non-autonomous and rather tricky for investigation. In the phase-frequency domain In the simplest case, when The standard engineering assumption is that the filter removes the upper sideband frequency from the input but leaves the lower sideband without change. Thus it is assumed that the VCO input is This makes a Costas loop equivalent to a phase-locked loop with phase detector characteristic corresponding to the particular waveforms and of the input and VCO signals. It can be proved that filter outputs in the time and phase-frequency domains are almost equal. Thus it is possible to study the simpler autonomous system of differential equations . The Krylov–Bogoliubov averaging method allows one to prove that solutions of non-autonomous and autonomous equations are close under some assumptions. Thus, the Costas loop block diagram in the time domain can be asymptotically changed to the block diagram on the level of phase-frequency relations. The transition to the analysis of an autonomous dynamical model of the Costas loop (in place of the non-autonomous one) allows one to overcome the difficulties related to modeling the Costas loop in the time domain, where one has to simultaneously observe a very fast time scale of the input signals and slow time scale of signal's phase. This idea makes it possible to calculate core performance characteristics - hold-in, pull-in, and lock-in ranges. Frequency acquisition The classical Costas loop will work towards making the phase difference between the carrier and the VCO become a small, ideally zero, value. The small phase difference implies that frequency lock has been achieved. QPSK Costas loop The classical Costas loop can be adapted to QPSK modulation for higher data rates. The input QPSK signal is as follows Inputs of low-pass filters LPF1 and LPF2 are After synchronization, the outputs of LPF1 and LPF2 are used to get demodulated data ( and ). To adjust the frequency of the VCO to the reference frequency, signals and are limited and cross-multiplied: Then the signal is filtered by the loop filter and forms the tuning signal for the VCO , similar to BPSK Costas loop. Thus, QPSK Costas can be described by a system of ordinary differential equations: Here are parameters of LPF1 and LPF2 and are parameters of the loop filter. References Electronic oscillators Communication circuits