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Since the introduction of the diversity-multiplexing tradeoff (DMT) by Zheng and Tse for ML reception in frequency-flat MIMO channels, some results have been obtained also for the DMT of frequency-selective MIMO channels and for the DMT of suboptimal receivers such as linear (LEs) and decision-feedback equalizers (DFEs) for frequency-selective SIMO channels or frequency-flat MIMO channels. We have recently extended these results to the case of linear receivers for frequency-selective MIMO channels. However, the diversity properties of linear receivers turn out to be fairly catastrophic. In this paper we show that full diversity can be restored by the introduction of a convolutive linear MIMO precoding scheme that we showed earlier to allow to attain the optimal DMT for ML or DFE detection (in the frequency-flat case). The precoder needs to be used with a moderate amount of redundancy in the form of zero-padding, and with a MMSE design for the linear equalizer. A MMSE-ZF design also benefits substantially from the precoding. The proposed scheme is a significant extension of an earlier SISO result by Tepelenlioglu to the MIMO case.