Distribution of instantaneous frequency for signal plus noise

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
Q
-functions, with well-known properties. The results are applicable in the performance analysis of
n
-ary FM and PM data transmission systems. Illustrative examples of the application of the formula are worked out for binary FSK.