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We study the design and analysis of adaptive linear-quadratic (LQ) receivers for time-varying, frequency-selective CDMA environments. We propose a simple, intuitively appealing cost function that can be maximized to find linear-quadratic (LQ) detectors that are optimal in a certain sense. We discuss the motivation and derivation of the proposed LQ cost function and derive an adaptive algorithm for LQ detectors that can be viewed as a generalization of the minimum-output-energy algorithm for the MMSE linear detector. We also discuss an approach to adaptive signal design for time-varying channels based on the proposed LQ cost function. Finally, we present the results of a simulation study that clearly demonstrates the potential performance advantage of the proposed LQ receivers vis-g-vis MMSE linear receivers.