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In this paper, we propose an adaptive linear-receiver structure based on interpolated finite-impulse response (FIR) filters with adaptive interpolators for direct-sequence code-division multiple-access (DS-CDMA) systems in multipath channels. The interpolated minimum mean-squared error (MMSE) and the interpolated constrained minimum-variance (CMV) solutions are described for a novel scheme, where the interpolator is rendered time-varying in order to mitigate multiple-access interference and multiple-path propagation effects. Based upon the interpolated MMSE and CMV solutions, we present computationally efficient stochastic gradient and exponentially weighted recursive least squares type algorithms for both receiver and interpolator filters in the supervised and blind modes of operation. A convergence analysis of the algorithms and a discussion of the convergence properties of the method are carried out for both modes of operation. Simulation experiments for a downlink scenario show that the proposed structures achieve a superior bit-error-rate convergence and steady-state performance to previously reported reduced-rank receivers at lower complexity.