Phase noise measurements and effective noise bandwidth

[dnessett]

Hello,

This is an advanced question. I am attempting to measure and characterize the phase noise of oscillators. After a rather long preparatory period, I am now able to carry out measurements using a setup involving a mixer generating a signal for which variations in voltage represent variations in oscillator frequency. I won't go into detail how this setup works because it is not germane to the question. However, those interested can read the discussion starting here.

The measurements sample this signal using a PicoScope 4262. Since the version of the PicoScope software I am using (on Linux) does not support output in dBm @ 50 ohms, I have to make measurements in dBV units, then convert to dBm. The measurement setup includes selecting a Blackman-Harris window. This is the foundation of the question.

From what I have read in the literature, using a window to shape the input to a DFT (hereafter, FFT) calculation requires correction when stochastic signals like noise are involved. In particular the window, which is applied in the time domain, creates a transfer function in the frequency domain that "bleeds" power into adjacent bins of the FFT. This requires correction in computing the power spectral density (PSD) of the noise.

This is discussed in this paper, from which I now quote:

"Due to the width of the window in the frequency domain, each frequency bin collects not only the noise in that frequency bin, but also from adjacent bins. Dividing the result by the effective noise bandwidth corrects for this phenomenon." (page 15, paragraph at top of page).

This applies specifically to white noise. So, if an FFT calculation involves computing the spectrum of white noise, to get the correct value, it is necessary to divide the result by the effective noise bandwidth associated with the window. For Blackman-Harris, the divisor is 1.7.

However, phase noise is not comprised only of white noise. Various noise sources combine to generate phase noise. For example, variations in very small frequency offsets may occur due to FM flicker noise, at higher offsets the source of these variations occur due to random walk phase noise, then there may be a region of offsets where the noise is generated by flicker phase noise. Finally, further out in frequency offset, white noise may be the source of frequency fluctuations.

It is not my intention to open a discussion of these noise processes. Rather, they are mentioned only to establish that the PSD of phase noise is not simply that of white noise.

This brings us to the question. How should FFT calculations of phase noise be adjusted to correct for "bleeding" associated with a windowing function? Since, the portion of the spectrum generally of interest (i.e., that close to the fundamental frequency of the oscillator) is generated by noise, but not by white noise, is it correct to divide the spectrum values by the constant associated with that window? Or, should some other strategy by used?