and the noise covariance matrix.

The CRB [ 1 ] provides a lower bound on the covariance ma- trix of any unbiased estimator. In other words, the difference

between the covariance and the CRB is a positive semi- definite matrix. The CRB is expected to be a good predictor of estimation performance for large values of N (i.e., many snapshots). In fact, under mild regularity conditions, the MLE achieves the CRB asymptotically, as the number of snapshots tends to infinity.

Given N independent samples of a zero mean Gaussian process x whose statistics depend on a parameter vector 4, the elements of the Fisher information matrix (FIM) are given by

where R is the covariance matrix of x. The CRB equals the inverse of F = [F,,], the FIM [l].

In the sequel we will always compute the H M and the CRB for N = 1.The results for N > 1 can always be obtained by dividing the CRB values by N , or by dividing the standard deviations (the square roots of the CRB) by 0.

Substituting (1 1 ) in (3) we obtain the relation

A

L J

L = O J =O

where, for notational convenience, we introduce

and

We assume that the signals and the noise are Gaussian and uncorrelated. However, the signals may be correlated or even coherent (perfectly correlated.)

We define our parameter vector as

cp = [er,pT.$']T (22)

where is the vector of L signal DOAs, p is a parameter vector that specifies the entries of P, and q is a vector of the noise Fourier coefficients. The FIM can be partitioned into blocks, where each block is associated with one or two of the parameter vectors included in 4. In the appendix, we show that these blocks are given by

Fee =2Re (PAHR-lAP) x (AHR-'A)?

)T} (23)

{ +(PAHR-~A) x (PAHR-~A

+ ( A H R - ~ ; I ) ~ 8 ( P A H R - ~ A ) } Q ~ (24)

Fep =Q3{ (AHR-'AP)T 8 (A"R-'A)

Feq, =2Re -U diag PANR-'X:,R- 'A)} (25)

F p p =Qt [ (AHR-'A)* 8 (AHRelA)] QF (26)

Fpr), =Qt[(R-'A)'@ (E,R-'A)H]vec{I} (27)

Fr), 9, =tr{R-'XtR-'XJ}. (28)

1560 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 43, NO. 7, JULY 1995

We used here x to denote the Hadamard product of two matrices (i.e., their element by element product also referred to as the Schur product). The symbol 63 denotes the Kronecker product, vec{B} denotes a vector consisting of a concatenation of the columns of the matrix B, and diag{B} is a column vector containing the diagonal elements of the matrix B. The operators ( . ) T , (.)*, and ( . ) H denote transposition, conjuga- tion, and conjugate transposition, respectively. Finally, Qt and Qs are constant matrices whose definitions can be found in [ I l l , and

Equations (23)-(28) provide a compact closed form set of formulas for computing the CRB.

How Many Noise Parameters Can be Estimated? Our ba- sic assumption in this work is that both the noise and the signals are realizations of Gaussian processes. Therefore, all the information about the parameters of interest is embedded in the data covariance matrix R. Intuitively, any estimation scheme should look for parameter values that "best" fit the model of R. The number of independent parameters that define R is M 2 since the covariance matrix is Hermitian. Therefore, intuitively, the maximum number of parameters that can be estimated is M 2 . On the other side, the number of parameters that should be estimated is as follows. The L directions of arrival, the L2 independent parameters of P, and the J = J + 1 noise parameters. Thus, the problem is well posed only if

M 2 > L + L 2 + J I . (30)

Therefore, we can not expect to estimate the directions of arrival properly, if there are more than M 2 - L2 - L noise parameters. This intuitive observation is supported by the CRB. The FIM becomes singular when the relation (30) is violated.

{ q j } . If we collect N independent data samples the log likelihood function is

Q(q5) = -Nlog{det{rR}} - Ntr{R-'D} (32)

where

x( k)XH ( I C ) . k = l

N (33)

The maximizer of Q(q5) is the MLE for the problem at hand. One can try to maximize (32) directly, or find ways to reduce the dimension of the multidimensional search implied by (32). In general, the dimension of the search is L+ L2 + 5. However, we can reduce the search to L + J by solving for P in terms of 0 and r ) and then substituting back to Q(q5). The derivation follows the steps in [6].

