As part of finishing up my doubling phase noise paper, I had to revisit this question:
"How much information do we need from a reported Allan variance/deviation in order t compare it to a phase noise power spectral density"
This is important to me because some of the low noise measurement people only report (and some only measure) Allan deviation of their (clock) signal.
One of the results I wanted to compare with was in the paper: Optical Frequency Synthesis and Comparison with Uncertainty at the 10^-19 level (Science), (figure 2B, curve (ii))...
As described in elog:1074, you can directly convert from a phase/frequency noise PSD to an Allan deviation...if you look at the equation for conversion (section titled: Time response and frequency filter properties), you can see that in the case of white phase noise (frequency noise which goes up as f in amplitude), the Allan deviation level at all times depends on measurement bandwidth (increases linearly with bandwidth).
To figure out what the heck is going on, I made some pretend noise spectra in MATLAB, and forced a 1/f^4 roll-off at different frequencies, and looked at different noise shapes...
(1) White Phase Noise with different frequency roll-offs
(2) White Frequency Noise with different roll-offs
(3) 1/f^0.5 Phase Noise with different roll-offs
From the wiki page:
**PSD** Shape |
ADEV Shape |
White Freq |
1 / sqrt(Tau) |
White Phase |
1 / Tau |
1/f Phase [rad^2/Hz] |
sqrt(ln(Tau)) / Tau |
The first two are obviously consistent. I'm not sure about the third by inspection, but it easily could be consistent |