Optimal Loop Filter of the Discrete-time PLL in the Presence of Phase Noise

The design of the loop filter for the linearized model of the discrete-time phase-lock loop is investigated in the paper. Phase noise, which always affects oscillators, is modelled as a stationary random waveform and is taken into account in the design. Being the phase noise a predictable random process, optimization of the loop filter is worked out in the light of linear prediction. The main result of the paper is that the optimal loop filter is the causal and minimum phase factor of the spectrum of phase noise plus white additive noise. From this result they descend a lower bound on the variance of the phase error and a closed-form expression for the optimal bandwidth of the loop. Exploiting our analysis, we show that the popular perfect second order loop is optimal in tracking oscillators affected by random frequency walk plus random phase walk. Also, similarities and differences between the discrete-time PLL and the continuous-time PLL are investigated in the paper.