Maximization of (32) with respect to 0 , q and P is equiv- alent to the following two-step procedure. We first maximize the function with respect to P for a fixed 0,s and substitute for the resulting P as a function of 6,9 and the data D back into the log-likelihood function, resulting in a function to be maximized over only the 0 ,q parameters. A necessary condition for an extremum of Q(q5) with respect to P is that the partial derivatives of Q(q5) with respect to the entries of P be equated to zero. Jaffer [6] shows that this condition is equivalent to

AHR-'[D - R]R-lA = 0. (34)

In order to proceed we need an expression for the inverse of R. Define

Using a well known identity, we get

and V. ESTIMATION ALGORITHMS

Once we have a parametric model for the problem we can R-1 = E-1 - E-1/2A(pAHA + I)-lpAHE-1/2 (37)

apply various estimation techniques. Each of these algorithms has different advantages and limitations compared with the others. In this section, we derive two such algorithms: an

which yield

R-'A = E-1/2A[I - [PAHA + I)-'PAHAl . %

exact ML algorithm, and a suboptimal but computationally = X:-1/2A(PAHA + I)-l[(PAHA + i) - PAHA] more efficient version.

= E-'/ZA(pAHA + 1)-1. (38)

Using this relation, the condition in (34) becomes A. Maximum Likelihood Approach

It is well known that the MLE is asymptotically (with large number of samples) unbiased and statistically efficient. (pAHA+1)-1AE-1/2(D -R)E-1/2A(pAHA+1)-1=0 Since we assumed that the signals and the noise processes are

by

(39)

(40)

Gaussian, the probability density function of the data is given and reduces to

AE-l12(D - R)E-'12A = 0.

Substituting for R from (3) we get

where 4 is the parameter vector. In our case q5 includes the AHEC-1/2DE-'/2A =AHE-l12(ApA + E)x-1'2A directions of arrival, the entries of P and the noise parameters =AHApAHA + A H A . (41)

FRIEDLANDER AND WEISS: DIRECTION FIM)ING USING NOISE COVARIANCE MODELING 1561

Thus, the solution for P is given by by expressing the noise parameters estimates using only the directions of arrival. Unfortunately, we could do this only for a single noise parameter, in which case we get

p = ( A H A ) - ' [ A H D A - A H A ] ( A H A ) - I ,

D !!Z-1/2DE-1/2 (42) tr{HEil/2SE,1'2}. (56)

1 70 = -

Define M - L

R e A P A ~ + E. (43) The generalization of this result is still an open problem.

Then, the maximization of Q(4) is reduced to the maximiza- tion of

Ql(e,q) = -log{det{R}} - tr{R-'D} (44)

with respect to 0 ,q . Unfortunately, (44) requires a matrix inversion. We can simplify (44) following the steps described in [6] with the required modifications. Using (37), we have

& - I D = z - 1 ~ - ~ - 1 / 2 A ( p A f f A + 1)-1pAHx-1/2~.

(45) Substituting (42) in PAHA + I we get

(46) P A H A + I = ( A H A ) - p D A .

VI. FAST SUBOPTIMAL ALGORITHM

In the previous section, we described the exact ML al- gorithm for the problem at hand. The MLE is known to be asymptotically unbiased and efficient. Unfortunately, this method requires a search in a parameter space whose di- mension is L + 5. We now derive a suboptimal algorithm, using intuitive arguments, whose performance is acceptable in most cases. The search associated with this algorithm is of dimension L.

We first note that

lim D = R = A P A ~ + C . (57) N+m

Taking the inverse of this result yields Therefore, a reasonable cost function for minimization is

Q(+) = )ID - A P A H - Ell2 = Ild - Bg - rg1I2 (47) (58) (PAHA + 1 ) - 1 = ( A H D A ) - 1 AHA.

Substituting (47) in (45) and evaluating the trace of the where matrices, we get

tr{R-'D} = tr{D} - t r{AWAP}.

