Analysis of Nonlinearity-Induced Boundary Spur Families in Fractional- N Frequency Synthesizers Using the Periodic Nonlinearity Noise Concept

Spurious tones arise in the output spectra of fractional- N frequency synthesizers due to nonlinear distortion of the accumulated quantization error introduced by the digital $\Delta \Sigma $ modulator (DDSM). Both empirical results and rigorous mathematical foundations presented in previous works have demonstrated that such nonlinearity-induced spurs can be predicted by using the Periodic Nonlinearity Noise (PNN) concept. Using the continuous-time extension of the PNN, this paper establishes the distinguishing features of integer- and fractional-boundary spur families. Furthermore, it shows how to predict the locations and amplitudes of fractional boundary spurs in common multi-stage noise shaping (MASH) DDSMs.