Analysis of direct spectrum measurement of a sinusoidal signal impaired by either fractional Gaussian phase noise or fractional Brownian phase motion

This article discusses the empirical spectrum (periodogram) of a sinusoidal signal that is phase modulated by either fractional Gaussian noise (fGn) or fractional Brownian motion (fBm). These two cases are of note because they are frequently used to model oscillator phase instabilities known as "1/f noise" and "1/f³ noise," respectively. This work demonstrates that an fGn phase noise may result in a 1/f-shaped spectrum, as might be expected. However, we also show that a sinusoid impaired by an fBm phase noise with 1/f³ noise properties will yield either a Gaussian shaped spectrum or a 1/f²-shaped spectrum, depending on measurement duration. We also prove that the empirical spectrum in both cases is ergodic. Thus, it is guaranteed that a spectrum measurement performed on a single sinusoidal source will result in the same spectrum shapes as listed above, providing that sufficient averaging of consecutive captures is conducted.