Using (46), one gets

t,r{i'AHA} = tr{(AHA)-'AHDA - I} = tr

where

G A(AHA)-'AH.

Using this result, (48) becomes

n (48) d = vec{D}

B ~ A * ~ A

GD} - L (49)

The minimization of (58) with respect to q is obtained by (50) choosing

i j = (rHr)- lrH(d - Bg). (63) A tr{R-lD} = tr{HD} + = I - G. (51)

The matrices G and H are recognized as projection matrices on the column space of A and on the null space of AH, respectively. we get

Substituting (42) in (43) and multiplying by E-1/2 yields

Substituting (63) back to (58) and using the definition

n 5 I - r(rHr)-lrH

Q = IlWd - Bg1I2.

(64)

(65) ~ - 1 / 2 R ~ - 1 / 2 = GDG + H. (52) The minimization of (65) with respect to g is obtained by choosing

In addition

(53) g = (BHrIB)-IBHnd. (66) det{C-l/2RC-1/2} = ~ det { R}

det{E} '

Hence, (53) and (52) yield Substituting (66) back to (65) we get

det{R} = det{E}det{GDG + H}. (54) &(e) = (III(1- B(BHIIB)-'BHII)dl12. (67) Substituting (54) and (51) in (44) and eliminating the constant L that does not affect the maximization, we get The minimization of (67) requires a search of dimension L

that may be considerably less than the search dimension of Q 2 ( 8 , q ) =- log{det{E}}-log{det{GDG+H}}-tr{HD}.

The maximization of Q2 requires a reduced dimension search. We have tried to reduce the search dimension even more

the MLE. It is clear from the method of derivation that the (55) estimation error reduces to zero as N + cc. Therefore,

our estimates are asymptotically unbiased. The finite data performance of the estimator is examined in the next section.

1562 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 43, NO. 7, JULY 1995

RANDOM ARRAY, 5 SENSORS, 2 SOURCES 103

g 10'

0 2 4 6 8 IO 12 14 16 18 10 '

RANDOM ARRAY, 5 SENSORS, 3 SOURCES 102

t 10' r

100 :

I 2 4 6 8 10 12 14

I O ' 1

NUMBER OF NOISE PARAMETERS NUMBER OF NOISE PARAMETERS

DOA standard deviation as a function of number of noise parameters. Fig. 1. Source at 0" (solid line) and at 72" (dashed line).

DOA standard deviation as a function of number of noise parameters. Fig. 3. Source at 0" (solid line) and at 72" (dashed line).

ARRAY CONFIGURATION

considered earlier. It is clear from Fig. 3 that in this case the sensitivity to the number of noise parameters is increased as compared to Fig. 1, due to the presence of more sources.

Fig. 4 depicts the results of Monte Carlo experiments with the MLE. The root mean square error (RMS) is compared

algorithms were tested. One algorithm estimates the noise

0 4 -

0 3 -

0 2 - - E 0 0 1 - - with the CRB (solid line) developed in this paper. Two ML L c 0 -

X AXIS (half wavelength units]

parameters and is using the correct noise model (which has three parameters). The other algorithm assumes that the noise covariance is the scaled identity matrix. The RMS of the first algorithm is shown by small circles, while the RMS of the second algorithm is shown by small "plus" symbols. We used a four element array whose configuration is shown in Fig. 5. The spatial noise intensity is shown in Fig. 6. One source is daced at azimuth 0" and the other at different directions from

Fig. 2. Array configuration for Figs. 1 and 3. Source directions are " r e d counterclockwise from the z-axis.

100 to 700 in steps of 100. The are uncorrelated with each other and with the noise. The signal power is the same for

VII. NUMERICAL EXAMPLES

In this section, we present some numerical examples to gain more insight into the dependence of DOA estimation accuracy on various system parameters, and to validate the results of our analysis by computer simulation.

