This dissertation studies the signal processing aspect of multi-input multi-output (MIMO) communications. The contribution of this dissertation is twofold.; First, this dissertation presents a new perspective to the MIMO communications: any MIMO scheme can be regarded as a MIMO channel decomposer, which decomposes (in an information lossy or lossless manner) a MIMO channel into multiple scalar sub channels. Based on this perspective, this dissertation presents three novel MIMO transceiver designs, the geometric mean decomposition (GMD) scheme, the uniform channel decomposition (UCD) scheme, and the tunable channel decomposition (TCD) scheme. All these schemes deploy either a decision feedback equalizer (DFE) at the receiver or a dirty paper precoder (DPP) at the transmitter. These transceiver designs represent a paradigm shift from the conventional linear MIMO transceiver designs to the nonlinear ones. The superior performance of the GMD and UCD schemes unveils the practical significance of making transmitter and receiver cooperate with each other. That is, such cooperations facilitate achieving the optimal tradeoff between the diversity gain and multiplexing promised by the MIMO communication theory. The TCD scheme represents a unifying solution to a considerably wide range of problems, including designing the precoder for orthogonal frequency division multiplexing (OFDM) communications and the optimal code division multiple access (CDMA) sequence design.; Second, this dissertation introduces two novel matrix decomposition algorithms, i.e., the geometric mean decomposition (GMD) and the generalized triangular decomposition (GTD). The two matrix decompositions form the cornerstones of the three transceiver designs proposed in this dissertation. Moreover, the two decompositions have significant implications in the matrix analysis community. For instance, the GTD is a new solution to the inverse eigenvalue problem.
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