On the statistical properties of phase noise in Fractional-N frequency synthesizers using successive requantizers

The noise-shaping quantizer is commonly used in Fractional-N Frequency Synthesizers to provide a control signal to the feedback divider with a fractional mean component and high-pass spectral properties. The Digital Delta-Sigma Modulator (DΔΣM) is typically used to carry out this function. Unfortunately, divide ratio modulation is a significant component of the PLL phase noise, in particular due to interactions between modulation-induced phase noise and nonlinear loop dynamics. A recently proposed modulator, named the Successive Requantizer, claims to offer better spurious component and phase noise performance compared to the traditional DΔΣM, while offering similar noise-shaping properties. This paper describes a method of deriving the statistical properties of both the modulated control signal and the corresponding phase noise, for given state transition matrices. Convergence of the noise PMF is also described.