Calculation of the cycle length in a HK-MASH DDSM with multilevel quantizers

The HK-Multi stAge noise SHaping (MASH) Digital Delta-Sigma Modulator (DDSM) uses output feedback to make the quantization step appear prime, thereby maximising the cycle lengths for constant inputs. This structure uses a cascade of modified first-order error feedback modulator (EFM1) blocks with one-bit quantizers. Multibit quantizers are often used in DDSMs as the increased number of levels allows the quantizer to behave more linearly, thereby improving the stability of the modulator. In this paper, we analyse a HK-MASH DDSM with multi-level quantizers and show that the cycle length is always N^L, regardless of initial conditions and the input value, where N is the cycle length of each stage and L is the number of stages in the DDSM.