Calculation of cycle lengths in higher-order MASH DDSMs with constant inputs

A mathematical analysis is performed to investigate the periodic behavior of Multi stAge noise SHaping (MASH) Digital Delta-Sigma Modulators (DDSMs). The analysis is performed on fourth- and fifth-order MASH DDSMs with an odd initial condition on the first stage and all other states initially zeroed. We prove that the maximum cycle length for the fourth- and fifth-order MASH DDSM is 2^N+2 where N is the wordlength of the modulator. In the case of fourth-order modulators, the maximum cycle length can be achieved when odd digital inputs are applied. In the case of fifth-order modulators, the cycle length is 2^N+2 for all digital inputs.