Linear Periodically Time-Variant Digital PLL Phase Noise Modeling Using Conversion Matrices and Uncorrelated Upsampling

This paper introduces a conversion matrix method for linear periodically time-variant (LPTV) digital phase-locked loop (DPLL) phase noise modeling that offers precise and computationally efficient results to enable rapid design iteration and optimization. Unlike many previous studies, which either assume linear time-invariance (LTI) and therefore overlook phase noise aliasing effects, or solve LPTV systems with noise folding and multiple sampling rate conversions that heightens modeling and computational complexity, the proposed conversion matrix method allows the designer to represent LPTV systems using intuitive LTI-like transfer functions with excellent accuracy. Additionally, the uncorrelated upsampling method addresses the cross-correlated spectrum of cyclostationary noise sources by a simple matrix multiplication. This eliminates the need to consider the beat frequency of the upsampled noise source and the system with different sampling rates, thus improving computational efficiency. The proposed algorithm is applied to modeling a integer-
$N$
DPLL with time-varying proportional loop gain, and the modeling accuracy is validated with Simulink transient simulations.