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The maximization of the ergodic capacity for single-stream beamforming, which is a (constrained) transmission scheme referred to as “optimum beamforming,” has been extensively addressed in the open literature for multiple-input-single-output (MISO) Rayleigh fading channels and spatially uncorrelated MISO Rician fading channels with a unit transmit covariance matrix, and closed-form solutions have been derived for these cases. However, optimum beamforming for spatially correlated or uncorrelated MISO Rician fading channels with a nonunit transmit covariance matrix has received less attention and remains a complex multidimensional optimization problem. This paper first proves that this convex constrained optimization problem can be reduced to only one dimension; hence, it can be solved very fast using standard 1-D search algorithms. Then, simulations mainly performed for linear equispaced antenna arrays demonstrate that: 1) the proposed method for the calculation of the optimum beamformer has significantly lower computational complexity compared with other currently used multidimensional algorithms; and 2) the optimum beamformer further improves capacity compared with the (single-stream) beamforming transmission that maximizes the signal-to-noise ratio (SNR) at the receiver, whereas in some operational environments, it achieves ergodic capacity that is very close or equal to the maximum ergodic capacity.
Date of Publication: Feb. 2013