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The demand for high-speed mobile wireless communications is rapidly growing. MIMO technology promises to be a key technique for achieving the high date capacity and spectral efficiency for wireless communication systems of the near future. This dissertation presents an overview of Multiple Input Multiple Output (MIMO) systems. A simple mathematical model is used to describe MIMO systems. Systems' capacity improvement is derived. Various Benefits achievable by MIMO system are discussed. History and open problems in this field are presented as well. An investigation of Space Time Trellis Code (STTC) design criteria is presented.There are four sets of design criteria which apply to different channel models. Design Criteria Set I and Criteria Set II apply to slow fading channels and design Criteria Set III and Criteria Set VI apply to fast fading channels. If a code has small diversity order, design Criteria Set I and Criteria Set III will apply and in the case of large diversity order, Criteria Set II and Criteria Set VI apply. Based on these design criteria, this dissertation developed a construction method of optimal STTC codes for fast fading channels.when the diversity order is large,trace criteria is applicable. Set partition idea is extended to symbol group partition. A set of design rules are developed to construct optimal STTC. The optimality is proved as well. This dissertation also provides simulation results of optimal STTC over other claimed STTC. The simulation results showed that the optimal STTC outperformed all other codes. And at the same times, using the simulation results, the effect of number of transmitting antennas and number of receivering antennas on the performance of STTC are also studied. Finally, open problems for future research are discussed.