The transmit beamforming technique in a wireless communication system is a beamforming technique applied to the downlink channel where the downlink is the transmission link from the transmitter to a single receiver or multiple receivers.

In order to describe beamforming process efficiently, we assume that the transmitter has $ M $ transmit antennas and each receiver has a single antenna. The received signal at the receiver in the transmit beamforming system is modelled as

$ \mathbf{x} = \sum_{i=1}^K \mathbf{w}_i s_i = \mathbf{W} \mathbf{s} $

where $ s_i $ in $ \mathbf{s} = [s_1, s_2, \ldots, s_K] $ is the message signal (chosen from a complex Gaussian alphabet with a constraint of power $ P/M $) intended for the $ i $th receiver and $ \mathbf{w}_i \in \mathbb{C}^{M \times 1} $ in $ \mathbf{W} = [\mathbf{w}_1, \mathbf{w}_2, \ldots, \mathbf{w}_M] $ is the corresponding beamforming vector.

$ y=ax+b $

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