Question 19.13: A clock’s zero crossings are modulated by a sinusoidal distu......
A clock’s zero crossings are modulated by a sinusoidal disturbance at a frequency f_{m} so that
\phi _{k} = Φ_{0}\sin (2\pi f_{m}t) (19.71)
What is the resulting rms absolute jitter, period jitter, and adjacent period jitter?
The phase noise S_{\phi }^{2}(f) will have a discrete spectral tone at f_{m} .
S_{\phi }(f) = \frac{Φ_{0}^{2}}{2}δ(f – f_{m})
Key Point: Periodic variations in the phase of a clock result in discrete tones in phase noise spectra and are called spurs.
Discrete tones in phase noise spectra are called spurs because on a plot of S_{\phi }^{2}(f) versus frequency they appear as vertical spikes. The integral in (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)
is easily evaluated to give the absolute jitter,
\sigma _{\tau }^{2} = \left\lgroup\frac{T_{0}}{2\pi } \right\rgroup^{2} \left\lgroup\frac{Φ_{0}^{2}}{2} \right\rgroup = \frac{1}{8} \left\lgroup\frac{T_{0}Φ_{0}}{\pi } \right\rgroup^{2} (19.72)
The period jitter is reduced slightly by the extra term in the integrand of (19.64)
\sigma _{J}^{2} = \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)
because the change in absolute jitter from one transition to the next is less than the total range over which it deviates.
\sigma _{J}^{2} = \frac{1}{2} \left\lgroup\frac{T_{0}Φ_{0}}{\pi } \right\rgroup^{2} \sin ^{2}(\pi f_{m}T_{0}) (19.73)
Note that if the phase modulation is slow (i.e. f_{m}T_{0}~ «~ 1 ) the period jitter is small since \phi _{k} varies very little from one cycle to the next. The same is true of adjacent period jitter.
\sigma _{C}^{2} = 2\left\lgroup\frac{T_{0}Φ_{0}}{\pi } \right\rgroup^{2} \sin ^{4}(\pi f_{m}T_{0}) (19.74)