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Substantial progress has been made in the receiver signal-processing algorithms for wireless communications to minimize the requirements on signal-to-noise (and/or interference) power ratio and computational complexities for the same quality of service. In cellular infrastructure systems, one of the key system design objectives in the base stations is to maximize the receiver sensitivity, so that the required signal level from the mobile stations can be minimized. The use of advance signal-processing algorithms, based on maximum a posteriori (MAP) estimation, iterative (turbo) channel estimation, equalization, and decoding, allows for a reduction of the required transmitter power by one-third to one-half. Lower computational complexities in the terminals, which implies a reduced power drain on the digital circuits, can be achieved by using techniques that adapt the state complexity of the receiver to the propagation channel. We give an in-depth review of these algorithms, and discuss their performance and implementation requirements.