# Definition of Phase Noise

I'm confused with the definition of phase noise. I'm referring to this IEEE document. A noisy oscillator output is $v(t) = A\sin(\omega t + \phi)$, where phi is random phase deviation. I assumed no amplitude noise. So carrier power is $P = \frac{A^2}{2*R_L}$.

I can find the $\phi$, in radians and compute its onesided power spectral density with periodogram in matlab. Thus, I have $S_\phi$ in rad$^2$/hz.

Now, the IEEE document says, phase noise is defined as one half of $S_\phi$, but also says its unit is dbc/hz. Should not we dividing $S_\phi$ with P to get dbc/hz, but then $S_\phi$ unit is rad$^2$/hz not power, i.e. P is in Watts?

How can I correctly compute single side band phase noise spectrum in dbc/hz from $S_\phi$ which is in rad$^2$/hz?

Edit: the document I posted is old (1999). This is the active one. But still, my understanding seems to be correct.

• Instead of calculating the power spectral density of the phase phi you should be calculating the power spectral density of the Noise surrounding the carrier resulting from the phase deviations. Phi is a time domain parameter while the carrier power is described in the frequency domain. No wonder you get confused ! – Bimpelrekkie Dec 3 '15 at 14:50
• If I understand it correctly, that would be equal to half of s_phi, one sided power spectrum of instantaneous phase deviation. So I don't have to divide s_phi to carrier power, phase noise is simply half of what I mistakenly computed (s_phi). Am I right? – Aenid Dec 4 '15 at 6:44
• I don't think so because you need to convert that phase into frequency through a sin(x) function. If you search for them you should be able to find matlab examples with phasenoise. Have a look at those for some ideas. – Bimpelrekkie Dec 4 '15 at 7:05
• I see. That's why document says L(f) is approximation. sin(x) = x when x is small. – Aenid Dec 4 '15 at 9:47

$L(f) = \frac{S_\phi(f)}{2}$; $S_\phi(f)$ : One-sided spectral density of the phase deviation $L(f)$: One-half of the phase instability
From the RF spectrum, one can compute $L(f)$ by computing noise power around the carrier in dbc/hz giving $\sin(\phi)$. Since $\phi$ is small, $\sin(\phi) = \phi$, you got the spectrum of phase noise. So one does not need to compute phase deviations to compute phase noise.