Phase-locked Loop and Costas loop are electronic circuits widely used in communication systems for carrier recovery and demodulation.
However, rigorous analysis of stability for these circuits is often omitted. Analytical methods such as Lyapunov functions and frequency-domain stability criterions can be effectively used to obtain various stability ranges (PLL bandwidth): pull-in and lock-in.
Another application of the analog PLLs is generation of chaotic signals, which are inherently (ultra) wideband signals. Chaotic PLLs can be used to generate bandpass wideband signals, therefore, they are ideal candidates to generate a wideband carrier. The generated chaotic signal should be a bandpass signal centered about a prescribed carrier frequency and they should have a flat (uniform) power spectral density. Here third and fourth-order analog PLLs are considered for generation of desired wideband signals.