A Study on Noise Folding Due to Nonlinearity in Fractional-N CP PLL

This paper investigates the phenomenon of modulator-contributed phase noise folding to low frequency due to charge pump nonlinearity in fractional-N frequency synthesizers with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Sigma \Delta $ </tex-math></inline-formula> modulators. To address this issue, we derive the noise transfer function of the loop’s equivalent current under <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Sigma \Delta $ </tex-math></inline-formula> modulation by modeling the phase-locked loop based on linear system theory. Since the input signal is a random sequence that cannot be directly applied to a linear model analysis, this paper introduces a new mathematical model that functionalizes the input sequence. This functionalized random variable retains its original probability distribution and power spectrum, while providing a functional expression that can be integrated directly into the system’s transfer function for calculation. Furthermore, the model incorporates the input frequency information, which aids in analyzing the effects of nonlinearity. Using this functionalization method, we conduct a detailed analysis of the quantization noise folding phenomenon and derive exact equations for the noise floor and noise corner frequency under charge pump mismatch. Additionally, we present the complete power spectral density of the output noise under nonlinear effects. To validate the proposed model, a behavioral-level simulation is performed using MATLAB, with results showing strong agreement between the simulation and theoretical predictions, demonstrating the accuracy and efficiency of the proposed approach.