Expected Number of Spikes of Phase Locked Loop Demodulators

Abstract

A new method is presented for finding the expected number of spikes in a phase locked loop of any order, with or without modulation. The procedure can also be employed to determine the threshold of FMFB, FM discriminators and the Maximum Likelihood Estimator. The low pass equivalent gaussian noises x(t), y(t) in the differential equation describing the system (PLL or FMFB) are replaced by the deterministic time functions (Conditional Expectations) 1) E[x(t)/x(0), ẋ(0)] 2) E[y(t)/y(0), ẏ(0)] and solved on a digital computer. The mid spike time (t=0)is taken to be the time when x(0) (quadrature noise)=0, and a surface or surfaces in ẋ(0), y(0), ẏ(0) space are determined which indicates the region A where spikes in the demodulator are obtained. From this the expected number of spikes per second is calculated. Results are presented for the first, second, and third order phase locked loops, and for an ordinary FM discriminator (which can be shown to be equivalent to a PLL of infinite gain). The second order loop used a constant plus integral filter, while the third order loop used a constant plus integral plus double integral filter.