Pull-In Frequency for a Pulse-Tracking Phase-Locked Loop with Imperfect Integration

Automatic detection using correlation relies on a phaselocked loop for synchronization. If the loop contains an imperfect integrator, only a finite error frequency (pulse rate) can be tracked. This concise paper yields the transcendental equations (and some specific numerical results for a maximally flat loop response) that relate duty-cycle, loop-bandwidth, integrator imperfection, and pull-in frequency when the input is noiseless. Analysis is based on piecewise-linear methods which are exact, although solutions are found numerically. One exemplary result shows that for a duty cycle of 5 percent and an integrator pole
(b)
to loop zero
(a)
ratio of of
b/a = 0.005
, the pull-in frequency is 1.05 times the undamped natural loop frequency.