Detection of Gaussian Signal Using Adaptively Whitened Data

The adaptive matched filter, like many other adaptive detection schemes, uses in its test statistic the data under test whitened by the sample covariance matrix <inline-formula><tex-math notation="LaTeX">$\mathbf {S}$</tex-math></inline-formula> of the training samples. Actually, it is a generalized likelihood ratio test (GLRT) based on the conditional (i.e., for given <inline-formula><tex-math notation="LaTeX">$\mathbf {S}$</tex-math></inline-formula>) distribution of the adaptively whitened data. In this letter, we investigate detection of a Gaussian rank-one signal using the marginal (unconditional) distribution of the adaptively whitened data. A first contribution is to derive the latter and to show that it only depends on a scalar parameter, namely the signal to noise ratio. Then, a GLRT is formulated from this unconditional distribution and shown to have the constant false alarm rate property. We show that it bears close resemblance with the plain GLRT based on the whole data set (data under test and training samples). The new detector performs as well as the plain GLRT and even better with multiple cells under test and low training sample support.