Diss / (Springer Series in Information Sciences 25) S. Haykin, J. Litva, T. J. Shepherd (auth.), Professor Simon Haykin, Dr. John Litva, Dr. Terence J. Shepherd (eds.)-Radar Array Processing-Springer-Verlag

20 Z. Zhu and S. Haykin

background of white spatial noise. This is also a consequence of the fact that the required spatial processing in both coherent and noncoherent radars is the same as can be seen from (2.23) and (2.65), or (2.25) and (2.66).

Next, we consider the detection of multiple targets with known directions. When the number of targets K > 1, an averaged likelihood ratio test may be too complicated to implement; moreover, it will no longer be invariant with respect to the unknown signal powers. As in the case of a coherent radar, a generalized likelihood ratio test based on the maximum likelihood principle is the preferred method. When K targets are present in a range gate of a noncoherent radar, the observation vector may be described as in (2.7), repeated here for convenience:

where the direction matrix A is known, but the signal-in-space vector Sn is unknown.

The binary hypothesis testing problem to be considered is

(2.67)

for n = 1, ... , N.

Let us consider the deterministic signal model. In this case, the likelihood function for hypothesis HI is

In (2.68), the unknown parameter vector 8 includes Sl" •• ,SN' We have assumed that (12CO is known. The maximum likelihood estimate sn of the signal- in-space vector s" may be obtained by the minimization of

Equation (2.69) describes the summation of N quadratic forms, each of which is positive definite, due to the positive definiteness of Co and CO' 1 • In addition, the observation vectors Xn are independent for different n, and so are the signal-in- space vectors Sn' Hence, the minimization of J is equivalent to N independent least-squares problems involving N quadratic forms. The signal-in-space vector s" enters the model linearly. We may thus develop a closed-form solution for the maximum likelihood estimate of Sn as

(2.70)

We assume K::;; M, i.e., the number of targets is less than or equal to the number of array elements. The M x K matrices A and DA are of full column

n=l

where

(2.72)

Substituting (2.70) and (2.72) into (2.68), we get the likelihood function for H 1 when 8 equals 8 as

(2.73)

where 8 is the maximum likelihood estimate of 8.

The generalized likelihood ratio may be obtained from (2.21) and (2.73) as

Jc(X) = f1(XI8) fo(X)

= exp(~;)

U n=l

And from (2.74), we obtain the test statistic

(2.75)

Equation (2.75) indicates that, in a noncoherent radar used to detect multiple targets with known directions, the observation vector in every snapshot should first be put into a quadratic form associated with the matrix COl A(AHC0 1A)-l A HC0 1 in the spatial domain, and then integrated noncoherently in the time domain. It is instructive to compare the operations described here with that for a coherent radar, in Sect. 2.2.2. There, we have seen that the observation vectors in all snapshots are integrated coherently first in the time domain, and then a quadratic form is computed in the spatial domain. In both cases, the matrix associated with the quadratic form is the same.

22 Z. Zhu and S. Haykin

We will not discuss the generalized likelihood ratio test for detecting multiple targets of the Swerling II model. The related problem of estimating spectral parameters of Gaussian signals in wavefields has been solved in Ref. [2.6].

2.3.3 Detection of Targets with Unknown Directions: Deterministic Signal

The deterministic signal is considered in this section, while the Gaussian model is reserved for the next section.

First, we consider the detection of targets of known number with unknown directions. The binary hypothesis testing problem is described by (2.67) which is rewritten as

(2.67)

for n = 1, ... , N.

In the present case, when K targets are present in a range gate, the unknown

°parameter vector 8 includes not only Sl" •• , SN' as in Sect. 2.3.2, but also

1 " •• , OK contained in A.

We will use two different maximum likelihood estimates in a generalized likelihood ratio test to solve the problem of (2.67). One of them is the maximum likelihood estimate of signal-in-space vectors Sn' and directions oftargets OK in a true sense. The other one, which we refer to as the pseudo maximum likelihood estimate, is the maximum likelihood estimate of the whole matrix AS under a relaxed constraint that will be mentioned later.

