# Multi-User Linear Equalizer and Precoder Scheme for Hybrid Sub-Connected Wideband Systems
## 4 Multi-User Equalizer Design
### Proposed Method Derivation  
Due to the non-convex nature of the merit function and the constraint imposed in problem (18), it is difficult or even impossible to obtain an analytical solution to the optimization problem at hand. Moreover, as we are considering a multi-user scenario, the resulting average BER is a weighted function of the average BER for each user, making it even harder to obtain a solution to the aforementioned problem. Hence, instead of an exact solution to problem (18) we will derive, in the following, an algorithm to obtain an approximate solution to the previous optimization problem. Using the exponential upper bound of the Q-function \((\mathcal{Q}(x)\leq(1/2)e^{-x^{2}/2})\), we obtain \(\mathcal{Q}(x)\leq(1/2)(1+x^{2}/2)^{-1}\), and as a consequence we have  
\[\text{BER}\leq\frac{1}{2U}\sum_{u=1}^{U}\frac{1}{1+\text{SINR}_{u}}. \tag{19}\]  
Replacing Equation (16) in Equation (19), and after some mathematical manipulations, we obtain,  
\[\text{BER}\leq\frac{\sigma_{n}^{2}}{2U\text{I}\text{N}_{c}}\sum_{u=1}^{U}\sum_ {k=0}^{N_{c}-1}\biggl{[}\mathbf{H}_{\alpha\
eq k}^{H}\mathbf{W}_{a}\Bigl{(} \mathbf{W}_{a}^{H}\mathbf{W}_{a}\Bigr{)}^{-1}\mathbf{W}_{a}^{H}\mathbf{H}_{ \alpha\
eq k}+\sigma_{n}^{2}\mathbf{I}\biggr{]}_{u.u}^{-1}, \tag{20}\]  
and then an approximate solution to the optimization problem (18) may be obtained from the following simplified optimization problem  
\[\mathbf{W}_{a}=\arg\min_{\mathbf{W}_{a}}\sum_{u=1}^{U}\sum_{k=0}^{N_{c}-1} \biggl{(}\mathbf{h}_{\alpha\
eq u,k}^{H}\mathbf{W}_{a}\Bigl{(}\mathbf{W}_{a}^ {H}\mathbf{W}_{a}\Bigr{)}^{-1}\mathbf{W}_{a}^{H}\mathbf{h}_{\alpha\
eq u,k}+ \sigma_{n}^{2}\biggr{)}^{-1},\,\text{s.t.}\,\mathbf{W}_{a}\in\mathcal{W}_{a}\,, \tag{21}\]  
where \(\mathbf{h}_{\alpha\
eq u,k}\in\mathbb{C}^{N_{rx}}\) represents the equivalent channel of user \(u\). To solve it, we propose to iteratively compute matrix \(\mathbf{W}_{a}\) column by column, which in practice corresponds to iteratively adding RF chains to the receiver. Let matrix \(\mathbf{W}_{a}^{(i)}\) and vector \(\mathbf{w}_{a}^{(i)}\) denote the first \(i\) columns and column \(i\) of matrix \(\mathbf{W}_{a}\), respectively; then we can define \(\mathbf{W}_{a}^{(i)}=\bigl{[}\mathbf{W}_{a}^{(i)},\mathbf{w}_{a}^{(i)}\bigr{]}\). The aim is to compute \(\mathbf{w}_{a}^{(i)}\) iteratively instead to compute the overall matrix \(\mathbf{W}_{a}\) at once, and thus the optimization problem can be modified as  
\[\biggl{(}\mathbf{w}_{a}^{(i)}\biggr{)}_{\text{opt}}=\arg\min_{\mathbf{w}_{a}^ {(i)}}\sum_{u=1}^{U}\sum_{k=0}^{N_{c}-1}\biggl{(}\mathbf{h}_{\alpha\
eq u,k}^{ H}\mathbf{W}_{a}^{(i)}\biggl{(}\biggl{(}\mathbf{W}_{a}^{(i)}\biggr{)}^{H} \mathbf{W}_{a}^{(i)}\biggr{)}^{-1}\biggl{(}\mathbf{W}_{a}^{(i)}\biggr{)}^{H} \mathbf{h}_{\alpha\
eq u,k}+\sigma_{n}^{2}\biggr{)}^{-1},\,\text{s.t.}\, \mathbf{W}_{a}\in\mathcal{W}_{a}\,. \tag{22}\]  
From the definition of \(\mathbf{W}_{a}^{(i)}\) and the Gram-Schmidt orthogonalization follows \(\mathbf{W}_{a}^{(i)}\bigl{(}\mathbf{W}_{a}^{(i)}{}^{H}\mathbf{W}_{a}^{(i)} \bigr{)}^{-1/2}=\bigl{[}\mathbf{U}^{i-1},\mathbf{P}^{(i-1)}\mathbf{w}_{a}^{(i) }\bigr{]}\)[35], where \(\mathbf{P}^{(i-1)}=\mathbf{I}-\mathbf{U}^{(i-1)}\mathbf{U}^{(i-1)}{}^{H}\), and \(\mathbf{U}^{(i-1)}=\mathbf{W}_{a}^{(i  
with equality if \(\mathbf{W}_{a}^{(i)}(\mathbf{W}_{a}^{(j)^{H}}\mathbf{W}_{a}^{(i)})^{-1}\mathbf{W}_{a }^{(j)^{H}}=\mathbf{I}\). Therefore, if one of the terms of Equation (23) is large, the other one must be small and the following approximation follows,  
\[\mathbf{S}_{u,k}^{(i)}\approx\max\left\{ \begin{array}{l}\mathbf{h}_{eq,u,k}^{ H}\mathbf{U}^{(i-1)}\mathbf{U}^{(i-1)^{H}}\mathbf{h}_{eq,u,k^{\prime}},\\ \mathbf{h}_{eq,u,k}^{H}\mathbf{P}^{(i-1)}\mathbf{w}_{a}^{(i)}\mathbf{w}_{a}^{( i)^{H}}\mathbf{P}^{(i-1)^{H}}\mathbf{h}_{eq,u,k}\end{array} \right\}+\sigma_{n}^{2}.