Clock and data recovery/Structures and types of CDRs/The (slave) CDR based on a second order PLL/The jitter transfer function

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[edit] Jitter transfer function of the second order slave CDR

The transfer function for a sinusoidal input (that is the Jitter Transfer function!) is:

\tfrac{y(s)}{x(s)}=\tfrac{1}{\left ( \tfrac{s^2}{\omega_n^2}+\tfrac{2s\zeta}{\omega_n}+1\right ) }

The magnitude of the jitter transfer function of jω tells, at each frequency f = ω/2π , the amplitude of the output jitter for an input jitter with the amplitude of 1 radian ≈ 57.3°.

\left | \tfrac{y(j\omega)}{x(j\omega)}\right | =\tfrac{1}{\sqrt{\left ( 1 - \tfrac{\omega^2}{\omega_n^2}\right )^2+\left ( \tfrac{2\omega\zeta}{\omega_n}\right )^2}}

The following figure is the Bode magnitude plot of the jitter transfer function. Curves for different values of the parameter ζ (damping ratio) are shown:

The magnitude of jitter filtering, for different values of the damping ratio.
Peak amplification for low values of the damping ratio.


It can be seen that the CDR is essentially a low-pass filter for the phase jitter. There is no amplification of the input jitter but for values of the damping ratio smaller than \sqrt{2} = 0.707, at some frquencies at and around the resonant frequency. The peak amplification occurs at \left(\tfrac{\omega_r}{\omega_n}\right) = \sqrt{\left ( 1 - 2 \zeta^2 \right ) } and is:

\tfrac{1}{\sqrt{\left ( 2 \left ( \tfrac{\omega_r}{\omega_n}\right ) ^2 \left ( 1- \left ( \tfrac{\omega_r}{\omega_n}\right )^2 \right ) +\left (\left (1-\left (\tfrac{\omega_r}{\omega_n}\right )^2\right )\right )^2\right ) }}
.



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