Detection of spectral lines using eigencoefficients

We discuss a detection problem based on a Cramer (1940) spectral representation for a widesense stationary random process. Development of the Cramer representation culminates in determination of a smoothed power spectral density estimate expressed completely in terms of observable expansion coefficients where the basis functions consist of discrete prolate spheroidal wave functions. Using these coefficients, we develop a least squares estimate for sinusoidal components. An F-test statistic is then introduced to determine the presence of sinusoidal components with fixed frequencies, but with random amplitudes and phases, embedded in a widesense stationary random process. We show that the magnitude of the F-test statistic provides an estimate of the confidence with which it can be stated that a signal is present, and develop the theoretical probability of detection associated with this technique under a false alarm constraint. The probability of detection for faint components using this test statistic yields near optimal results.