# Multiuser Equalizer for Hybrid Massive MIMO mmWave CE-OFDM Systems
## 3 Precoder and Equalizer Design
### Hybrid Equalizer
#### 3.2.2 Design of Fixed Analog Feedforward Equalizer  
In this section, the analog part of the hybrid equalizer is designed. From (11) and after obtaining the solution for \(\mathbf{B}_{d,k}^{(i)}\) and \(\mathbf{W}_{d,k^{\prime}}^{(i)}\), we have the following optimization problem:  
\[\mathbf{W}_{a}= \begin{array}{ll}\operatorname*{argmin}\sum\limits_{k=1}^{N_{c }}\text{MSE}_{k}\\ \text{s.t}\mathbf{W}_{a}\in\mathcal{W}_{a},\end{array}  \tag{22}\]  
where the power constraint of (11) or (18) is not considered, since it is assured by \(\mathbf{\Omega}_{d}\), given by Equation (21), computed in the digital feedforward equalizer. Since the analog part is computed in the first iteration, for simplicity, we removed the index \(i\), and then \(\text{MSE}_{k}\) corresponds to \(\text{MSE}_{k}^{(1)}\). The problem (22) is nonconvex because of the constraint \(\mathbf{W}_{a}\in\mathcal{W}_{a}\). Then, we propose a solution based on OMP algorithm, where a set of vectors is selected from \(\mathcal{W}_{a}\). We assume in this paper that \(\mathcal{W}_{a}\) is composed by the set of receive array response vectors, \(\mathbf{a}_{rx,u}\Big{(}\bigoplus\limits_{q,t^{\prime},t^{\prime}}^{rx} \sigma_{q,t^{\prime}}^{rx}\Big{)}\), \(q=1,\ldots,N_{cl}\), \(l=1,\ldots,N_{my}\), presented in the channel model (4). The concatenated matrix with this set of vectors is \(\mathbf{A}_{rx}=[\mathbf{A}_{rx,1},\ldots,\mathbf{A}_{rx,U}]\in\mathbb{C}^{N_ {rx}\times N_{cl}N_{my}U}\), where:  
\[\mathbf{A}_{rx,u}=\Big{[}\mathbf{a}_{rx,u}\Big{(}\big{(}\big{(}\big{(}\big{(} \big{(}\big{(}\big{)}\big{)}^{rx}_{1,1},t^{\prime}\big{)},\sigma_{1,1}^{rx} \Big{)}\ldots,\mathbf{a}_{rx,u}\Big{(}\big{(}\big{(}\big{(}\big{(}\big{(}\big{(} \big{(}\big{(}\big{(}\big{(}\big{(}\big{(}\big{(\big{}}\big{(\big{(\big{(\big{(\big{( }}}}}\big{(\big{(\big{( }}}}\big{(\big{(\big{( }}}\big{(\big{(\big{( }}}}\big{((\big{( }}\big{( (\big{(\big{( }}}}(( \big{( (\big)}}}((( ( \big{( (\big{( (\big))}}}}(( (((\big{( ( (\big{( (\big))}((( (\big{( (\big))}}(( ((\big{( (\big{\big))}} ((( (\big{(\big{\big{\big{\)}}} ( (\big{\big{\big{\big{\}}} ( (\big{\big{\}}} (( \big{\big{\big{\}}} ( ((\big{\big{\big{\big{\}}} ( \big{\big{\}}  
\[\mathbf{W}_{ad,k,r}=\mathbf{W}_{ad,k,r-1}+\mathbf{w}_{d,k,r}(\mathbf{w}_{a,r})^{H}, \tag{24}\]  
for \(r=1,\ldots,N_{rx}^{RF}\). Replacing (24) in (17), the problem (22) simplifies to:  
\[ \begin{array}{rl}\mathbf{w}_{a,r}=&\operatorname*{argmin}_{k=1}^{N_{c}} \operatorname*{MSE}_{k,r}\\ &\operatorname*{s.t.}\mathbf{w}_{a,r}\in\mathcal{F}_{a},\end{array}  \tag{25}\]  
where \(\mathcal{F}_{a}\) represents the dictionary defined by columns of \(\mathbf{A}_{rx}\), \(\mathbf{W}_{res,k,r-1}=\overline{\mathbf{W}}_{fd,k}^{(1)}-\mathbf{W}_{ad,k,r-1}\) is the residue matrix, and  
\[\operatorname*{MSE}_{k,r}=\big{|}\Big{(}\mathbf{w}_{d,k,r}\big{(}\mathbf{w}_{ a,r}\big{)}^{H}-\mathbf{W}_{res,k,r-1}\Big{)}\Big{(}\overline{\mathbf{R}}_{k} \Big{)}^{1/2}\big{|}_{F}^{2}, \tag{26}\]  
is the MSE using \(r\) RF chains. Making \(\partial\Big{(}\sum\limits_{k=1}^{N_{c}}\operatorname*{MSE}_{k,r}\Big{)}/ \partial\Big{(}\mathbf{w}_{d,k,r}\Big{)}=\mathbf{0}\), we obtain:  
\[\mathbf{w}_{d,k,r}=\mathbf{W}_{res,k,r-1}\overline{\mathbf{R}}_{k}\mathbf{w}_{ a,r}\Big{(}(\mathbf{w}_{a,r})^{H}\overline{\mathbf{R}}_{k}\mathbf{w}_{a,r} \Big{)}^{-1}. \tag{27}\]  
Replacing Equation (27) in Equation (26), we obtain  
\[ \begin{array}{rl}\operatorname*{MSE}_{k,r}&=\|\overline{\mathbf{W}}_{res,k, r-1}\big{(}\overline{\mathbf{R}}_{k}\big{)}^{-1/2}\big{|}_{F}^{2}\\ &+\|\overline{\mathbf{W}}_{res,k,r-1}\mathbf{w}_{a,r}\Big{(}\mathbf{w}_{a,r}^ {H}\overline{\mathbf{R}}_{k}\mathbf{w}_{a,r}\Big{)}^{-1/2}\big{|}_{F}^{2}, \end{array}  \tag{28}\]  
where \(\overline{\mathbf{W}}_{res,k,r-1}=\mathbf{W}_{res,k,r-1}\overline{\mathbf{R} }_{k}\). As mentioned, the analog part of equalizer should remain constant due to hardware constraints, thus, we need to fix the coefficients of matrix \(\mathbf{\Psi}^{(i)}\) to calculate \(\mathbf{W}_{a}\).