Exact sequence lengths in MASH digital delta-sigma modulators for fractional-N frequency synthesizers

The formulae for calculating the exact sequence lengths of multi-stage noise shaping (MASH) digital delta-sigma modulator (DDSM) for any irreducible rational input constant without initial condition are presented in this paper. The sequence lengths are determined by the order of MASH DDSM and the denominator of the input. The factor of the sequence length as a function of the denominator of the input has period that is determined by the order of MASH DDSM. The periods of the factors of the sequence lengths are 1, 2, 6, 6, 30 and 30 for the first 6 orders respectively. The minimum sequence length is equal to the denominator of the input for all orders. The maximum sequence lengths are 1, 2, 6, 12, 60 and 60 times the denominator of the input for the first 6 orders respectively. The formulae provided are suitable for MASH DDSM of orders from 1 to 6 and can be extended to cover higher orders. The simulation and measurement results are provided to support our conclusions.