Averaging Techniques for the Analysis of Event Driven Models of All Digital PLLs

In this paper, we introduce a statistical approach for studying a special class of nonlinear dynamical systems such as ADPLLs and ADPLL networks, where the process driving the adjustment of the DCO frequency can be seen as ΣΔ modulation. We showed that, by applying the Frobenius-Perron operator to the governing equation, it is possible to find the invariant probability density which is valid for dynamically changing input of ΣΔ modulator. By using this, we show that the average behaviour of the corresponding complex system can be dramatically simplified and studied analytically.