Distribution of the phase angle between two vectors perturbed by Gaussian noise II

The Case I, Case II, and Case III distributions for the phase angle between two vectors perturbed by Gaussian noise are restated in terms of integrals with integrands containing only exponentials, Added to these are a Case IV distribution in which one of the vectors is noise free; and Case V, Case VI, and Case VII distributions for the instantaneous radian frequency that results when the time between the two vectors goes to zero. In addition, existing and new forms for the probability density functions corresponding to each of the distributions are given. The results are applied to digital FM with a limiter/discriminator receiver to easily obtain the bit error probability without coding and the corresponding probability density function needed in a Chernoff bound approach to study the bit error probability with coding.