This paper proposes statistical analysis of phase-only correlation functions between two signals with stochastic phase-spectra following bivariate circular probability distributions based on directional statistics. We give general expressions for the expectation and variance of phase-only correlation functions in terms of joint characteristic functions of the bivariate circular probability density function. In particular, if we assume bivariate wrapped distributions for the phase-spectra, we obtain exactly the same results between in case of a bivariate linear distribution and its corresponding bivariate wrapped distribution.

This paper proposes statistical analysis of phase-only correlation functions with phase-spectrum differences following wrapped distributions. We first assume phase-spectrum differences between two signals to be random variables following a linear distribution. Next, based on directional statistics, we convert the linear distribution into a wrapped distribution by wrapping the linear distribution around the circumference of the unit circle. Finally, we derive general expressions of the expectation and variance of the POC functions with phase-spectrum differences following wrapped distributions. We obtain exactly the same expressions between a linear distribution and its corresponding wrapped distribution.

This paper proposes statistical analysis of phase-only correlation functions based on linear statistics and directional statistics. We derive the expectation and variance of the phase-only correlation functions assuming phase-spectrum differences of two input signals to be probability variables. We first assume linear probability distributions for the phase-spectrum differences. We next assume circular probability distributions for the phase-spectrum differences, considering phase-spectrum differences to be circular data. As a result, we can simply express the expectation and variance of phase-only correlation functions as linear and quadratic functions of circular variance of phase-spectrum differences, respectively.