Matrix theory plays an important role in precoding methodology for multiple input multiple output (MIMO) systems.\nIn this paper, an improved block diagonal (BD) precoding scheme is proposed for a MIMO multicast channel with two\nusers, where the unitary precoding matrix is constructed in a block-wise form by joint triangularization\ndecomposition. In order to reduce large signal-to-noise ratios (SNRs) spread across different transmitted data streams\nand users, the combination of joint equi-diagonal triangularization (JET) and joint geometric mean decomposition\n(JGMD) is applied to submatrix construction in the inner process of this precoding scheme. An elaborate\nimplementation is presented, and the existence condition of JGMD is also investigated for two complex-valued\nmatrices with two columns, where the analytical result reveals the connection with the particular channel realization\nand essentially determines when to consider JGMD for submatrix construction. In addition, the properties of the\ndiagonal elements generated by joint triangularization decomposition are discussed as well as the computational\ncomplexity of the proposed scheme. Simulation results indicate that in general, JGMD is employed with high\nprobability in the hybrid model, and the proposed scheme readily outperforms the JET scheme in terms of bit error\nrate (BER) performance in the moderate to high SNR regimes.
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