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

14 Z. Zhu and S. Baykin

Finally, the test statistic is

(2.45)

Comparing (2.41) and (2.45), we see that when the directions of the targets of interest are unknown, the test statistic of the generalized likelihood ratio test is no longer in closed-form. Besides, it is a more complicated function of x than the quadratic form that arises due to the dependence of Aon x.

Next, we discuss the more general problem ofdetecting multiple targets of an unknown number with unknown directions. In this case, in addition to the decision as to whether targets are present or not, we also have to determine the number of targets. Here, we note that there is no unique method for determining the number of targets; we recommend two possible methods. One method considers the problem as a multiple alternative hypothesis testing problem. The other method treats it as a model-order selection problem. Both methods are considered in the sequel.

a) The Multiple Alternative Hypothesis Testing Problem

Let H K be the hypothesis that the number of targets is K, and let!K(XI 8 K ) be the likelihood function for hypothesis H K • The unknown parameter vector 8 K includes Sl' ... , SK; and 81 , ... , 8K· Using 8K to denote the maximum likelihood estimate of 8 K, we find from (2.43) that the likelihood function for HK when 8 K equals 8K is

where J~KJ.ax represents J2max of (2.42) with AK substituted for A,°and AK denotes Aassuming the number of targets is K. Obviously, J~~~x = for K = 0.

When K = M, the M x M matrix A is of full rank. From (2.42) we know J~Af,!ax = NxHC(jl x, which is independent of A. Thus we see that, if K = M, A and hence s do not have unique solutions. From (2.46), we then get the likelihood function for H M when 8 M equals 8M as

Let K = 0, 1, ... , P, where P < M, and define a sequence of binary hypothesis tests

H1:HM ,

(2.48)

Ho:HK' K = 0, 1, ... , P.

From (2.42, 46 and 47), the generalized likelihood ratio at the Kth step is

l(X) = fM(XI~M)

fK(XI8K)

=exp [ -:2(NXHColX-J~~ax)J

We reject the null hypothesis HK if the test statistic qK exceeds a threshold, determined so that the test has a specific size under the null hypothesis HK • Starting from K = 0, HK (K = 0,1, ... , P) are tested sequentially. The value of K, for which HK is first accepted, is selected as the estimate Kof the number of targets.

b) The Model-Order Selection Problem

The model-order selection problem can be described, in general, by recognizing that there is a family of models, i.e., a parameterized family of probability density function fK(XI8K), where 8 K is the unknown parameter vector of the model.

incoming data. Two useful criteria for model-order selection may be used to solve this problem. One criterion uses An Information Criterion (AIC) due to Akaike. The other criterion is based on the Minimum Descriptive Length (MDL) principle due to Rissanen. Specifically, the two criteria are as follows:

a) When using AIC, we select the model-order that minimizes the criterion (2.51)

where In denotes the "natural logarithm".

b)When using MDL, the model-order is determined by minimizing the criterion

(2.52)

Here, @Kdenotes the maximum likelihood estimate of the 8 K, andfK(XI@K) is the likelihood functiop. for the model when 8 equals @K' Note that apart from a factor of 2, the first term in (2.52) is identical to the corresponding one in (2.51), while the second term has an extra factor oft In N. It should be noted that both the AIC and the MDL criterion assume that the number of samples N is large.

16 Z. Zhu and S. Haykin

If we visualize the number of targets as the model-order, we may determine the number of targets by using (2.51) or (2.52). Because Ok is a real parameter, and Sk is a complex parameter, the number of adjustable parameters is clearly

(2.53)

Substituting (2.42, 46 and 53) into (2.51), and omitting terms irrelevant to K, we obtain

The difficulty with the use of the likelihood ratio test is in determining the threshold. Because of the sequential nature of the testing procedure, the probability of accepting the null hypothesis HK at each step depends not only on the threshold of that particular test, but also on the probability of rejecting the preceding null hypothesis. On the other hand, when using the model-order selection approach to determine the number of targets, no subjective judgement for deciding on threshold is required, but the error probabilities, including the false alarm probability, cannot be controlled.

Both of these methods solve the detection and estimation subproblem simultaneously, in that once the number of targets is determined, the directions of targets and their signal intensities are also estimated.

2.3 Noncoherent Radar Detection

In this section we consider the case of noncoherent radar detection. We begin by describing the pertinent signal and noise model.

2.3.1 Signal and Noise Model

For a noncoherent radar, the signal sample of the kth target and the nth snapshot may be expressed as

where the amplitude bkll and the phase cJ>kll are unknown variables that depend on n. Two models, a deterministic model and a narrowband Gaussian model, will be considered for the signal. The first model recognizes that the signal is unknown without any a priori information·about it, while the second model

assumes that the signal may be described as a narrowband Gaussian process and temporal samples of the signal are pulse-to-pulse independent, that is, equivalent to the Swerling II fluctuating target model in radar terminology. The Doppler information reflecting the radial velocity of a target cannot be retained in the phase 4Jk" due to the random initial phase of the transmission carrier of a noncoherent radar and due to the fluctuation of the target.

The noise model is the same as in the coherent radar case. The noise vector w" is assumed to be a complex, stationary and ergodic Gaussian vector process with zero mean and a positive definite covariance matrix (12 Co. Furthermore, we assume that the spatial covariance (12Co of the noise is known, and the elements of the time series describing the noise are independent and identically distributed, and that the noise is independent of the signals.

2.3.2 Detection of Targets with Known Directions

As in the case of a coherent radar, the known direction of a target may be considered as the steered direction of the array antenna. Here, again, we first discuss the single target case.

Assuming K = 1 and omitting the subscript k in (2.4) and (2.56), we may express the observation vector in a noncoherent radar for a single target as

(2.57)

and the signal sample as

(2.58)

In the present case, fJ and a(fJ) are known. As for the signal S,,' if it is unknown and deterministic, we prefer the generalized likelihood ratio test, as developed later for multiple targets (including single target) with known directions. If we can prescribe a probabilistic model, then we may try to explicitly derive an optimum detector using an averaged likelihood ratio test. For example, we may assume the Swerling II target model. In this model, the amplitude b" fluctuates according to the Rayleigh distribution with the probability density function

where B2 is the average signal power. The phase 4J" is uniformly distributed over the interval [0, 2n]. The time series of b" and 4J" are independent and identically distributed, respectively. Initially, we assume that B2 is known. We may then consider the following binary hypothesis testing problem:

(2.60)

for n = 1, ... , N.