When using a generalized likelihood ratio test based on a true maximum likelihood estimate of signals and directions of targets, the detection and estimation subproblems are solved simultaneously, as mentioned in Sect. 2.2. The likelihood function for H1 has been expressed by (2.68). The least-squares problem of (2.69) arises when we seek the maximum likelihood estimate of 8. We assume K :::;;; M. The linear part of the least-squares problem has a closedform solution, as shown by (2.70), which yields sn. The nonlinear part of the least-squares problem may be solved by a K-dimensional search to minimize J of (2.69) with sn substituted for Sn ,or, equivalently, to maximize J 2 of (2.72). The solution of the nonlinear part of the least-squares problem is maximum likelihood estimates of directions of targets 01 " •• , OK' We use A to denote A with 01 " •• , OK being its arguments. From (2.72), the maximum value of J2 is

(2.76)

The likelihood function for Hl' when 8 equals 8, can be obtained from (2.73) with J2max of (2.76) substituted for J 2 as

From (2.21) and (2.77), we obtain the generalized likelihood ratio

A(X) = f1(XI8)

The test statistic is

(2.79)

We have seen that, in the case of a deterministic signal model, when the directions of targets are known, the closed-form test statistic of (2.75) is the sum of quadratic forms. Equation (2.79) indicates that when the directions of targets are unknown, the observation vector Xn should be processed according to a function which is more complicated than a quadratic form, due to the fact that A is dependent on X, and the test statistic no longer has a closed-form expression due to the presence of A.

Next, we introduce a generalized likelihood ratio test using a pseudo maximum likelihood estimate of the whole matrix AS, under a relaxed constraint based on the eigendecomposition method. We will see later that this approach decouples the detection and estimation subproblem, i.e., the detection part of a combined detection-estimation problem may be solved without explicitly estimating the signals and directions of targets of interest.

as the signal part of the observation matrix X, where A is the direction matrix of (2.9), and S is the signal matrix of (2.12). Instead of seeking the true maximum

likelihood estimates of Sl" •• , SN' and 01" •• , OK' we will find the pseudo maximum likelihood estimate of U.

Using (2.80), the likelihood function for Hi of (2.68) may be written as

24 Z. Zhu and S. Haykin

The M x K matrix A is of full column rank, due to the fact that column vectors a(On), n = 1, ... ,K are linearly independent. In a noncoherent radar environment with K target signals that are not fully correlated with each other, the row vectors of the signal matrix S are also linearly independent, and the K x N matrix S is of full row rank. Hence, when target signals are not fully correlated with each other, the rank of U equals K when K < M ~ N.

Now, we are going to find the pseudo maximum likelihood estimate of U under the constraint that the rank of U equals K [2.4J. We define the transformed observation matrix

where D is the spatially whitening operator, as defined in Sect. 2.2.2.

From (2.81) we see that when u2CO is known, the maximization offl (X 18) is equivalent to the minimization of the following expression:

where II 112 signifies the squared Euclidean norm of a matrix. By the principle of orthogonality, if the minimum of (2.84) is attained for Uy = Uy, then Uy must satisfy

U~(Y - Uy ) = 0,

or

It is a straightforward matter to verify that

(2.86)

satisfies (2.85). The matrix EK is the M x K matrix with any K eigenvectors ofthe matrix

as columns. By the spectral representation theorem, Sy may be expressed as (2.88)

where L = diag(ll' ... ,1M ) is the matrix of the eigenvalues of Sy with

II > 12 ... > 1M > 0, and E = [El ... EM] is the matrix of corresponding eigenvectors. From (2.82) and (2.87), we have

(2.89)

(2.90)

is the sample covariance matrix of xn •

We see from (2.86) that the rank of Uy is K when K < M ~ N. But in (2.86), the matrix EK may be formed by usiJlg any set of K of the M eigenvectors. Clearly, there are (~) such sets. From (2.84, 85, 86, and 88), we have

II y - Uy l1 2 = tr{(Y - Uy)Hy}

=tr{yHy} - tr{ yHEKE~ Y}

=tr{ yyH} - tr{ yyHEKE~}

=Ntr{Sy} - Ntr{SyEkEr}

=Ntr{ELEH} - Ntr{E~SyEd

=Ntr{L} - Ntr{E~ELEHEd

M

= N 'L li-N'Lli

where I is the ~et of indices of K eigenvalues. In order to find 0y with the smallest II y - Uy II, we must select I as the set 1, 2, ... , K. 1

So far, we have proved that under the constraint that the rankof Uy equals K, the pseudo maximum likelihood estimate of Uy may be expressed by (2.86), with EK composed of K eigenvectors associated with largest K eigenvalues of Sy. Hence, we have

where U= DUy is the pseudo maximum likelihood estimate of U under the constraint that the rank of U equals K.

