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This letter addresses the issue of joint space-time trellis decoding and channel estimation in time-varying fading channels that are spatially and temporally correlated. A recursive space-time receiver which incorporates per-survivor processing (PSP) and Kalman filtering into the Viterbi algorithm is proposed. This approach generalizes existing work to the correlated fading channel case. The channel time-evolution is modeled by a multichannel autoregressive process, and a bank of Kalman filters is used to track the channel variations. Computer simulation results show that a performance close to the maximum likelihood receiver with perfect channel state information (CSI) can be obtained. The effects of the spatial correlation on the performance of a receiver that assumes independent fading channels are examined.