Consider a transmitter and receiver that are in relative motion as in Problem 10.1. Once again assume that the linearized loop equations hold. Under this assumption determine the PLL phase error as a function of time for the all-pass and low-pass loop filters. Demonstrate that the validity of the assumption of the linearized loop equations depends on the value of the gain K0.
A transmitter is sending an unmodulated tone of constant energy (a beacon) to a dis-tant receiver. The receiver and transmitter are in motion with respect to each other such that d(t) = D[1— sin (mt)] + D0, where d(t) is the distance between the transmitter and receiver (possibly this represents an aircraft doing “figure-eight” maneuvers over a ground station), and D, m, and D0, are constants. This relative motion will cause a Doppler shift in the received transmitter frequency of
Where ∆ω0 is the Doppler shift, wo is the nominal carrier frequency, V(t) = d(t) is the relative velocity between the transmitter and receiver, and c is the speed of light. Assuming that the linearized loop equations hold and that the receiver's I'LL is in lock (zero phase error) at t = 0, show that an appropriately designed first-order loop can maintain frequency lock.