Rigorous analysis of delta-sigma modulators for fractional-N PLL frequency synthesis

In this paper, rigorous analyses are presented for higher order multistage noise shaping (MASH) Delta-Sigma ((delta) (sigma)) modulators, which are built out of cascaded first-order stages, with rational DC inputs and nonzero initial conditions. Asymptotic statistics such as the mean, average power, and autocorrelation of the binary quantizer error are formulated using a nonlinear difference equation approach. An important topic of interest considered here is the fractional-N phase-lockedloop frequency synthesis applications, where the input to the modulator has to be a rational constant. It has been mathematically shown that, regardless of the initial conditions, first-order and second-order MASH (delta)(sigma) modulators with rational DC inputs cannot sufficiently randomize the quantization error samples, and, therefore, are not suitable for fractional-N synthesis applications. An irrational initial condition imposed on the first accumulator of a third or higher order MASH modulator, on the other hand, annihilates the tones throughout the whole output spectrum, and provides a very smooth noise shaping. Simulation results are provided to support the theoretically derived results. Implementation issues of the irrational initial condition in the digital domain are also discussed and investigated together with the effect of finite accumulator size on the noise-shaping quality factor.