Calculation of sequence lengths in MASH 1-1-1 digital delta sigma modulators with a constant input

Knowledge of exact sequence lengths of undithered digital delta sigma modulators (DDSM) with respect to the input, initial conditions, and quantizer modulus, enables designers to predict spur intervals. It is necessary to know the sequence length for some applications such as fractional-N frequency synthesizers. In this paper, sequence lengths of multi stAge noise SHaping (MASH) DDSMs comprising first order error feedback modulators (EFM) up to three stages are calculated. Exact formulae for calculating sequence lengths with all possible inputs, initial conditions and moduli are provided for first and second order modulators. In the case of a third order modulator, the sequence length is found for the special case when it is not divisible by 3 but it is divisible by 4.