Some Statistical Properties of Signal Plus Narrow Band Noise Integrated over a Finite Time Interval

Abstract

The variance of narrow band noise plus c.w. signal after detection and integration over a finite time interval is derived. The problem is solved for Ideal, Gaussian, and High Q Single Tuned filters with square law and linear detection, the latter for large predetection signal-to-noise ratio. In addition to these filters the analysis is carried through for a model which considers the envelope of narrow band noise plus signal as a function which is constant for intervals of duration equal to the reciprocal of the noise band width and statistically independent in different intervals. Graphs of the mean square deviation divided by the mean power for noise alone are given for all filters and the model, with square law detection and predetection signal-to-noise ratios 0, 0.5, 1, 2.