and the noise is MVN with mean ScjJ and covariance R = a 2 R o:where J-L = 0 under H o and J-L > 0 under H 1 • This is the standard detection problem wherein the polarity of the signal x is assumed known. Near the end of Section V we replacewhere polarity is unknown.We shall assume that the signal x obeys the linear subspaceThe detection problems to be studied in this paper may be described as follows. We are given N samples from a real, scalar time series {y(n), n = 0,1, 0 0 " N -I} which are assembled into the N-dimensional measurement vectorBased on these data, we must decide between two possible hypotheses regarding how the data was generated. The null hypothesis H o says that the data consist of noise tI only. The alternative hypothesis H 1 says that the data consist of a sum of signal J-LX and noise tI; that is, (2.2) (2.1)to include subspace interferences. These problems involve unknown parameters in the mean and covariance of a multivariate normal (MVN) distribution. For each problem in the class, we establish invariances for the GLR and find that they are identical to the natural invariances for the problem. We show that a monotone function of the GLRT equals one of the uniformly most powerful invariant (UMP-invariant) tests derived in [1]. This means that the GLRT is itself UMPinvariant. In addition to tying up the theories of invariance and the GLRT, our results generalize and extend previous work on these problems published in [1]-[6]. We begin our development by establishing the invariances of the GLRT in the MVN problem. We then specialize our results for structured means in order to derive UMP-invariant GLRT detectors for matched subspace filtering in subspace interference. The GLRT produces an UMP-invariant detector, which is CFAR if the noise variance is unknown. As we shall find, the optimum detector may be interpreted as a null steering or interference rejecting processor followed by a matched subspace detector.

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