Efficient and Reliable Small-Signal Estimate of Quantization Noise Contribution to Phase Noise in $\Delta \Sigma $ Fractional- $N$ PLL

The effect of the quantization noise introduced by <inline-formula> <tex-math notation="LaTeX">$\Delta \Sigma $ </tex-math></inline-formula> modulators in fractional-<inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> phase locked loop is studied in the time domain through a variational model of the corresponding analog mixed signal circuits. The latter can be interpreted as hybrid dynamical systems involving discontinuity boundaries and switching in the vector field. Discontinuity mapping is used to define an accurate high order model to propagate perturbations at discontinuity boundaries. This novel approach is more efficient than computing power spectral density of large-signal waveforms from long lasting time domain simulations. Furthermore, with respect to established macro-modeling techniques, it provides a better understanding of how electric elements influence the noise performance. The inherent features of the circuit components turn out to be directly embedded both in the system fundamental matrix and in the discontinuity mapping involved in its dynamical evolution. The validity of the approach is verified through numerical simulations.

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