Fig. 1 depicts the standard deviation of the estimated directions of arrival of two sources, based on the CramCr-Rao results. In this example, we used an array with a random configuration, as shown in Fig. 2. The sources in this case are at 0" and 72". The variance of each of the signals is 10. The noise parameters (Fourier coefficients) are = 1,1/4,1/9,1/25 .... We assume that 100 snapshots were used to produce the estimates. It is clear from Fig. 1 that as the number of noise parameters is increased, the accuracy of DOA estimates degrade. However, this degradation is mild for up to 13 noise parameters.

Fig. 3 depicts the standard deviation of the estimated directions of arrival of three sources, based on the CRB. In this example we used the same array whose configuration is shown in Fig. 2. The sources in this case are at O", 30°, and 72". All the other parameters are the same as in the two source example

- - both sources and it is 10 times the noise power at each sensor (SNR=lOdB). Each experiment is based on 100 snapshots of data. We have performed 50 Monte Carlo experiments for each point of the plot. It can be seen that there is a very good agreement between the performance of the MLE with the correct noise model, and the CRB. The agreement between the CRB and the MLE with white noise assumption is not as good. This result is, of course, expected.

Fig. 7 depicts the results of Monte Carlo experiments with the suboptimal estimator. The RMS (small circles) is compared with the CRB (solid line). We used again the four element array whose configuration is shown in Fig. 5. The spatial noise intensity is shown in Fig. 6. One source is placed at azimuth 0" and the other at different directions from 10" to 70" at steps of 10". The sources are uncorrelated with each other and with the noise. The signal power is the same for both sources and it is 10 times the noise power at each sensor (SNR=IOdB). Each experiment is based on 100 snapshots of data. We have performed 50 Monte Carlo experiments for each point of the plot. It can be seen that there is a good agreement between the performance of the estimator and the CRB, except for sources that are closely spaced. The results for the closely

FRIEDLANDER AND WEISS: DIRECTION FINDING USING NOISE COVARIANCE MODELING 1563

0 3 , -

1

FIRST SOURCE :-_I d .

1 0

I O 20 30 40 50 60 70 SECOND SOURCE DOA ldegl

SECOND SOURCE 5 ,

, -2 -1 -0.8 -0.6 -0.4 -0 2 0 0.2 0.4 0 6 0 8 1

X AXIS [half wavelength units]

-031

Fig. 5 . counterclockwise from the .r-axis.

Array configuration for Fig. 4. Source directions are measured

NOISE INTENSITY VS. DIRECTION

-1 , 1

-$OO -150 -lW -50 0 50 100 150 200

DIRECTION [degl

Fig. 6 Noise power density function for Fig. 4

spaced emitters can be improved considerably by using more snapshots.

Fig. 8 depicts the CRB for estimating DOA for two sources using the array of Fig. 2. The solid line corresponds to

FIRST SOURCE 10

2 I O 20 30 40 50 7 -i__ .A 60 70

SECOND SOURCE DOA ldegl

SECOND SOURCE

2 - 0

0 0 ’

I O 20 30 40 50 60 70 SECOND SOURCE DOA [degl

Fig. 7. the array in Fig. 5 ) .

Performance of the suboptimal estimator compared to CRB (using

RANDOM ARRAY, 5 SENSORS, 2 SOURCES I I

-i i

....~~~.. .~~~~~ . . . ~ ~ ~ ~ ~ I O I --.A

(I 2 4 6 8 10 12 I 4 16 18

NUMBER OF NOISE PARAMETERS

Fig. 8. Comparison between estimation performance with known noise covariance (dashed line) and unknown noise covariance (solid line) as a function of the number of noise parameters. based on the CRB (using the array in Fig. 2 ) .

unknown noise parameters, while the dashed lines correspond to perfectly known noise covariance. From this plot it is clear that the optimal (unbiased, minimum variance) performance of DOA estimates is only slightly affected by the absence of knowledge of the noise covariance, if it can be described by a small number of parameters. In the above case. as long as the noise covariance can be described by less than 13 parameters, DOA estimation accuracy is essentially unaffected by the lack of knowledge of the noise parameters. The number of noise parameters that can be estimating without degrading DOA estimation accuracy decreases rapidly as the number of sources increases.

