Constant-time discontinuity map for forward sensitivity analysis to initial conditions: Spurs detection in fractional-N PLL as a case study

The constant-time discontinuity map is presented and used to compute the nonlinear effects of quantization noise injected in fractional-N phase locked loops by delta-sigma modulators. The effects of quantization noise are studied through sensitivity analysis of the nominal trajectory of a noiseless integer-N phase locked loop with respect to a small additive perturbation. It is shown that, to accurately measure the effects of quantization noise injection, a simple linearized model is not sufficient. A nonlinear constant-time discontinuity map across the switching points of the vector field allows the sensitivity analysis to be significantly improved, at a small computational cost.