Mathematical calculation of sequence length in LFSR-dithered MASH digital delta-sigma modulator with odd initial condition

Multi-stAge noise SHaping (MASH) digital delta-sigma modulator (DDSM) with linear feedback shift register (LFSR) dither and odd initial condition is analyzed mathematically to investigate its periodic behavior. We derive the nonlinear equation of the quantization error in each stage to determine the corresponding sequence length. And five useful lemmas are introduced to help cope with the complicated calculation caused by the pseudo-random dither signal.