Stability Analysis of a Digital Rate-Locked Loop

We analyze the locking properties of a digital rate-locked loop that may be used to synchronize a slave oscillator with a master. Formulating the dynamics of this system by a nonlinear differentialintegral equation, we rigorously prove that in the absence of filtering in the loop the slave oscillator output waveform has periodic zero crossings occurring at exactly the same rate as those of the master oscillator. The main requirement for this is only that the quantization of the staircase comparator in the slave oscillator not be too coarse. A similar proof for a more general loop remains elusive.