Phase noise in oscillators: DAEs and colored noise sources

Oscillators are key components of electronic systems. Undesired perturbations, i.e. noise, in practical electronic systems adversely affect the spectral and timing properties of oscillators resulting in phase noise, which is a key performance limiting factor, being a major contributor to bit-error-rate (BER) of RF communication systems, and creating synchronization problems in clocked and sampled-data systems. In this paper, we first present a theory and numerical methods for nonlinear perturbation and noise analysis of oscillators described by a system of differential-algebraic equations (DAEs), which extends our recent results on perturbation analysis of autonomous ordinary differential equations (ODEs). In developing the above theory, we rely on novel results we establish for linear periodically time-varying (LPTV) systems: Floquet theory for DAEs. We then use this nonlinear perturbation analysis to derive the stochastic characterization, including the resulting oscillator spectrum, of phase noise in oscillators due to colored (e.g.,,1/f noise), as opposed to white, noise sources. The case of white noise sources has already been treated by us in a recent publication. The results of the theory developed in this work enabled us to implement a rigorous and effective analysis and design tool in a circuit simulator for low phase noise oscillator design.