Folding of Phase Spectrum in Phase-Locked Loops

In charge-pump phase-locked loops used as frequency synthesizers, the phase error between the reference and the divider output is sampled at the reference rate. This operation is an inherent sub-sampling of the VCO phase and it causes conversion of VCO noise and disturbances from highto low-frequency offsets, which may dominate over reference and charge-pump noise. Starting from the discrete-time linear model of PLLs, this paper proposes a model, which takes into account spectrum folding and allows deriving closed-form expressions of output noise as a function of noise contributors.