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I need to estimate averaged signal of a generator with known phase noise and the amplitude noise can be neglected.

The averaged signal of the generator is expressed as: $$ \langle A\exp{i(\Omega t + \varphi)}\rangle + \mathrm{c.c.} = Ae^{i\Omega t} \langle \exp{i\varphi} \rangle + \mathrm{c.c.}, $$ with $\Omega$ the generator carrier frequency and $\varphi$ the phase noise, $\langle\varphi\rangle = 0$.

I suppose that phase noise is stationary. Its spectral density $ S_{\varphi\varphi} = \frac 1{2\pi} \int_{-\infty}^\infty\mathrm d\omega \, e^{-i\omega t} \langle \varphi(0)\varphi(t) \rangle$ is given.

To obtain $\langle \exp{i\varphi} \rangle$ I suppose $\varphi$ to have Gaussian statistics. It is easy to show for that case that $$ \langle \exp{i\varphi} \rangle = e^{-\frac{\langle\varphi^2\rangle}2}. $$ Then, one finds $$\langle\varphi^2\rangle = \int_{-\infty}^\infty\mathrm d\omega \, S_{\varphi\varphi}(\omega). $$

I've taken $S_{\varphi\varphi}$ of the Agilent N5173B at 20GHz carrier, see the picture below and the generator's datasheet. (Note the two-sided phase noise spectral density is called $\mathcal L$ in industry, see the IEEE standart.) Agilent N5173B phase noise

I approximate the integral of the spectral density as the square under the connected points shown in the table: $$ \begin{array}{|r|r|} \hline f, \mathrm{Hz}& 1& 10& 100& 10^3& 10^4& 10^5& 10^6& 1.1\cdot10^6& 1.4\cdot10^6\\ \hline S_{\varphi\varphi}, \mathrm{dBc}/\mathrm{Hz}& -40& -50& -80& -90& -95& -95& -125& -130& -135 \\ \hline \end{array} $$ First plotting the spectral density and approximating its integral by hand, than using Python scipy trapz integration, I obtain $$\sqrt{\langle\varphi^2\rangle} \sim 10\,\mathrm{rad}$$ which is hard to believe for such a costly generator. And, according to the abovesaid, it will actually diminish the generator output, as $e^{-\frac{\langle\varphi^2\rangle}2} \sim 0.01$.

So, 0) What the hell? 1) Probably I've misunderstood the IEEE definitions? I guess there should be some narrower frequency cutoffs? 2) Is it valid to assume the phase noise of a generator to be gaussian? 3) Probably better approaches exist for estimating the averaged signal of a generator with phase noise?

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  • $\begingroup$ (I can't help) but maybe you should ask on SE Electrical Engineering. $\endgroup$ – George Herold Sep 5 '14 at 15:19
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What's so hard to believe about this? You have an awful low frequency phase noise of -40dBc/Hz at 1Hz. Lock it to a rubidium standard like the SRS FS725 and that problem should go away. The phase noise of the FS725 is given as <-100dBc/Hz at 1Hz. In other words... there is a VERY good reason why the N5173B has a 10MHz reference input... its internal reference isn't that great. And if that isn't good enough... get yourself a real atomic clock!

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  • $\begingroup$ Sounds reasonable. Then when I'll ensemble average generator's output I'll have sum of sinusoids each with different phase -> almost null as result. $\endgroup$ – Andrii Sep 6 '14 at 10:25
  • $\begingroup$ @Andrii: If you compare the outputs of two generators with large phase noise (by mixing them down to a difference frequency close to zero), you get something that resembles strong amplitude noise (possibly overlaid by a drifting beat frequency). If you use one low noise master oscillator for the reference, the beat goes away and the noise in the mixing product should greatly diminish (in this case by 60dB!). This, of course, assumes, that the PLLs in these instruments are designed correctly, and that all the phase noise comes from the internal reference. $\endgroup$ – CuriousOne Sep 6 '14 at 13:38

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