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This book is concerned with synchronization in communications systems. Synchronization failure may have catastrophic effects on the performance of a communication system. The Kalman filter has a great potential as a tool for synchronization. Although not widely used in the field of communications, the Kalman filter is popular in many other fields such as radar and navigation. Using the Kalman filter has many advantages. For the application, only the state space model of the problem is required. High performance is achieved with relatively low complexity. A single Kalman filter can solve a number of estimation problems encountered. In this work, the application of the Kalman filter to a number of synchronization and channel estimation problems is investigated. Both single and multi-carrier systems are considered. For the multi- carrier sytem (OFDM), a Kalman filter is applied to solve most of the synchronization problems, estimating frequency, phase and timing offsets, as well as estimating the multipath channel. For single carrier systems, the carrier synchronization task is proposed to be performed using an array of Kalman trackers.