Question 19.12: What is the rms period jitter of the sinusoid in additive wh......
What is the rms period jitter of the sinusoid in additive white noise from Example 19.9?
Again assuming the noise is much smaller than the amplitude of oscillation, \sigma _{n} ~«~ A , it was shown in Example 19.9 that the absolute jitter is also white. Hence, the phase noise is constant. Combining (19.57)
\sigma ^{2}_{\tau } = \frac{\sigma ^{2}_{n}}{A^{2} \omega _{0}^{2}} (19.57)
and (19.60)
\sigma ^{2}_{\tau } = \left\lgroup\frac{T_{0}}{2\pi } \right\rgroup ^{2} \int\limits_{0}^{1/2T_{0}}{S_{\phi }(f)df} (19.60)
therefore yields
S_{\phi }(f) =\left\lgroup\frac{8\pi ^{2}}{T_{0}} \right\rgroup \left\lgroup\frac{\sigma _{n}^{2}}{A^{2}\omega _{0}^{2}} \right\rgroup = \frac{2T_{0}\sigma _{n}^{2}}{A^{2}} (19.69)
Integration in (19.64)
\sigma ^{2}_{J} = \left\lgroup\frac{T_{0}}{\pi } \right\rgroup ^{2} \int\limits_{0}^{1/2T_{0}}{\sin ^{2}(\pi fT_{0})S_{\phi }(f)df} (19.64)
gives the variance of the period jitter.
\sigma ^{2}_{J} = \frac{2\sigma _{n}^{2}}{A^{2}\omega _{0}^{2}} (19.70)
The rms period jitter is the square root of (19.70).