# Generator phase noise influence on its averaged signal

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.)

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?

• (I can't help) but maybe you should ask on SE Electrical Engineering. – George Herold Sep 5 '14 at 15:19