Wiener´s Analysis of the Discrete-Time Phase-Locked Loop With Loop Delay

The discrete-time phase-locked loop (PLL) operating at the steady state is considered in this paper. Phase noise, which always affects oscillators, is modeled as a stationary random waveform and optimization of the loop filter is worked out in the light of Wiener´s theory. Delay in the loop, which may affect the numerical implementation of the PLL, is considered. The main results and novelties of this paper are the optimal loop filter and the minimum mean-square error (MSE) that can be achieved for a given spectrum of phase noise and for a given delay in the loop. Closed-form expressions for the loop filter and for the MSE are given for the case where the phase noise is characterized as a second-order disturbance. Application of these results to carrier recovery based on the Costas loop is presented in this paper.