Finally, we examine how the shape of the noise spatial power density function affects the DOA estimation. We have chosen the following power density function

E = 1 + 77,. cos(H) (68)

where ‘rn. is referred to as the “modulation index.” We used a circular array with eight elements, and inter-element spacing of half a wavelength. We placed a single source at three different

EEE TRANSACITONS ON SIGNAL. PROCESSING. VOL. 43, NO. 7, JULY 1995

0 6

o s

0 4

03

0 2

O I l i 0 0 0 2 0 3 0 4 0 5 06 0 7 08 0 0 1 09

MODULATION INDEX

Fig. 9. circular array.

DOA standard deviation versus “modulation index,” eight element

locations (DOA’s) and checked the effect of the “modulation index” on the standard deviation. The local SNR is kept fixed (at 10dB) during the experiment.

The results are shown in Fig. 9. The solid line shows the results for the source at azimuth O”, the dashed line represents the results for the source at azimuth 90” and the dash-dot line represents the results for the source at azimuth 180”. We observe that the estimation errors of the source at azimuth 90” is essentially unaffected by the “modulation index.” The estimation accuracy of the source at the peak of the colored noise, at 0”, is improving as the “modulation index” increases. This can be explained by the fact that the array acts as a spatial filter, focused on the source. As m increases, the effective SNR (the ratio between the signal power and the noise power at the output of the spatial filter) decreases slightly due to the cosine shape of the power density function. This phenomenon is reversed for the source at azimuth 180”, since this is the lowest point of the power density function.

Fig. 10 depicts similar results for a 32 element circular array. We observe that due to the improved spatial filtering of the array the effect of m is reduced. We conclude that the effects of colored noise can be studied by concentrating only on the noise that passes the spatial filter of the array, when the array is steered towards the sources.

VIII. CONCLUSIONS

In this paper, we presented and analyzed a class of direction finding methods that operate in the presence of correlated noise with an unknown covariance matrix. Our approach is based on joint estimation of the directions of arrival and the parameters of a model for the noise covariance matrix. This can be done using an MLE, a suboptimal version derived in Section II- B, or by using other estimation algorithms. As long as the noise model is identifiable, it is possible in principle to obtain asymptotically unbiased estimates of the DOAs, in spite of the fact that the noise covariance matrix is unknown. We present formulas for evaluating the maximal number of identifiable noise parameters.

Using the CRB we can study the degradation in DOA estimation accuracy due to the estimation of the noise param-

ooll 0 0 1 02 03 0 4 05 0 6 0 7 O S 0 9

MODULATION INDEX

Fig. 10. DOA standard deviation versus “modulation index,” 32 element circular array.

eters in addition the DOA estimates. By studying a number of test cases we conclude that DOA estimation accuracy is not degraded significantly as long as the number of noise parameters is sufficiently small. We believe that in many cases of practical interest, the noise covariance matrix can be approximated quite well with a few parameters. Thus, the approach presented here may offer a practical solution for the problem of direction finding in an unknown noise environment.

APPENDIX A DERIVATION OF THE FISHER INFORMATION MATRIX

In this appendix, we derive the blocks of the FIM associated

Derivatives with Respect to 8: In the following develop- with the problem defined in the main text.

ment, we make use of the notational device

to denote the partial derivative of the matrix A with respect to 8j . We write the partial derivative of the covariance matrix with respect to 8, as

dR - dej = A8,PAH + A P A E .

Noting that

t r (AH) = {tr(A)}* (A.3)

where * denotes the complex conjugate operation, we obtain

+ tr{ A8 PAHR-lAd,PAHR-l}}. (A.4)

Observe that

where the unit vector ei is the ith column vector of the identity matrix, and A is the matrix of derivatives defined by

N

i=l

FRIEDLANDER AND WEISS: DIRECTION F'INLXNG USING NOISE COVARIANCE MODELING 1565

Using (AS), (A.4) becomes

Ae,ef.PAHR-'APe,eTAHR-')

+tr { Ae,eTPA R- Ae, eFPAH R-

=2Re{ef.PAH R- APe,eTA R-l Ae,

+ef.PAH R-' Ae, eTPAH R- ' Ae, } . (A.7)

Hence 7

(PAHR-lAP) x (kHR-'A)

where F88 is the submatrix of the FIM associated with the derivatives with respect to 8 , and x denotes the Hadamard product of two matrices, defined by

('4.9) A

(A x B),] = A,,B,, .

