In this paper, a general problem of the distributed detection of a constant multidimensional signal with unknown parameters in a background of a zero-mean Gaussian noise with unknown varying covariance matrix is considered. This problem is encountered in many situations of decentralized processing involving a large number of sensors, where noisy processes at these sensors have different covariance matrices. We discuss test statistics at the sensors, where a hypothesis testing results in a sequence of 1 (if the signal-plus-noise waveform exceeds a predetermined threshold) and 0 (otherwise), and at the fusion center, where the k out of m decision rule regarding the presence or the absence of a signal is used. Test statistics at the sensors are obtained by means of a generalized maximum likelihood ratio (GMLR) test. This test is invariant to intensity changes in the noise background and achieves a fixed probability of a false alarm. No learning process is necessary in order to achieve the constant false alarm rate (CFAR). Operating in accordance to the local noise situation, the test is adaptive. It is shown that this test is uniformly most powerful invariant (UMPI) and robust against departures from normality in the following sense. It is still UMPI in a broad class of distributions, and the null distribution under any member of the class is the same as that under normality.
