Though I have gone through a communications course in EE, I am still very blur about the Phase-locked loop system in performing coherent demodulation of AM signal. The problem with direct coherent detection straight from the received AM signal from the receiver antennae is that the local oscillator cannot reproduce the exact same carrier frequency to be hecterodyned with the AM signal to bring it back to baseband. Many reasons colud have caused that, for example, the phenomenon of frequency shifting due to relative motion of the Tx or the Rx (What is the name I forgot), etc etc. So there needs to be a negative feedback to ensure the frequency of the local oscillator approaches the frequency of the carrier in the AM signal. But I really do not understand the mathematics analysis of the system. Can anyone give me an insight to that?