Oscillator Phase Noise - Oscillators and Frequency Correlation

Oscillators and Frequency Correlation

With driven cyclostationary systems that have a stable time reference, the correlation in frequency is a series of impulse functions separated by f_o = 1/T. Thus, noise at f₁ is correlated with f₂ if f₂ = f₁ + kf_o, where k is an integer, and not otherwise. However, the phase produced by oscillators that exhibit phase noise is not stable. And while the noise produced by oscillators is correlated across frequency, the correlation is not a set of equally spaced impulses as it is with driven systems. Instead, the correlation is a set of smeared impulses. That is, noise at f₁ is correlated with f₂ if f₂ = f₁ + kf_o, where k is close to being an integer.

Technically, the noise produced by oscillators is not cyclostationary. This distinction only becomes significant when the output of an oscillator is compared to its own output from the distant past. This might occur, for example, in a radar system where the current output of an oscillator might be mixed with the previous output after it was delayed by traveling to and from a distant object. It occurs because the phase of the oscillator has drifted randomly during the time-of-flight. If the time-of-flight is long enough, the phase difference between the two becomes completely randomized and the two signals can be treated as if they are non-synchronous. Thus, the noise in the return signal can be taken as being stationary because it is 'non-synchronous' with the LO, even though the return signal and the LO are derived from the same oscillator. If the time-of-flight is very short, then there is no time for the phase difference between the two to become randomized and the noise is treated as if it is simply cyclostationary. Finally, if the time-of-flight significant but less than the time it takes the oscillator’s phase to become completely randomized, then the phase is only partially randomized. In this case, one must be careful to take into account the smearing in the correlation spectrum that occurs with oscillators.