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

30 Z. Zhu and S. Haykin

Besides, A is of full column rank. Then, the rank of 'P;x, and hence, the rank of 'Py, are equal to K when K < M. Thus, the spectral representation of Ry is

(2.110)

where A = diag(A1" .. , AK) is the matrix of the K largest eigenvalues of Ry , with A1 ~ A2 ... ~ AK > (12, and V = [V1 ... VK] is the M x K matrix of the corresponding eigenvectors. Now, the unknown parameter vector 8 contains A and V.

Using the spectral representation theorem, we may express the positive definite matrix Syas

Sy = ELEH ,

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

11 > 12 ... > 1M > 0, and E = [E1 ... EM] is the matrix of the corresponding eigenvectors.

Following Anderson [2.17], the maximum likelihood estimates of Ai and Vi

are

for i = 1, ... , K. Thus, the pseudo maximum likelihood estimate of Ry , under the assumption that only the rank of K of 'Py is known, and regardless that A is a known function of 01> ... , OK' may be expressed as

fly = Ediag(l1 - (12, ... , IK - (12,0, ... , O)EH + (121

(2.113)

Hence, we obtain

IRylA = (12(M-K) nK Ii . i=1

From (2.111) and (2.113), we have

Using the above relations, from (2.107) we obtain the log-likelihood function

- Nln (nM.nli) .

• =1

Letting Y = DX in the above equation, we get the log-likelihood function for HI when 'l'x equals its pseudo maximum-likelihood estimate as

The log-likelihood function for Ho may be obtained by letting K = 0 in (2.114) as

which is in closed-form.

Now, let us consider the detection of Swerling II targets of an unknown number by a noncoherent radar. Two possible ways, based on the pseudo maximum likelihood estimate, will be described under the assumption that target signals are not fully correlated with each other.

a) The Multiple Alternative Hypothesis Testing Problem

Let H K be the hypothesis that the number of targets is K in a certain range gate.

(2.118)

The likelihood function for Ho can be written from (2.21) as

In a manner similar to (2.128), we can find the maximum likelihood estimate of 0"2 under H 0 as

We see, therefore, that when the noise power 0"2 is unknown, the test statistic for detecting emitters with known directions is a ratio of two sums of quadratic forms, the computation of which is more complicated than (2.75) for known 0"2. Even so, the test statistic of (2.135) is still in closed-form.

The generalized likelihood ratio test for detecting emitters of narrowband Gaussian model with known directions, may be derived on the basis of [2.6].

2.4.3 Detection of Emitters with Unknown Directions: Deterministic Signal

The deterministic signal is considered in this section. First, we discuss the case of a known number of emitters.

36 Z. Zhu and S. Haykin

The binary hypothesis testing problem of (2.67) is reproduced here as

for n = 1, ... , N. In the present case, however, both signals and directions of the emitters are unknown. Moreover, the noise power (J2 is also unknown.

As in Sect. 2.3.3, there are two approaches that we make in applying a generalized likelihood ratio test to solve the problem of (2.67). The first approach is based on a true maximum likelihood estimate. Here, we also have a least-squares problem. The linear part of the problem has a solution that is the maximum likelihood estimate of signals as shown by (2.125). The nonlinear part of the problem may be solved by a K-dimensional search to find the maximum likelihood estimate of emitter directions. As in Sect. 2.4.2, we can find the maximum likelihood estimate of the noise power (J2 under H 1 as

n=1

and

(2.138)

where A denotes A taking maximum likelihood estimates of directions as its arguments. We thus get the likelihood function for hypothesis H1 when 8 equals @as

(2.139)

From (2.133) and (2.139), the generalized likelihood ratio is

(2.140)

The test statistic is

Equation (2.141) is more complicated than (2.135), due to the fact that the denominator of (2.141) is a function more complex than the quadratic form given in (2.135). The reason for this is the dependence of A on X. Note that the test statistic of (2.141) is in a concise form but not a closed-form.

The second approach uses a pseudo maximum likelihood estimate based on an eigendecomposition method.

In a similar way to (2.81), and noting that Co = I, we can express the

When the signals of the emitters are not fully correlated with each other, and the condition K < M ::; N holds, we may write, e.g. see (2.91):

where Ii is the eigenvalue of the sample covariance matrix Sx = (ljN)XXH in decreasing order with respect to i.

Taking the natural logarithm of(2.143), and setting the derivative of it, with respect to 0"2, to zero, we obtain the maximum likelihood estimate of 0"2 under

H1 as

i= 1

i=K+1

which is in closed-form.

Finally, we consider two possible ways for detecting emitters of unknown numbers with unknown directions by the multiple alternative hypothesis testing.

a) The Method Based on a True Maximum Likelihood Estimate

Let H K denote the hypothesis that K emitters exist. The likelihood function for HK is fK(XI6K). The unknown parameter vector 6 K contains S1" •• ,SK; (}1" •• , (}K; and (12. The maximum likelihood estimate of 6 Kis denoted by 8K. From (2.139) we have

(2.150)

where J:!fJ may be expressed from (2.137) and (2.138) as

(2.151)

In (2.151), AK signifies Aassuming that the number of emitters is K. For K = 0,

When K = M, we have J:!t!) = 0; then, (2.150) makes no sense if K = M. We consider K = M -1. From (2.130) we have

MN