Stability Analysis of an Nth Power Digital Phase-Locked Loop--Part II: Second- and Third-Order DPLL´s

The behavior of a digital phase-locked loop (DPLL) which tracks the positive-going zero crossings of the incoming signal can be characterized by a nonlinear difference equation in the phaseerror process. This equation was first presented by Gill and Gupta for the CW loop, and modified by Osborne and Lindsey for the th power loop. Some stability results had been previously obtained by other authors for first- and second-order loops by linearizing the equation about the steady-state solution. In Part I, a mathematically more rigorous and powerful approach was introduced whereby the acquisition behavior was studied by formulating the equation as a fixedpoint problem. Stability results can be obtained by using theorems such as Ostrowski\´s Theorem and the Contraction Mapping Theorem. Part I then presented some new stability results (and rederived some previously obtained results) for first-order DPLL\´s. In Part II, we present the results for the second- and third-order DPLL\´s. It is shown how both second- and third-order DPLL\´s have gain-dependent limitations on the frequency offsets which can be withstood. These restrictions imply that the higher order DPLL\´s cannot track a frequency ramp, regardless of the ramp slope, and are in contrast to the results for analog loops. Important restrictions on the signal power are also noted for both loops.