# Multiuser Equalizer for Hybrid Massive MIMO mmWave CE-OFDM Systems
## 3 Precoder and Equalizer Design
### Hybrid Equalizer
#### 3.2.1 Design of Iterative Digital Equalizer
From Equation (13) and from the Karush-Kuhn-Tucker (KKT) conditions, i.e., \(\partial\left(\sum\limits_{k=1}^{N_{c}}\text{MSE}_{k}^{(i)}\right)/\partial \left(\mathbf{B}_{d,k}^{(i)}\right)=\mathbf{0}\), it can be proven that the digital feedback matrix is given by:  
\[\mathbf{B}_{d,k}^{(i)}\Big{[}\mathbf{W}_{a},\mathbf{W}_{d,k}^{(i)}\Big{]}= \Big{(}\mathbf{W}_{d,k}^{(i)}(\mathbf{W}_{a})^{H}\mathbf{H}_{k}-\mathbf{I}_{U }\Big{)}\Big{(}\mathbf{V}^{(i-1)}\Big{)}^{H}. \tag{16}\]  
To compute the digital feedforward filter, we start by replacing Equation (16) in Equation (13), and obtain:  
\[\text{MSE}_{k}^{(i)}=\big{\|}\Big{(}\mathbf{W}_{d,k}^{(i)}(\mathbf{W}_{a})^{H }-\overline{\mathbf{W}}_{fd,k}^{(i)}\Big{)}\bigg{(}\overline{\mathbf{R}}_{k}^ {(i-1)}\Big{)}^{1/2}\big{\|}_{F}^{2}, \tag{17}\]  
where \(\overline{\mathbf{W}}_{fd,k}^{(i)}\) denotes a non-normalized version of full digital equalizer, i.e., \(\mathbf{W}_{fd,k}^{(i)}=\mathbf{\Omega}\left(\mathbf{I}_{U}-\big{|}\mathbf{V} ^{(i-1)}\big{|}_{2}\right)^{-1}\overline{\mathbf{W}}_{fd,k}^{(i)}\), with \(\mathbf{W}_{fd,k}^{(i)}\) given by Equation (A2). The formulation of problem and corresponding solution for the fully digital case is presented in the Appendix A. As can be shown by Equation (17), the MSE can also be expressed as the difference between the hybrid and the full digital equalizer matrices. Therefore, the optimization problem to compute the iterative digital feedforward equalizer is given by:  
\[ \begin{array}{l}\mathbf{W}_{d,k}^{(i)}[\mathbf{W}_{a}]=\text{argmin}\sum \limits_{k=1}^{N_{c}}\text{MSE}_{k}^{(i)}\\ \text{s.t.}\sum\limits_{k=1}^{N_{c}}\text{diag}(\mathbf{W}_{d,k}^{(i)}(\mathbf{ W}_{a})^{H}\mathbf{H}_{k})=N_{c}\mathbf{I}_{U},\end{array}  \tag{18}\]  
where Equation (18) is also a convex problem, since \(\mathbf{W}_{d,k}^{(i)}\) is independent of constraint \(\mathbf{W}_{a}\in\mathcal{W}_{a}\). To calculate \(\mathbf{W}_{d,k}^{(i)}\) as function of \(\mathbf{W}_{a}\), we can compute the associated Lagrangian of the problem (18) given by:  
\[ \begin{array}{l}\mathcal{L}\Big{(}\mu_{u},\mathbf{W}_{d,k}^{(i)}\Big{)}=\\ \sum\limits_{k=1}^{N_{c}}\big{\|}\Big{(}\mathbf{W}_{d,k}^{(i)}\Big{(}\mathbf{W}_ {a}^{(i)}\Big{)}^{H}-(\overline{\mathbf{W}}_{fd,k}^{(i)})_{opt}\Big{)}\bigg{(} \overline{\mathbf{R}}_{k}^{(i-1)}\Big{)}^{1/2}\big{\|}_{F}^{2}\\ +\sum\limits_{u=1}^{U}\mu_{u}\text{tr}\Big{(}\mathbf{W}_{d,k}^{(i)}\Big{(} \mathbf{W}_{a}^{(i)}\Big{)}^{H}\mathbf{H}_{k}\mathbf{e}_{u}\mathbf{e}_{u}^{H} \Big{)}\\ +\sum\limits_{u=1}^{U}\mu_{u}\bigg{(}\sum\limits_{k=1}^{N_{c}}\text{tr}\Big{(} \mathbf{W}_{d,k}^{(i)}\Big{(}\mathbf{W}_{a}^{(i)}\Big{)}^{H}\mathbf{H}_{k^{ \prime}}\mathbf{e}_{u}\mathbf{e}_{u}^{H}\Big{)}-N_{c}\bigg{)}.\end{array}  \tag{19}\]where the coefficients \(\mu_{u},u\in\{1,\ldots,U\}\) are the Lagrange multipliers, and \(\mathbf{e}_{u}\in\mathbb{C}^{U}\), and make \(\partial\mathcal{L}\Big{(}\mu_{u},\mathbf{W}_{d,k}^{(i)}\Big{)}/\partial\Big{(} \mathbf{W}_{d,k}^{(i)}\Big{)}=\mathbf{0}\). Therefore, the solution of (18) is given by:  
\[\mathbf{W}_{d,k}^{(i)}[\mathbf{W}_{a}]=\mathbf{\Omega}_{d}\mathbf{H}_{k}^{H} \mathbf{W}_{a}\Big{(}(\mathbf{W}_{a})^{H}\overline{\mathbf{R}}_{k}^{(i-1)} \mathbf{W}_{a}\Big{)}^{-1}, \tag{20}\]  
where \(\mathbf{\Omega}_{d}\) ensures the power constraint of (18) and it is given by:  
\[\mathbf{\Omega}_{d}=N_{c}\Bigg{(}\sum_{k=1}^{N_{c}}\text{diag}\Bigg{(}\mathbf{ H}_{k}^{H}\mathbf{W}_{a}\Big{(}(\mathbf{W}_{a})^{H}\overline{\mathbf{R}}_{k}^{(i-1)} \mathbf{W}_{a}\Big{)}^{-1}(\mathbf{W}_{a})^{H}\mathbf{H}_{k}\Bigg{)}\Bigg{)}^{ -1} \tag{21}\]  
The pseudo-code to compute the digital part is presented in the Algorithm 1. This pseudo-code uses the analog precoder matrix, \(\mathbf{W}_{a}\), as input, which is designed in the next section.