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In this paper, we focus on the design of adaptive receivers for nonhomogeneous scenarios. More precisely, at the design stage we assume a mismatch between the covariance matrix of the noise in the cell under test and that of secondary data. Under the above assumption, we show that the Wald test is the adaptive matched filter, while the Rao test coincides with the receiver obtained by using the Rao test design criterion in homogeneous environment, hence providing a theoretical explanation of the enhanced selectivity of this receiver.

In recent years adaptive detection of targets embedded in Gaussian disturbance with unknown spectral properties has received an increased attention in the signal processing community. In the seminal paper by Kelly [

However, the homogeneous environment is an assumption that might not be met in realistic situations: see, for example, ([

Another model of interest assumes that CUT and secondary data share the same covariance matrix of the thermal noise plus clutter, but, in addition, the CUT contains a noise-like interferer [

In this work, we use the Wald test and the Rao test design criteria to solve the latter detection problem. In particular, we show that these design criteria yield the decision statistics that have been obtained for the homogeneous environment,

The remainder of the paper is organized as follows: the next section is devoted to the problem formulation while Section III addresses detector designs. Finally, Section IV contains some concluding remarks and potential directions for future works.

Assume that a linear array formed by _{a}_{t}^{N}^{×}^{1}_{a}N_{t}_{k}^{N}^{×1},

The detection problem at hand can be formulated as follows:

^{N}^{×1} is the nominal steering vector;

_{k}^{N}^{×1}, ^{N}^{×}^{N}, i.e., _{k}_{N}

_{N}^{†}), where ^{N}^{×1} is due to a noise-like interferer and ^{†} denotes conjugate transpose. Moreover, we assume that

Notice that the probability density function (pdf) of _{1},…, _{K}_{1} (and assuming that

_{r}, α_{i}

^{T}

_{A}_{B}

^{2×1} is a vector that contains in univocal way the real and the imaginary parts of the elements of

As a preliminary step towards the derivation of the receivers, denote by

In this section we solve the binary hypothesis testing _{0}:

The Wald test is the following decision rule [_{A}_{,1} and _{B}_{,1} the maximum likelihood estimates of _{A}_{B}_{1}, respectively, and _{1} is the threshold to be set to achieve a predetermined probability of false alarm (_{fa}_{2} denotes the 2-dimensional identity matrix, and that
^{−1} in _{1} given by [

It is tedious but not difficult to show that

Finally, by substituting

Notice that the left-hand side of

The Rao test for the

_{1}) is the natural logarithm of the pdf of _{1} hypothesis (see

_{AA}_{0}) = [_{AA}_{0})]^{−}^{1}, where _{AA}

_{B},_{0} denoting in turn the maximum likelihood estimate of _{B}_{0};

_{2} is the threshold to be set in order to ensure the preassigned _{fa}

It is straightforward to show that
_{0}

It follows that

Gathering the above results, the Rao test can be recast as

which is the H-RAO [

This work addresses the adaptive detection of point-like targets in nonhomogeneous scenarios. In particular, at the design stage it is assumed that the CUT contains a noise-like interferer in addition to thermal noise, clutter, and to the possible signal of interest; a set of secondary data, free of signal components, is available: such data share a common covariance matrix that is equal to that of thermal noise plus clutter in the cell under test. Observe that the covariance mismatch between the CUT and the secondary data can lead to receivers that are less inclined to reveal a signal with the actual steering vector different from the nominal one (selective receivers). In addition, we assume that the noise-like interferer is orthogonal to the nominal steering vector in the whitened observation space. Under the above assumptions, we show that the Wald test coincides with the AMF, while the RAO test is the H-RAO. The latter result provides an alternative explanation of the good selectivity properties exhibited by the H-RAO [

As a final comment, it would be of interest to investigate under which conditions these design criteria are invariant with respect to different classes of detection problems. A preliminary step towards this direction is given in [

This work was supported by the National Natural Science Foundations of China under Grant No. 61172166.