Editing Phase noise (section)

==Definitions==
An ideal [[electronic oscillator|oscillator]] would generate a pure [[sine wave]].  In the frequency domain, this would be represented as a single pair of [[Dirac delta function]]s (positive and negative conjugates) at the oscillator's frequency; i.e., all the signal's [[power (physics)|power]] is at a single frequency.  All real oscillators have [[phase modulated]] [[Electronic noise|noise]] components.  The phase noise components spread the power of a signal to adjacent frequencies, resulting in noise [[sidebands]].  

Consider the following noise-free signal:

:<math>x(t)= A\cos(2 \pi f_0 t)</math>

Phase noise is added to this signal by adding a [[Stochastic Process|stochastic process]] represented by <math>\phi(t)</math> to the signal as follows:

:<math>x(t)= A\cos(2 \pi f_0 t + \phi (t))</math>

Different phase noise processes, <math>\phi(t)</math>, possess different power [[Spectral density]] (PSD). For example, a white noise PSD follows a <math>f^0</math> trend, a pink noise PSD follows a <math>f^{-1}</math> trend, and a brown noise PSD follows a <math>f^{-2}</math> trend. 

<math> \operatorname{S}_{\phi}(f) </math> is the single-sided (f>0) '''phase noise PSD''' <math> \left[ \frac{rad^2}{Hz} \right] </math>, given by the [[Fourier transform]] of the [[Autocorrelation]] of the phase noise. <ref>{{Citation |last=Rubiola |first=Enrico |year=2008 |title=Phase Noise and Frequency Stability in Oscillators |publisher=Cambridge University Press |isbn=978-0-521-88677-2}}</ref>

:<math> \operatorname{S}_{\phi}(f) = \mathcal{F}\left[ \operatorname{E} \left[ \phi(t)\overline{\phi(t+\tau)} \right]\right]  </math>

The noise can also be represented at the single-sided (f>0) '''frequency noise PSD''', <math>\operatorname{S}_{\Delta \nu}(f) \left[ \frac{Hz^2}{Hz} \right] </math>, or the '''fractional frequency stability PSD''', <math>\operatorname{S}_{y}(f) \left[ \frac{1}{Hz} \right] </math>, which defines the frequency fluctuations in terms of the deviation from the carrier frequency, <math>f_0</math>.

:<math> \operatorname{S}_{\Delta \nu}(f) = f^2\operatorname{S}_{\phi}(f)  </math>

:<math> \operatorname{S}_{y}(f) = \frac{\operatorname{S}_{\Delta \nu}(f)}{f_0^2} = \frac{f^2\operatorname{S}_{\phi}(f)}{f_0^2}</math>

The phase noise can also be given as the '''spectral purity''', <math>\mathcal{L}\left(f\right) \left[ \frac{dBc}{Hz} \right]</math>, the single-sideband power in a 1Hz bandwidth at a frequency offset, f, from the carrier frequency, <math>f_0</math>, referenced to the carrier power.

:<math> \mathcal{L}\left(f\right) = 10\log_{10} \left( \frac{\operatorname{S}_{\phi}(f)}{2} \right) </math>