On the instantaneous frequency of Gaussian stochastic processes

TL;DR

This paper derives the probability density function of the instantaneous frequency for nonstationary Gaussian processes, extending stationary process results and revealing that the IF has zero or infinite variance at fixed times.

Contribution

It provides the first explicit PDF of the IF for nonstationary Gaussian processes and links the mean IF to the Wigner spectrum's frequency moment.

Findings

IF has zero or infinite variance at fixed times

Mean IF equals the normalized first order frequency moment of the Wigner spectrum

Extends stationary process results to nonstationary processes

Abstract

This paper concerns the instantaneous frequency (IF) of continuous-time, zero-mean, complex-valued, proper, mean-square differentiable nonstationary Gaussian stochastic processes. We compute the probability density function for the IF for fixed time, which extends a result known for wide-sense stationary processes to nonstationary processes. For a fixed time the IF has either zero or infinite variance. For harmonizable processes we obtain as a byproduct that the mean of the IF, for fixed time, is the normalized first order frequency moment of the Wigner spectrum.