Influence of initial conditions on the fundamental periods of LFSR-dithered MASH digital delta-sigma modulators with constant inputs

A digital delta-sigma modulator (DDSM) with a constant input may produce a periodic output with a small fundamental period, resulting in strong tonal output behavior instead of the expected shaped white quantization noise. In practice, the problem is alleviated by dithering the DDSM. Pseudorandom dither generators based on linear feedback shift registers (LFSRs) are widely used to "break up" periodic cycles in DDSMs with constant inputs. Pseudorandom dither signals are themselves periodic and can lead to relatively short output sequences from dithered DDSMs. It is known that the fundamental period of the output signal depends not only on the input and the initial condition of the DDSM but also on the initial state of the LFSR. This brief shows that bad LFSR initial conditions can lead to ineffective dithering, producing short cycles and strong tonal behavior. Furthermore, it explains how to set the initial state of the DDSM as a function of the initial state of the LFSR in order to obtain a maximum-length dithered output