Further Results on the Detectability of Known Signals in Gaussian Noise

Abstract

The detection of a completely known signal which may or may not be present in a finite sample of gaussian noise is considered from two points of view. The first examines the performance of a maximum likelihood detector operating on a finite set of discrete measurements of the stimulus as the set becomes large. The stimulus is either signal plus noise or noise alone. Examples are presented for signals in bandlimited noise, using as measurements either equispaced amplitude samples or derivatives at one instant in time. For both, the detectability groups without bound as the number of measurements is increased. The second point of view bases detection on a continuous measurement (linear integral operator) which maximizes the detectability. Solutions have been obtained when the noise has a rational power spectral density. The detector utilizes a cross-correlation between stimulus and signal which is well known and a mechanism, designated extrapolation detection, which involves evaluation of derivatives of the stimulus. The contribution of the derivative measurements to the detectability is examined as the noise approaches bandlimited noise and is found in many cases to grow without bound.