that can he uniquely localized by a general array that satisfies some mild geometrical constraints. The conditions are in terms of the num- ber of sensors and the ...
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IEEE TRANSACTIONS ON ACOUSTICS. SPEECH, A N D SIGNAL PROCESSING. VOL. 37. NO. 7. J U L Y 1989
On Unique Localization of Multiple Sources Passive Sensor Arrays
Abstract-We present conditions for unique localization of narrowband sources having the same known center frequency by passive sensor arrays. The conditions specify the maximum number of sources that can he uniquely localized by a general array that satisfies some mild geometrical constraints. The conditions are in terms of the number of sensors and the rank of the correlation matrix of the sources.
1. INTRODUCTION key problem in the area of signal processing in passive sensor arrays is the estimation of the directions of arrival of narrow-band sources having the same known center frequency. A problem of crucial importance in this context is that of specifying the conditions under which the problem has a unique solution. In spite of its great practical and theoretical importance, this problem is far from being resolved. In fact, the only results available are for the special case of a uniform linear array (Di and Tian [2] and Bresler and Macovski [ 11). In this paper, we present conditions that guarantee a unique solution for a general array. The conditions specify the maximum number of sources that can be uniquely localized by a general array that satisfies some mild geometrical constaints. The conditions are in terms of the number of sensors and the rank of the correlation matrix of the sources. We present two different conditions. The first guarantees uniqueness for every batch of sampled data. The second, which is weaker, guarantees uniqueness for almost every batch of sampled data, with the exception of a set of batches of measure zero. We show as well that a condition which is slightly weaker than the second one is also necessary. The organization of the paper is as follows. In Section 11, we present the formulation of the problem. In Section 111, we derive the sufficient conditions and discuss their relation to the previous work. In Section IV, we derive the necessary conditions. Finally, in Section V, we present some concluding remarks.
known frequency, say q, impinge on the array from distinct locations 0 1 , . . . , 8,. For simplicty, assume that the sources and the sensors are located in the same plane and that the sources are in the far field of the array. In this case, the only parameter that characterizes the source location is its direction of arrival 8. Using complex envelope representation, the p X 1 vector received by the array can be expressed as
A
11. PROBLEM FORMULATION Consider an array composed of p sensors with arbitrary locations and arbitrary directional characteristics. Assume that q narrow-band sources, centered around a Manuscript received August 12, 1987; revised October 3 , 1988. The authors are with RAFAEL, Box 2250, Haifa 3 102 I , Israel. IEEE Log Number 8928135.
4
C a@,) S k ( f ) + +). k= 1
x(t) =
(la)
Here a ( 8 ) is the “steering vector” of the array towards direction 8: a(,g) =
[a,(e)e-~wo~l(~)
. . . , ~ , , ( 8 ) e - I ~ ~ ~ / ~ ( l~b))] ’ .
a , ( 8 ) the amplitude response of the kth sensor towards direct 8, 7,( 8 ) the propagation delay between the reference point and the kth sensor to a wavefront impinging from direction 8, s k (t ) the signal of the kth source as received at the reference point, n k ( t ) the noise at the kth sensor. In matrix notation, this becomes x(t) =
where A
( e )is the p
A ( 8 )s(t)
+ n(t),
(2a)
x q matrix of the steering vectors:
A ( @ )=
[~(e,),
,~(e,)].
(2b)
Suppose that the received vector x ( r ) is sampled at M time instants t l , . . , tM. From ( l ) , the sampled data can be expressed as
X
=
A(8)S
+N
(3a)
where X and N are the p x M matrices
and S is the q x M matrix
The localization problem can be stated as follows. Given the sampled data X, estimate the locations of the sources e , , ,
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0 1989 IEEE
WAX A N D ZISKIND: U N I Q U E L O C A L I Z A T I O N OF M U L T I P L E S O U R C E S
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This problem is of interest if the solution is unique. In what follows, we investigate the conditions on the array geometry, the number of sources, the number of sensors, and the rank of the matrix S for which uniqueness can be guaranteed.
Now (9) holds true if and only if the nullity of the matrix [ A ( 6 ) , A ( 6 ’ ) l is smaller than the rank of the matrix [-:,I. Denoting { = null [ A ( 6 ) , A ( 6 ’ ) ] = 2q
- rank [ A ( 6 ) , A ( 6 ’ ) ] ( 10)
111. SUFFICIENT CONDITIONS FOR UNIQUENESS Our analysis of the uniqueness problem is restricted to arrays satisfying the two following conditions. A l ) The array manifold, defined as the set { a @ ) , 0 E [ 0, 27r] 1, is known. A2) Any subset of p distinct steering vectors from the array manifold is linearly independent. A l ) can be fulfilled by either computing the array manifold analytically, if possible, or alternatively, by measuring it in the field. A2) imposes certain constraints on the array geometry. These constraints, however, do not pose a serious problem and can easily be come by. By its nature, the uniqueness problem is decoupled from the estimation problem. Thus, we ignore the noise in the following analysis and rewrite (3) as
X
=
A(6)S.
(4)
The left-hand side of (4) represents the given batch of data, while the right-hand side contains the unknown parameters (6, Our goal is to specify the conditions under which the solution S ) of the set of equations (4) is unique. To this end, let 7 denote the rank of the q x M signal matrix S :
s).
(e,
7
=
rank S .
and
it thus follows that to prove the theorem, it suffices to show that if (7) holds true, then {
< v.
(12)
To this end, first note that (1 1) implies that
v 1 q.
(13)
Next, by A2), we have
{ p , 2q - d ) . considered p < 2q - d ,
rank [A(€)), A ( 6 ’ ) ] = min
(14)
Hence, since for the case it follows from (14), (lo), and (13), provided that (7) holds true, that
< 7 Iv. Case 2) Oi = 0,’ f o r d pairs, q > d I2q { = 2q - p
j ( i , j = 1,
*
*
(15)
- p , of i and , 4 ) . In this case, we rewrite (8) as
(5)
Note that rank [ S S H ] = 7
(6)
where H denotes the conjugate transpose, namely, 7 is the rank of the sample covariance of the signals. We now state the conditions that guarantee uniqueness for every batch X. ilteorern 1: An array satisfying conditions A l ) and A2) can uniquely localize q sources provided that
a
