Distribution of instantaneous frequency for signal plus noise

Abstract

A concise formula is derived for the probability distribution of the instantaneous frequency of an arbitrary bandpass signal plus Gaussian noise. The results are presented in terms of Marcum's <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</tex> -functions, with well-known properties. The results are applicable in the performance analysis of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> -ary FM and PM data transmission systems. Illustrative examples of the application of the formula are worked out for binary FSK.