Mathematical analysis of digital MASH delta-sigma modulators for fractional-N frequency synthesizers

A mathematical analysis is performed to investigate the periodic behavior of digital MASH delta-sigma modulators (DSM). The analysis is performed on first, second and third order MASH digital DSMs with odd initial conditions on the first stage. We prove that the maximum sequence length for the first order modulator is 2^N and for the second order modulator is 2^{N + 1}, where N is the word length of the state. In the case of first and second order modulators, the maximum sequence length can be achieved when odd digital inputs are applied. In the case of the third order modulator, the sequence length is equal to 2^{N + 1}for all digital inputs