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This paper proposes a general framework of space-time-frequency codes (STFCs) for multi-band orthogonal frequency division multiplexing (MB-OFDM) ultra-wide band (UWB) communications systems. A great similarity between the STFC MB-OFDM UWB systems and conventional wireless complex orthogonal space-time block code (CO STBC) multiple-input multiple-output (MIMO) systems is discovered. This allows us to quantify the pairwise error probability (PEP) of the proposed system and derive the general decoding method for the implemented STFCs. Based on the theoretical analysis results of PEP, we can further quantify the diversity order and coding gain of MB-OFDM UWB systems, and derive the design criteria for STFCs, namely diversity gain criterion and coding gain criterion. The maximum achievable diversity order is found to be the product of the number of transmit antennas, the number of receive antennas, and the FFT size. We also show that all STFCs constructed based on the conventional CO STBCs can satisfy the diversity gain criterion. Various baseband simulation results are shown for the Alamouti code and a code of order 8. Simulation results indicate the significant improvement achieved in the proposed STFC MB-OFDM UWB systems, compared to the conventional MB-OFDM UWB ones.