Derivatives with Respect to P: Define two real vectors

(A. 10)

(A. 1 1 )

A

A

g = Re{vcc{P}}

h = Im{vcc{P}}

so that

v~c{P} = g + jh (A.12)

where vec{P} is a vector of all the columns in P concatenated. Thus, the FIM elements corresponding to g , and g, can be obtained from

- d P - - d P - A w,,,, = tr A - A ~ . A - A ~ = tr{C,C,} (A. 13)

{ dgm &n 1 where

A 2 R - ~ P A (A.14)

and

(A. 15)

Using the properties of the Kronecker product, denoted by @, we can write

where

B 2 A* SA. (A.17)

Note that the last equality in (A.16) is true only if the elements of g are all independent. Using the relation

tr{ATB} = VCT'{A}VPC{B} (A.18)

we can write (A.13) as

W,,,,, = e:BHBe, . (A. 19)

Thus, the matrix W is given by

W = B ~ B . (A.20)

In order to take into account the fact that the elements of g are not independent, since P is Hermitian, we define the following sets of indices:

J1 2[2 : N . N + 3 : 2N, 2N

Jz 2[iV + 1 : N : ( N - l ) N + 1 , 2 N

5 3 2 [l : N . N + 2 : 2 N . . . , N']

+4 : 3?V, ' ' ' , ( N - 1)N] (A.21)

+2 : N : ( N - l ) N + 2 , . . . , N2 - 11 (A.22)

(A.23)

where n : m denotes the set n. n + 1. . . . , m, and n : k : m denotes the set n. n + k . n + 2 k . . . . , m. The vector J1 defines all the elements below the main diagonal of P, the vector Jz defines the corresponding elements above the main diagonal and 5 3 defines the elements of P on and below the main diagonal.

We want to add the rows of W defined by J2 to the rows defined by J 1 and then add the columns defined by J2 to the columns defined by J1. This can be done with the help of the following matrix

Qi -1 1 + 1(Ji , J z ) (A.24)

where I is the identity matrix of size equal to the size of W(7.e.. N 2 x N 2 ) and l(J1.52) defines a matrix, with the same size, with elements corresponding to the indices defined by J1 and 5 2 equal to one and other elements are zero. For example, if the first element of J1 is p and the first element of J2 is q, then the p . q element of l (J1,Jz) is equal to one.

Finally, we want to select only the rows and columns corresponding to the indices defined by J3. This can be done using the matrix

Qz 2 1 ( j 3 , J 3 ) (A.25)

where

5 3 2 [1 : 1 : N ( N + l ) / 2 ] . (A.26)

Thus, we obtain

Fgg Q2QiWQTQF 2 QWQT (A.27)

where

Q 2 Q z Q i . (A.28)

Repeating the same set of considerations for the vector h, we have

and

(A.30) vcr{Cm} = (A* @A)- dh = jBem.

dhm Thus, (A.29) can be rewritten as

(A.3 1) Wm,, = eLBHBe,.

1566 lEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 43, NO. 7, JULY 1995

Equation (A.31) indicates that W = W. Since the elements of h are not independent, we want to subtract the rows of W defined by 52 from the rows defined by J1 and then subtract the columns defined by J2 from the columns defined by J1.

