Phase Noise and Transient Times for a Binary Quantized Digital Phase-Locked Loop in White Gaussian Noise

Digital phase-locked loops of the type employing discrete phase adjustments form an interesting class for which both steady-state and transient performance may be determined in the presence of additive white Gaussian noise. A general technique for obtaining this analysis is presented. A class of "sequential filters" is described that appears to be well suited to this type of loop. Their performance is characterized by a variable number of inputs per output (depending upon the input sequence) and the use of coarse quantization. Two specific examples are discussed. The closed-loop transient analysis shows these loops to have effectively a slew-rate limited phase adjustment, indicating that they are decidedly nonlinear. A digital loop "quasi-bandwidth" measure is defined in terms of this transient response. This definition allows the comparison of digital loops on a basis of equal signal-to-noise ratios within the loop bandwidth and, to a limited extent, makes possible a similar comparison of the digital loops with linear loops. Performance of the digital loops is found to be similar to that of the first-order linear phase-locked loop model for low loop bandwidth signal-to-noise ratio but reaches a limiting minimum phase error due to quantization of the phase adjustments for high signal-to-noise ratio. This limit, however, can be set as low as desired by choosing a small enough phase-correction quantum. The digitalloop bandwidth is found to vary with the signal energy per noise spectral density ratio rather than with the signal amplitude as in the case of the linear analog model.