Modeling of frequency random walk instability and effects on spectral spreading

Oscillator phase instability is addressed and a mathematical model is offered for the frequency random walk component of oscillatory motion. The covariance relations of the phase random process resulting from the frequency random walk model are derived and used to determine the auto-correlation function of the associated RF sinusoidal signal. The model also assumes initial conditions for phase and frequency. Expressions for the auto-correlation function and power spectral density are derived which involve related Airy functions. Exact as well as asymptotic expansions are presented along with error limits. It was found that the asymptotic expansion falls well within the precision capability of most computational devices. Plots of power spectral densities are presented for oscillators modulated by a CW or a finite pulse train as used in the radar case.