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# Phase Noisebasics

Phase noise is the frequency-domain representation of random fluctuations in the phase of a waveform, corresponding to time-domain deviations from perfect periodicity (jitter). See phase noise.

An ideal oscillator would generate a pure sine wave. In the frequency domain, this would be represented as a single pair of Dirac delta functions (positive and negative conjugates) at the oscillator's frequency; i.e., all the signal's power is at a single frequency. All real oscillators have phase modulated noise components. The phase noise components spread the power of a signal to adjacent frequencies, resulting in noise sidebands. Oscillator phase noise often includes low frequency flicker noise and may include white noise.

let denote the ideal signal, phase noise can be modeled by adding a stochastic process : , where can be composed of frequency components not related to f. E.g., thermal noise, shot noise, and 1/f noise (flicker noise).

Here we model as: , where:

• is a Gaussian variable denotes the noise components of the system, with standard deviation of σsync
• is the cumulative sum of a Gaussian variable denotes the phase deviation of the oscillator, which accumulates over time. The standard deviation of the un-accumulated Gaussian variable is σacc.
• cumsum is the cumulative summation, e.g. cumsum({1, 2, 3}) = {1, 3, 6}
Toolbox: phase noise generator
f =     σsync =     σacc =
Time domainFrequency domain