||In mobile communication systems, the Doppler shift is a very common cause of time-varying changes in the carrier-frequency offset. This thesis discusses the use and design of an adaptive line enhancer to aid in the process of detecting and tracking these time-varying changes. Specifically, it describes the use of an adaptive line enhancer in quadrature amplitude modulation (QAM), or phase shift keying (PSK), carrier-recovery systems. It shows that finite-impulse response (FIR) adaptive line enhancers are not well-suited for tracking changes in the carrier frequency offset, and that an infinite-impulse response (IIR) adaptive line enhancer is much more effective for this application. Then, based on the LMS algorithm, the IIR adaptive line enhancer algorithm is derived. Finally, an FIR filter is developed that is optimal for out-of-band noise rejection, and is used in conjunction with the IIR adaptive line enhancer to further improve its line-enhancement capabilities. The FIR and IIR adaptive line enhancer algorithms are then simulated under typical mobile communication scenarios, to compare the effectiveness of each algorithm. Then, it is shown that the IIR adaptive line enhancer, along with the optimal FIR filter, is much more effective at tracking changes in the carrier-frequency offset.