Abstract
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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N+2</sup> 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 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N+2</sup> for all digital inputs.
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