This paper presents a generalized nonlinear (Markov) analysis technique that is used to evaluate the statistical performance of uniforn~ly sampled digital phase-locked loop (DPLL) demodulators. Although the rigorous analysis of the statistical performance of the analog PLL for digitally modulated signals (i.e., Costas loops) remains an open problem, the discrete time analysis for MPSK modulation is tractable. This paper characterizes the first-order, decisiondirected DPLL based demodulator as a Markov chain. Traditional analytical techniques are used to evaluate the steady-state statistical performance. Numerical results for the steady-state density function are derived for BPSK, QPSK, and 8PSK. The resulting steady-state bit error probabilities for these modulations are also calculated. Traditional Markov analytical techniques (absorbing boundaries) permit a numerical evaluation of the transient characteristics of the DPLL. The numerical work focuses on loops for both unmodulated and BPSK, QPSK, andl 8PSK modulated input signals. The transient characteristics are shown to be a function of the loop bandwidth but converge to those predicted by a diffusion analysis as the loop bandwidth decreases. The susceptibility of small bandwidth loops to hangup is shown to be the largest reason for the acquisition performance difference. The cycle slipping characterization for 'MPSK modulated signals is also a function of the loop bandwidth and each increase (doubling) of the modulation alphabet size reduces the slipping performance by approximately 6dB.

Date of this Version

May 1994