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I have been trying to measure the phase noise of an oscillator with a homodyne interferometric measurement with an I Q mixer in the microwave domain. The output I and Q are digitized. So, after calculating the Fourier transform, the PSD is in the unit of V^2/Hz.

Is there any way to convert this unit to rad^2/Hz? I have referred to Rubiola's chart. But I don't understand these conversion methods since, in this case, the carrier frequency is removed. Furthermore, I found this reference, where the conversion is described by $$S_{\phi}(f) = (V_{\phi}(f)/K_{\phi})^2$$ where \$ K_{\phi} \$ is the phase detector constant in V/rad. And, \$ V_{\phi}(f) \$ is the equivalent phase noise measured in V^2/Hz. Is there a way to measure this phase detector constant of the digitizer, including the interferometric measurement setup? Or is there another method to convert the phase noise from V^2/Hz to rad^2/Hz? (what I have now is \$ V_{\phi}(f) \$).

I have tried different methods to convert it, but none of them seem to work, or maybe I am not able to find the right method.

Thanks in advance!

Andrew Morton
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Greta
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  • If you compute atan(Q/I) (note: make sure you use atan2 here), you get the phase of your signal with respect to your reference phase (the reference you used to obtain I and Q). The spectrum of the arctangent is the phase noise. If you solely look at the spectrum of I*cos(wt)+jQ*sin(wt), you can obtain the SSB phase noise spectrum, which is an approximation of (half) the true phase noise spectrum assuming the phase deviations are small and amplitude noise is negligible. – LetterSized Aug 11 '22 at 15:26
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    Also strictly speaking, SSB phase noise is specified in dBc/Hz. What you seem to be looking for specifically is the spectral density of the phase noise itself, which is in rad^2/Hz. That is, if we have a signal v=Acos(wt+phi), the SSB phase noise is the spectral density of skirts around v with respect to the total power in v (units dBc/Hz). The true phase noise is the spectral density of phi, in rad^2/Hz. If you do a Taylor series expansion of v around phi=0, you can show the relation between the two: SSB phase noise is 3dB smaller than the true spectral density of phi. – LetterSized Aug 11 '22 at 15:32
  • Thank you so much for your comment. @LetterSized. I was wondering; if I have the voltage equivalent of the phase noise power spectral density in the unit of V^2/Hz. Can I convert to rad^2/Hz using the phase detector constant K_phi? Is this factor the phase to the voltage gain of the mixer? If yes, may I please know how to measure this? – Greta Aug 13 '22 at 08:59
  • It's not clear to me how exactly you are doing the phase detection. But if you truly have a phase detector, then you should multiply by K_phi^2 to go between V^2/Hz and rad^2/Hz. This of course assumes K_phi is in units of rad/V. To answer your final question, it would be helpful to have a block diagram of your measurement, just to make sure we're on the same page about how exactly you are doing the phase detection. – LetterSized Aug 14 '22 at 16:29

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