Geometry of optimum incoherent detection

The problem of distinguishing between two narrow-band signals in the presence of additive Gaussian noise when the forms of both signals are known except for phase is basically four dimensional, as the lack of phase information turns signal vectors into signal circles. The geometry of signal vectors and of the planes of signal circles is shown to account for the optimality of matched-filter detection. In the case of equal-energy equiprobable signals, it leads to the conclusion that a large variety of different pairs Of filters will perform exactly like a pair of matched filters. One such pair has orthogonal impulse responses, and its performance is, therefore, relatively easy to determine.