Thus, define

Q1 A I - l (J1,Jz) . (A.32)

Finally, we want to select only the rows and columns corresponding to the indices defined by J 1 . This can be done using the matrix

a 2 l ( J 1 , J l ) (A.33)

where

51 2 [l : 1 : (N - 1)N/2] . (A.34)

Hence, the desired matrix is

Fhh = Q2Q1WQfQF = &WQH (A.35)

where

Q 2 - j Q 2 Q 1 . (A.36)

Following the same set of considerations we obtain for the cross term the matrix

Fgh = QwQH. (A.37)

Observe that the block of the FIM associated with P can be obtained by

where

(A.39) Q a [ H - H t - Q Q ] '

Cross Terms: We start with the 8-g cross terms

where

x 2 A P A ~ (A.41)

then

Note that

Z, = V(:> n) U(n, :) + [V(:, n) U(n, : ) I H (A.45)

and

vec{Zn} = U(r~,:)~@V(:,n)+v(:,n)*@U(n, : ) H (A.46)

where V(:, n) represents the nth column of V. Thus, vec{Zn} may be obtained by selecting the (n - l )N + n column of

B2 e UT @V + V* 18 UH. (A.47)

Thus

vec{Zn} = & e ( n - l ) ~ + n * (A.48)

Now, (A.40) may be rewritten as

(A.49) T H Vmn = emB B2e(n-1)~+n.

To obtain Fge, we define the set of indices

54 [l , N + 2,2N + 3, . . , N 2 ] (A.50)

(according to the formula (n - l )N + n for n between one and N) and the matrix

Qs = 1(J4,J4) (AS 1)

where 5 4 is simply the indices [l : 1 : NI. Finally

Fge = QBHB2Qr (A.52)

and similarly

Fhe = QBHB2Qr. (A.53)

The 8-p block of the FIM is given by:

F@ = QtBHB2QF = Qt[T; @ T2 + TZ @ TiJQf (A.54)

T1 e AHR-lAP , T2 6 AHR-lA. (A.55)

Noise-Related Derivatives: The noise-noise related FIM el-

where

ements are given by

Fq,qm = tr{R-'XnR-'Xm}. (A.56)

The cross qm4, elements are

Ft)men =tr{ R-'XmR-' (Ae,e:PAH + APene:AH)}

+ e; A" R-'X,R-lAPe,. (A.57)

=ezPAHR-lX,R-lAe,

Thus, (A.57) can be described compactly by

Feqm = 2Re { ( diag PAHR-'XmR-lA)) (A.58)

where diag{ X} is a column vector with the ith element equal to Xii. We turn now to evaluate the noise-signal covariance block:

FRIEDLANDER AND WEISS: DIRECTION FINDING USING NOISE COVARIANCE MODELING 1567

Define [9] K. M. Wong et al., “Estimation of the directions of arrival of signals in unknown correlated noise, part I: The MAP approach and its implementation,” IEEE Trans. Signal Processing, vol. 40, no. 8, pp.

[lo] J. P. Reilly and K. M. Wong, “Estimation of the directions of arrival of

A - d P - CTI = 1 (A ’ R-’A)* (A’60) 2007-2017, Aug., 1992.

dgrl

Thus signals in unknown correlated noise, part II: Asymptotic behavior and performance of the MAP approach,” IEEE Trans. Signal Processing. vol. 40, no. 8, pp. 2018-2028, Aug., 1992.

[l I] A. J . Weiss and B. Friedlander, “On the Cramtr-Rao bound for direction finding of correlated signals,” IEEE Trans. Signal Processing, vol. 41, no. 1, pp. 495499, Jan. 1993.

[I21 V. Nagesha and S. Kay, “Maximum likelihood estimation for array processing in colored noise,” in Proc. IEEE Int. Conf Acoust., Speech, Signal Processing, Minneapolis, MN, April 1993, pp. IV-24&243.

[I31 J. LeCadre, “Parametric methods for spatial signal processing in the presence of unknown colored noise fields,” IEEE Trans. Acoustics, Speech, Signal Processing, vol. 37, pp. 965-983, July 1989.

vec{C,} = (A* 8 zmA) - % = (A* CC E,A)e, &77

(A.61) and

‘qmg7~ = vecT{l}(A* @ ’ Ben (A’62)

so that we obtain the row vector

Yqmg = B. (A.63)

Taking into account the dependence of the elements obtain

Fqmg = BQT.

