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

An improved block diagonal precoding scheme for MIMO multicast channel with two users