Some Non-Linear Effects of Quantization in Phase-Locked Loops

The combination of feedback and quantization causes special problems. (Each on its own has its problems, like instability and round-off errors.) The main problem is the likely existence of multiple equilibria or limit cycles. Another is possible coherence between the input signal and quantization error or limit cycle, making the error indistinguishable from whatever signal characteristic we want to measure. In this paper the conditions in which limit cycles occur are described, and the coherence between limit cycles and incoming signals, which can lead to spurious measurements, are examined. Particular mention is made of the fact that the phase-locked loop (PLL) exhibits a non-linear and phase-dependent relationship between amplitude quantization and the resultant phase quantization. A remedy for some of these problems, in the form of a design rule specifying desirable quantization resolution at various points in a typical phase-locked loop, is described.