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Phase Noise Description & Equations

Phase noise distortion of digital signal - RF CafePhase noise measurements quantify the short term stability of a frequency source. That is because phase and frequency are mathematically related by a differential function [ω(t) = dΦ(t)/dt] so they are directly connected. Phase noise also includes amplitude instability due to atomic scale effects like shot noise and flicker noise, and possibly even voltage supply noise. When that source is used as a local oscillator in a frequency converter (up- or down-), the amount of instability (jitter) is modulated onto the transmitter or received signal. While not usually a major concern in analog systems, in high speed digital communication systems phase noise can degrade the ability of the receiver to correctly determine the difference between a "1" and a"0." That is because the "decision point" at which a circuit declares the waveform to represent a "1" or "0" can fall at a place where an incorrect decision is made. The drawing below shows how a digital waveform can be distorted by phase noise during the up- or down-conversion and modulation or demodulation process.

Phase noise is typically symmetrical about the primary signal (carrier) of a local oscillator so frequency and power values normally references a single sideband (SSB) in a 1 Hz bandwidth.

Phase noise multiplication in PLL - RF CafeWhen the reference oscillator is used by a phase locked loop (PLL) frequency source that produces an output frequency higher than that of the reference, the phase noise power levels are multiplied by a factor of 20*log (fout/fref), thereby degrading the final phase noise specifications.

Consider, as an example the USXTA-10MEX-SXXB, 10 MHz ovenized crystal oscillator (OCXO) manufactured by Bliley Technologies, right here in my hometown of Erie, Pennsylvania. The graph below is where I plotted the advertised phase noise of the OCXO (red) and the phase noise of a 2 GHz oscillator phase locked to the 10 MHz reference oscillator. You will see that the phase noise of the 2 GHz oscillator is consistently 46 dB [20*log(2*109 / 10*106)] higher than the 10 MHz reference, per the above equation. Because of the multiplication effect, many S-band and high oscillators use a 100 MHz reference oscillator in order to gain a roughly 20 dB [20*log(10/1)] improvement in phase noise.

Use the following equation to calculate the phase noise of a phase-locked oscillator:

Phase NoisePLL = Phase NoiseRef + 20*log (fPLL/fRef)  {dBc/Hz}



The following formulas are available in many textbooks and application notes. To be honest, very few people really use these equations. Their value to most people is for demonstrating the composition of phase noise.

Note: When using these formulas, be sure to keep dimension units consistent; i.e., do not mix kHz with MHz, mm with inches, etc. It is safer to use base units (e.g., Hz, m) for calculation, then convert result to desired units.


Phase noise voltage equation - RF Cafe,  where Signal amplitude - RF Cafe= signal amplitude
Signal frequency - RF Cafe= signal frequency
Signal phase - RF Cafe= signal phase
Signal amplitude variation - RF Cafe= signal amplitude variation
Signal phase variation - RF Cafe= signal phase variation

Noise power density - RF Cafe noise power density

Single sideband (SSB) phase noise - RF Cafe single sideband (SSB) phase noise

SSB Phase Noise in dB relative to the carrier - RF Cafe SSB phase noise in dB relative to the carrier



Posted October 9, 2018

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    Kirt Blattenberger,

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