Diophantine frequency synthesis

A methodology for fine-step, fast-hopping, low-spurs phase-locked loop based frequency synthesis is presented. It uses mathematical properties of integer numbers and linear Diophantine equations to overcome the constraining relation between frequency step and phase-comparator frequency that is inherent in conventional phase-locked loop based frequency synthesis. The methodology leads to fine-step, fast-hopping, modular-structured frequency synthesizers with potentially very low spurs, especially in the vicinity of the carrier. The paper focuses on the mathematical principles of the new methodology and the related number theoretic algorithms.