Further Insights into Spur Immunity in MASH-Based Fractional-N CP-PLLs with Polynomial Nonlinearities.

The interaction of the quantization error introduced by the digital delta-sigma modulator (DDSM) and nonlinear distortion in the loop generates spurious tones in the output phase noise spectrum of a fractional-N frequency synthesizer. Previous research has used the periodic nonlinearity noise (PNN) method to analyze the spur immunity of MASH DDSMs of order 2, 3 and 4. The calculation of the PNN was based on an empirically derived probability distribution of the accumulated quantization error. It was hypothesized that spur immunity is guaranteed in the case of a polynomial distortion with an order up to $(l-1)$ when an lth ordered MASH DDSM is applied. This paper presents a method to calculate theoretically the probability distribution of the accumulated quantization error, based on which it is proven that a 5th order MASH DDSM guarantees spur immunity when the order of the polynomial distortion is up to 4.