# Hybrid MU-MIMO Precoding Based on K-Means User Clustering
## 2 Precoding Schemes
### Linear Precoding  
Linear precoding schemes, as used in Long-Term Evolution (LTE), suppress the inter-user interference through channel inversion. Techniques like Zero Forcing (ZF) and Regularized Zero Forcing (RZF) for single-antenna User Equipment (UEs) or Block Diagonalization (BD) for multiple-antenna UE have reduced complexity [7]. The channel matrix with complex coefficients corresponding to \(N_{T}\) transmit antennas and \(K\) single-antenna users is described by \(\mathbf{H}\in\mathbb{C}^{K\times N_{T}}\). The precoder identified by the matrix \(\mathbf{W}\in\mathbb{C}^{N_{T}\times K}\)and represented in Figure 1 can be defined by the following equations:  
\[\mathbf{W}_{ZF}=\beta\mathbf{H}^{-1}, \tag{1}\]where \(\mathbf{x}\in\mathbb{C}^{K\times 1}\) is the set of complex symbols intended for every \(K\) user, \(\mathbf{y}\in\mathbb{C}^{K\times 1}\) is the received array of symbols by the \(K\) users, and \(\mathbf{n}\sim\mathcal{N}_{\mathbb{C}}\left(\mathbf{0},\sigma_{n}^{2}\mathbf{I }_{K}\right)\) is the vector of Additive White Gaussian Noise (AWGN) of variance \(\sigma_{n}^{2}\). It follows a circularly-symmetric complex Gaussian distribution with zero mean and correlation matrix \(\sigma_{n}^{2}\mathbf{I}_{K}\). \[ \begin{array}{ll}\mathbf{W}_{RZF}&=\beta\times\arg\min_{\mathbf{W}}\mathbb{ E}\left\{||\beta^{-1}(\mathbf{H}\mathbf{W}\mathbf{x}+\mathbf{n})-\mathbf{x}||^{2}\right\} \\ &=\beta\mathbf{H}^{H}\left(\mathbf{H}\mathbf{H}^{H}+\frac{N_{T}\sigma_{n}^{2}}{ \mathbf{I}_{Tx}}\mathbf{I}\right)^{-1},\end{array}  \tag{2}\]  
where \(P_{tx}=\mathbb{E}\left\{\mathbf{x}^{H}\mathbf{x}\right\}\) is the total transmit power, \(\mathbb{E}\left\{x\right\}\) is the expectation of a random variable \(x\), \(||\mathbf{x}||\) is the L\({}_{2}\)-norm of a vector \(\mathbf{x}\), superscript \(H\) denotes the transpose complex conjugate, and \(\beta\) is a normalization factor, such that the total transmit power does not change after precoding:  
\[\beta=\sqrt{\frac{N_{T}}{\text{tr}\left(\mathbf{W}\mathbf{W}^{H}\right)}}, \tag{3}\]  
where \(\text{tr}\left(\mathbf{X}\right)\) is the trace of a square matrix \(\mathbf{X}\). The previously-mentioned schemes are used for users with a single antenna. The signal transmitted for other users is treated as interference, and the precoder attempts to cancel it. When users are equipped with multiple antennas, similar methods can be used. In this case, the signal intended for the same user, but for different antennas, is also treated as interference. As there is much more interference to cancel, noise is amplified at the receiver side. In these scenarios, the BD precoder performs better. It works by decomposing the MU-MIMO downlink channel into multiple parallel independent single-user MIMO downlink channels. The precoding matrix is determined such that the intended signal is projected in the null space of all other users' channel matrices. The remaining interference after the inter-user precoding can be canceled with any decoding or precoding techniques. The received signal by \(K\) users equipped with \(N_{R}\) antennas is:  
\[