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In this paper, three novel stochastic gradient algorithms for adjustment of the widely linear (WL) minimum mean-squared error (MMSE) filter for multiple access interference (MAI) suppression for direct-sequence code-division multiple access (DS-CDMA) are introduced and analyzed. In particular, we derive a data-aided WL least-mean-square (LMS) algorithm, a blind WL minimum-output-energy (MOE) algorithm, and a WL blind LMS (BLMS) algorithm. We give analytical expressions for the steady-state signal-to-interference-plus-noise ratios (SINRs) of the proposed WL algorithms, and we also investigate their speed of convergence. Wherever possible, comparisons with the corresponding linear adaptive algorithms are made. Both analytical considerations and simulations show, in good agreement, the superiority of the novel WL adaptive algorithms. Nevertheless, all proposed WL algorithms require a slightly lower computational complexity than their linear counterparts.