# Multi-User Linear Equalizer and Precoder Scheme for Hybrid Sub-Connected Wideband Systems
## 4 Multi-User Equalizer Design
### Proposed Method Derivation
\tag{25}\]  
Nonetheless, due to the independence of the \(U\) user channels, if \(\mathbf{w}_{a}^{(i)}\) leads to a large value for the term \(\mathbf{h}_{eq,u,k}^{H}\mathbf{P}^{(i-1)}\mathbf{w}_{a}^{(i)}\mathbf{w}_{a}^{ (j)^{H}}\mathbf{P}^{(i-1)^{H}}\mathbf{h}_{eq,u,k}\) for user \(u_{i}=u\), then with high probability for the other users \(u\
eq u_{i}\), the value of this term will be small. For this reason, \(\mathbf{S}_{u,k}^{(i)}\) may be approximated as follows  
\[\mathbf{S}_{u,k}^{(i)}\approx\left\{ \begin{array}{l l}\mathbf{h}_{eq,u,k}^{ H}\mathbf{U}^{(i-1)}\mathbf{U}^{(i-1)^{H}}\mathbf{h}_{eq,u,k},&u\
eq u_{i}\\ \mathbf{h}_{eq,u,k}^{H}\mathbf{P}^{(i-1)}\mathbf{w}_{a}^{(i)}\mathbf{w}_{a}^{ (j)^{H}}\mathbf{P}^{(i-1)^{H}}\mathbf{h}_{eq,u,k}+\sigma_{n}^{2},&u=u_{i}\end{array}  \right.. \tag{26}\]  
Therefore, the optimization problem (22) can be approximated by  
\[\mathbf{w}_{a}^{(i)}=\arg\min_{\mathbf{w}_{a}^{(i)}}\left(\alpha^{(i)}+\sum_{ k=0}^{N_{c}-1}\!\left(\mathbf{h}_{eq,u,k}^{H}\mathbf{P}^{(i-1)}\mathbf{w}_{a}^{ (i)}\mathbf{w}_{a}^{(j)^{H}}\mathbf{P}^{(i-1)^{H}}\mathbf{h}_{eq,u,k}+\sigma_{ n}^{2}\right)^{-1}\right)\text{s.t.}\ \mathbf{w}_{a}^{(i)}\in\mathcal{W}_{a,u_{i}}\,, \tag{27}\]  
where \(\alpha^{(i)}=\sum_{k=0}^{N_{c}-1}\sum_{u\
eq u_{i}}\left(\mathbf{h}_{eq,u,k}^{ H}\mathbf{U}^{(i-1)}\mathbf{U}^{(i-1)^{H}}\mathbf{h}_{eq,u,k}+\sigma_{n}^{2} \right)^{-1}\) is a constant and thus it only depends on the channels \(\mathbf{h}_{eq,u,k}\) and matrix \(\mathbf{U}^{(i-1)}\) computed in the previous iteration. \(\mathcal{W}_{a,u_{i}}\) represents the column \(i\) of the elements of set \(\mathcal{W}_{a}\). Therefore, the optimization problem (22) may be simplified to  
\[\mathbf{w}_{a}^{(i)}=\arg\min\,\sum_{k=0}^{N_{c}-1}\!\left(\mathbf{h}_{eq,u_{ i},k}^{H}\mathbf{P}^{(i-1)}\mathbf{w}_{a}^{(i)}\mathbf{w}_{a}^{(i)^{H}}\mathbf{P}^{( i-1)^{H}}\mathbf{h}_{eq,u_{i},k}+\sigma_{n}^{2}\right)^{-1}\text{s.t.}\ \mathbf{w}_{a}^{(i)}\in\mathcal{W}_{a,u_{i}}\,. \tag{28}\]  
In spite of the previous simplifications, the optimization problem is still non-convex and hard to solve due to the constraint \(\mathbf{w}_{a}^{(i)}\in\mathcal{W}_{a,u_{i}}\). Therefore, to further simplify it, we replace the set \(\mathcal{W}_{a,u_{i}}\) by the codebook \(\mathcal{F}_{a,u_{i}}=\mathbf{D}_{u_{i}}\mathbf{A}_{\pi,u_{i}}\), where \(\mathbf{D}_{u_{i}}\) is a block diagonal matrix where all blocks are zero except \(u_{i}\), which is equal to the identity matrix, i.e., the elements of the codebook are the normalized receiver array response vectors, which leads to the simpler optimization problem  
\[\mathbf{w}_{a}^{(i)}=\arg\min\,\sum_{k=0}^{N_{c}-1}\!\left(\mathbf{w}_{a}^{(i) ^{H}}\mathbf{P}^{(i-1)^{H}}\mathbf{h}_{eq,u_{i},k}\mathbf{h}_{eq,u_{i},k}^{H} \mathbf{P}^{(i-1)}\mathbf{w}_{a}^{(i)}+\sigma_{n}^{2}\right)^{-1}\text{s.t.}\ \mathbf{w}_{a}^{(i)}\in \mathcal{F}_{a,u_{i}}\,. \tag{29}\]  
Notice that we need to follow some criterion to associate a given user to iteration \(i\). We propose to do this association using the following mapping between users and iterations \(u_{i}=\text{imod}U\), i.e., the \(U\) users are interleaved along the iterations. For example, for \(N_{RF}=4\) and \(U=2\), we have \([u_{1},u_{2},u_{3},u_{4}]=[1,2,1,2]\). The procedure to obtain the analog part of the equalizer matrix is presented in Algorithm 1. It can be summarized as follows: Firstly we start with user 1, \(\mathbf{U}^{(0)}=0\) (line 1) and compute the projection matrix \(\mathbf{P}^{(1)}\) (line 4). After that, we compute the merit function of the optimization problem (29) for each element of the codebook \(\mathcal{F}_{a,u_{1}}\) (lines 5-9). Vector \(\mathbf{w}_{a}^{(1)}\) is set to be equal to the element of codebook \(\mathcal{F}_{a,u_{1}}\) with the lowest value (lines 10-11). Then, the column vector \(\mathbf{P}^{(0)}\mathbf{w}_{a}^{(1)}\) is added to matrix \(\mathbf{U}^{(0)}\) to form \(\mathbf{U}^{(1)}\) (line 12).