On the multistage design of optimal-NTF ΔΣ modulators-The case of fractional synthesizers

Digital ΔΣ modulators can be conveniently implemented as multi-stage noise shaping (MASH) architectures whenever some power of the first-difference differentiator is acceptable as the noise transfer function (NTF). Yet, other types of NTF can be more performant in some applications. In fact, strategies for a full adaptation of the NTF to the modulator environment have recently been studied. In this work, the possibility of implementing such optimized NTFs with MASH setups is considered. Although stability cannot be guaranteed as in conventional MASH designs, splitting the modulator in stages can still provide advantages. The case of fractional frequency synthesizers is taken as an example. Through the discussion, an empirical invariant on the overall number of output levels required for stable operation is identified.