In signal processing, 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). Generally speaking, radio-frequency engineers speak of the phase noise of an oscillator, whereas digital-system engineers work with the jitter of a clock.
Definitions 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. Consider the following noise-free signal:
x ( t ) = A cos ( 2 π f 0 t ) {\displaystyle x(t)=A\cos(2\pi f_{0}t)}
Phase noise is added to this signal by adding a stochastic process represented by ϕ ( t ) {\displaystyle \phi (t)} to the signal as follows:
x ( t ) = A cos ( 2 π f 0 t + ϕ ( t ) ) {\displaystyle x(t)=A\cos(2\pi f_{0}t+\phi (t))}
Different phase noise processes, ϕ ( t ) {\displaystyle \phi (t)} , possess different power Spectral density (PSD). For example, a white noise PSD follows a f 0 {\displaystyle f^{0}} trend, a pink noise PSD follows a f − 1 {\displaystyle f^{-1}} trend, and a brown noise PSD follows a f − 2 {\displaystyle f^{-2}} trend.
S ϕ ( f ) {\displaystyle \operatorname {S} _{\phi }(f)} is the single-sided (f>0) phase noise PSD [ r a d 2 H z ] {\displaystyle \left[{\frac {rad^{2}}{Hz}}\right]} , given by the Fourier transform of the Autocorrelation of the phase noise, as stated in the Wiener–Khinchin theorem.
S ϕ ( f ) = F [ E [ ϕ ( t ) ϕ ( t + τ ) ¯ ] ] {\displaystyle \operatorname {S} _{\phi }(f)={\mathcal {F}}\left[\operatorname {E} \left[\phi (t){\overline {\phi (t+\tau )}}\right]\right]}
The noise can also be represented at the single-sided (f>0) frequency noise PSD, S Δ ν ( f ) [ H z 2 H z ] {\displaystyle \operatorname {S} _{\Delta \nu }(f)\left[{\frac {Hz^{2}}{Hz}}\right]} , or the fractional frequency stability PSD, S y ( f ) [ 1 H z ] {\displaystyle \operatorname {S} _{y}(f)\left[{\frac {1}{Hz}}\right]} , which defines the frequency fluctuations in terms of the deviation from the carrier frequency, f 0 {\displaystyle f_{0}} .
S Δ ν ( f ) = f 2 S ϕ ( f ) {\displaystyle \operatorname {S} _{\Delta \nu }(f)=f^{2}\operatorname {S} _{\phi }(f)}
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