An analytical formulation of phase noise of signals with Gaussian-distributed jitter

The output of many oscillatory systems can be approximated by a stochastic square-wave signal with noise-free amplitude and Gaussian-distributed jitter. We present an analytical treatment of the phase noise of this signal with white and Lorentzian jitter spectra. With a white jitter spectrum, the phase noise is nearly Lorentzian around each harmonic. With a Lorentzian jitter spectrum, it is a sum of several Lorentzian spectra, a summation that has a 1/f⁴ shape at far-out frequencies. With a combination of the two, it has 1/f⁴ and 1/f² shapes at close-in and far-out frequencies, respectively. In all cases, the phase noise at the center frequency and the total signal power are both finite. These findings will improve our understanding of phase noise and will facilitate the calculation of phase noise using time- domain jitter analysis.