In a similar way, we obtain

Fq,h = BQT

and the FIM block associated with the qm-p derii given by

Fqmp = BQF.

These results are summarized in the text.

REFERENCES

of g we

(A.64)

(A.65)

ratives is

(A.66)

H. L. Van Trees, Detection, Estimation, and Modularion Theory, Part 1. New York: Wiley, 1968. W. S. Burdic, Underwater Acoustic System Analysis. Englewood Cliffs, NJ: Prentice Hall, 1984. R. 0. Schmidt, “Multiple emitter location and signal parameter es- timation,” in Proc. RADC Spectrum Estimation Workshop. Griffiths AFB. New York, 1979, pp. 243-258. Reprinted in IEEE Trans. Antenn. Propagar., vol. AP-34. no. 3. pp. 276-280, March 1986. A. Paulraj and T. Kailath, “Eigenstructure methods for direction of arrival estimation in the presence of unknown noise fields,” IEEE Trans. Acousr., Speech, Signal Processing, vol. 34, no. 1, pp. 13-20, Jan., 1986. J. F. Bohme and D. Kraus, “On least squares methods for direction of arrival estimation in the presence of unknown noise fields,” in Proc. IEEE Int. Conf Acousr., Speech, Signal Processing, New York, NY,

A. G. Jaffer, “Maximum likelihood direction finding of stochastic sources: A separable solution,” in Proc. IEEE Int. Conf Acoust., Speech, Signal-Processing, New York, NY, April 1988, pp. 2893-2896. A. H. Tewfik, “Direction finding in the presence of colored noise by candidate identification,” IEEE Trans. Signal Processing, vol. 39, no. 9, pp. 1933-1942, September 1991. M. Wax. “Detection and localization of multiple sources in noise with unknown covariance,” IEEE Trans. Signal Processing, vol. 40, no. 1, pp. 245-249, Sept. 1991.

April 1988, pp. 2833-2836.

Benjamin Friedlander (S’74-M’76-SM’82-F 87) received the B.Sc. and the M.Sc. degrees in elec- trical engineering from the Technion-Israel Institute of Technology in 1968 and 1972, respectively, and the Ph.D. degree in electrical engineering and the M.Sc. degree in statistics from Stanford University in 1976.

From 1976 to 1985, he was at Systems Control Technology, Inc., Palo Alto, CA. From November 1985 to July 1988, he was with Saxpy Computer Corporation, Sunnyvale, CA. Currently, he is with

the University of California at Davis, and Signal Processing Technology, Ltd. in Palo Alto, CA.

Dr. Friedlander is the recipient of the 1983 ASSP Senior Award, the 1985 Award for the Best Paper of the Year from the European Association for Signal Processing (EURASLP), and the 1989 Technical Achievement Award of the Signal Processing Society.

Anthony J. Weia (S784-M85-SM’86) received the B.Sc. from the Technion-Israel Institute of Tech- nology, Haifa, Israel, in 1973 and the M.Sc. and Ph.D. degrees from Tel-Aviv University, Tel-Aviv, Israel, in 1982 and 1985, all in electrical engineer- ing.

From 1973 to 1983, he was involved in research and development of numerous projects in the fields of communications, trackmg systems, command and control, and emitter localization. In 1985, he became a faculty member of the Department of Electronic

Systems, Tel-Aviv University. During the academic year 1986-87, he was a visiting scientist with the Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA. In 1987-88, he was with Saxpy Computer Corporation, Sunnyvale, CA. Since 1988, he spent his summers and sabbatical leaves with Signal Processing Technology, Ltd., Palo Alto, CA. His research activities include detection and estimation theory, signal processing, and array processing, with applications to radar, sonar, and passive surveillance systems.

Dr. Weiss held a Rothschild Foundation Fellowship from 1986 to 1987 and a Yigaal Alon Fellowship from 1985 to 1988. He was a co-recipient of the IEEE Acoustics, Speech, and Signal Processing Society’s 1983 Senior Award for a paper excerpted from his Masters thesis, “Fundamental Limitations in Passive Time Delay Estimation.”