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Two different iterative receivers for space-time coded data transmitted through frequency selective fading channels are analyzed. The first one considers a supertrellis to perform data detection and decoding in a joint manner, while the second system utilizes separate trellises for these two tasks. We compare the two schemes in terms of bit error rate and computational effort. Our results show that, in many cases, the supertrellis approach for combined detection and decoding can be a better compromise solution than the separate system.