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.

[1]  F. L. Walls,et al.  Characterization of frequency stability in precision frequency sources , 1991, Proc. IEEE.

[2]  S.C. Gupta,et al.  Phase-locked loops , 1975, Proceedings of the IEEE.

[3]  Ali H. Sayed,et al.  A survey of spectral factorization methods , 2001, Numer. Linear Algebra Appl..

[4]  Johannes W. M. Bergmans.  Effect of loop delay on phase margin of first-order and second-order control loops , 2005, IEEE Transactions on Circuits and Systems II: Express Briefs.

[5]  P. Schultz,et al.  Fundamentals of geophysical data processing , 1979 .

[6]  Monson H. Hayes,et al.  Statistical Digital Signal Processing and Modeling , 1996 .

[7]  Arnaldo Spalvieri.  Optimal Loop Filter of the Discrete-time PLL in the Presence of Phase Noise , 2006, 11th IEEE Symposium on Computers and Communications (ISCC'06).

[8]  Umberto Mengali,et al.  Synchronization Techniques for Digital Receivers , 1997, Applications of Communications Theory.

[9]  Ara Patapoutian.  Application of Kalman filters with a loop delay in synchronization , 2002, IEEE Trans. Commun..

[10]  Poras T. Balsara,et al.  All-Digital PLL With Ultra Fast Settling , 2007, IEEE Transactions on Circuits and Systems II: Express Briefs.

[11]  Peter F. Driessen.  DPLL bit synchronizer with rapid acquisition using adaptive Kalman filtering techniques , 1994, IEEE Trans. Commun..

[12]  G. S. Christiansen.  Modeling of PRML timing loop as a Kalman filter , 1994, 1994 IEEE GLOBECOM. Communications: The Global Bridge.

[13]  Heinrich Meyr,et al.  Synchronization in digital communications , 1990 .

[14]  Ara Patapoutian.  On phase-locked loops and Kalman filters , 1999, IEEE Trans. Commun..

[15]  Jan W. M. Bergmans.  Effect of loop delay on stability of discrete-time PLL , 1995 .

[16]  Bosco Leung,et al.  PLL-based frequency discriminator using the loop filter as an estimator , 2002 .