Distribution of the Phase Angle Between Two Vectors Perturbed by Gaussian Noise

The probability distribution of the phase angle between two vectors perturbed by correlated Gaussian noises is studied in detail. Definite integral expressions are derived for the distribution function, and its asymptotic behavior for large signal-to-noise is found for "small," "nearpi/2," and "large" angles. The results are applied to obtain new formulas for the symbol error rate in MDPSK, to calculate the distribution of instantaneous frequency, to study the error rate in digital FM with partial-bit integration in the postdetection filter, and to obtain a simplified expresion for the error rate in DPSK with a phase error in the reference signal. In the degenerate case in which one of the vectors is noise free, the results lead to the symbol error rate in MPSK.