Resolving Near-Carrier Spectral Infinities Due to 1/f Phase Noise in Oscillators

In this paper, we derive an expression for the near-carrier power spectral density of an oscillator having 1/f phase noise. Motivated by empirical metrics such as the Allan variance, we develop a rigorous mathematical analysis and derive a closed-form expression for the oscillator autocorrelation function in the case of exactly 1/f phase noise that is smoothed using a rectangular time window. We show that this smoothed 1/f phase noise results in a finite variance noise process and preserves oscillator stationarity. Furthermore, in agreement with experimental data, we explain how a quadratic and a logarithmic term appear in the autocorrelation function and establish the relationship between the logarithmic term and the 1/f characteristics of the oscillator random process.