The effects of noise in oscillators

An explicit expression for the output signal from an oscillator with several noise sources in the circuit is derived. This formula describes qualitatively and quantitatively the manner in which thermal and shot noise act to corrupt the performance of an ideal oscillator. The statistical properties of the signal are then evaluated, as it emerges from the oscillator stage, after passage through an output filter and after being operated on by an ideal n-times multiplier. Expressions are derived for the short term frequency stability, the power spectral density, and the power spectrum of the signal, as well as for the spectral density of the signal phase. The key to the results reported is an apparently novel perturbation technique which does not require smoothing of the instantaneous nonlinearity in the basic differential equation. Discussion of the solutions shows that the instantneous nonlinearities cause the device to act simultaneously like a linear AGC oscillator and like a high Q passive tuned circuit, with each aspect accorded one half the total noise excitation. Possible implications of this effect for other types of transient conditions in oscillators are indicated briefly.