ON THE INSTANTANEOUS FREQUENCY OF GAUSSIAN STOCHASTIC
PROCESSES

Patrik Wahlberg

Peter J. Schreier

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

2000 AMS Mathematics Subject Classification: Primary: 60G15, 62M15; Secondary:
60G35, 94A12.

Keywords and phrases: Continuous-time Gaussian stochastic processes, instantaneous
frequency, probability density function, Wigner spectrum.