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The linear minimum mean-squared error (MMSE) criterion is known to provide adaptive algorithms for interference suppression in direct-sequence (DS) code-division multiple-access (CDMA) systems. However, standard MMSE adaptation is not robust to fast fading, being unable to compensate for rapid channel variations. In this paper, we provide a framework for deriving robust adaptive algorithms in this setting based on a new differential MMSE (DMMSE) criterion, which is a constrained optimization problem in which the quantity to be tracked is the ratio of the data appearing in two successive observation intervals. When applied to a DS-CDMA system with short spreading waveforms (i.e., with period equal to the symbol interval) operating over a flat-fading channel, the DMMSE criterion avoids tracking the fades, exploiting the negligible variation of the fading gain over two consecutive symbols. For frequency-selective fading, the DMMSE criterion is extended to provide a new eigenrake receiver which provides interference suppression and diversity combining without requiring explicit information regarding the desired user's propagation channel.