1The result derived herein may be interpreted as the singular value decomposition (8VD) solution U, for a lower rank approximation of the matrix Y.

26 Z. Zhu and S. Haykin

Substituting the previous relation into (2.81), we obtain the likelihood function for HI when U equals the pseudo maximum likelihood estimate -0 of

The likelihood function for Ho may be expressed from (2.21) as

for detecting K targets with unknown directions when the target signals are assumed to be unknown and deterministic.

In this approach, we see that the detection part of a combined detection-estimation problem is solved only on the basis of eigenvalues of the transformed sample covariance matrix DS"DH • There is no requirement to explicitly estimate signals and directions of targets. On the other hand, unlike the true maximum likelihood estimate of signals and directions of targets, the pseudo maximum likelihood estimate of U is determined without using the a priori information that the direction matrix A is a known function of directions of targets. The constraint that is imposed on the pseudo maximum likelihood estimate is relaxed in comparison with that for the true maximum likelihood estimate. This should account for some deterioration in the performance of eigendecomposition-based methods compared with the maximum likelihood method, in a true sense. However, an eigendecomposition-based method has an advantage in that it can provide the closed-form test statistic, and it is simpler to implement than a true maximum likelihood method because the K-dimensional search (an essential feature of the latter method) is no longer needed.

The detection of targets of unknown number with unknown directions is discussed next. In the coherent radar case, two possible methods for determining the number of targets are applied. One method involves multiple alternative

28 Z. Zhu and S. Haykin

If qK exceeds a threshold at the Kth step, we reject the null hypothesis H K. Starting from K = 0, we test HK(K = 0, 1, ... , P) sequentially. The value of K for which HK is first accepted is selected as the estimate K of the number of targets. Note that the qk of (2.99) is not in closed-form due to the presence of AK •

b) The Method Based on a Pseudo Maximum Likelihood Estimate

Assume that the target signals are not fully correlated and the condition K < M ~ N holds. When a deterministic signal model is assumed, the likelihood function for the hypothesis H K that the number of targets is K, when U takes its pseudo maximum likelihood estimate, is given by (2.86). Using adequate notations, we rewrite (2.86) as

Starting from K = 0, and setting K = 0, 1, ... , P, where P < M, we compute the binary hypothesis test of HM against H K • The first time that we accept the null hypothesis H K, the estimate of the number of targets K is determined. Note that the qK of (2.101) is in closed form, whereas that of (2.99) is not.

H should be remembered that both of these methods suffer from the difficulty of determining the thresholds. Moreover, procedure (a) is more complicated to implement, but could provide better performance as compared to procedure (b).

2.3.4 Detection of Targets with Unknown Directions: Gaussian Signal

Having dealt with the deterministic signal model, we discuss the case of the Swerling II target model in this section.

Again, the case of a known number of targets is considered first. We rewrite the binary hypothesis testing problem of (2.67) here as

(2.67)

for n = 1, ... , N. This time, the signal-in-space vector sn is a complex, stationary and ergodic Gaussian vector process with zero mean and an unknown

covariance matrix

(2.102)

We have assumed in Sect. 2.3.1 that the temporal samples are independent. Thus, the likelihood function for H 1 may be expressed as

is the covariance matrix of x,., and the M x M matrix Sx = (ljN)XXH is the sample covariance matrix of X n , as defined in (2.90). From (2.67), we have for the hypothesis H 1 that

The generalized likelihood ratio test, based on a true maximum likelihood es~imate of 8, for detecting multiple targets of the Swerling II model with unknown directions may be developed on the basis of [2.6]. Here, we only discuss the pseudo maximum likelihood estimate of 'Px.

In terms of the transformed observation matrix Y of (2.82), the likelihood function for H 1 may be expressed as

(2.107)

where Sy is defined by (2.87). We need N ~ M to keep Sy positive definite. From (2.89) we know that

which is the covariance matrix of the transformed observation vector Yn. The M x M matrix 'Py is

'Py = D'PxDH

(2.109)

As in the preceding section, we suppose an environment with K target signals that are not fully correlated with each other. In such a case, R. is nonsingular.