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Multiple-input/multiple-output (MIMO) frequency selective fading channels introduce both the intersymbol interference (ISI) and the inter-(sub)channel interference (ICI) that are the major performance limiting factors for MIMO wireless data communications. This paper focuses on joint suppression of such two interferences in receiver design under the criterion of the minimum mean-squared error (MMSE). The well-known Kalman filtering arises as a natural design tool to achieve the optimum joint suppression of the ISI and ICI in the sense of MMSE that results in a two-stage design method that suppresses the ISI and ICI in series. Specifically, a state-space solution is derived for the optimum decision-feedback equalizer (DFE) based on temporal Kalman filtering, and is shown to be equivalent to a conventional finite length DFE for ISI suppression. A novel spatial Kalman filtering scheme is proposed as the second stage to successively cancel the ICI. It is shown that the proposed two-stage method yields the realizable and optimum joint ISI and ICI suppression, while it admits lower computational complexity than the conventional finite length DFE for ISI suppression and the MMSE V-BLAST receiver for ICI suppression. The results are applicable to any linear channels that admit state-space realizations, including MIMO frequency selective time-varying channels, and are thus the most general. Simulation results show the effectiveness of the proposed two-stage method and its robustness to channel estimation uncertainties that may arise in practical systems.