# 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  
In this section we derive the digital part of the equalizer, which is composed by the feedforward, \(\mathbf{W}_{d,k^{\prime}}^{(i)}\), and feedback, \(\mathbf{B}_{d,k^{\prime}}^{(i)}\) coefficients. The hybrid equalizer is designed by minimizing the sum of the \(\text{MSE}_{k}^{(i)}=\mathbb{E}[\|\mathbf{c}_{k}^{(i)}-\mathbf{c}_{k}\|^{2}]\) of all subcarriers. Mathematically, the optimization problem may be formulated as:  
\[\left(\mathbf{W}_{a},\mathbf{W}_{d,k^{\prime}}^{(i)}\,\mathbf{B}_{d,k}^{(i)} \right)=\text{argmin}\sum_{k=1}^{N_{c}}\text{MSE}_{k}^{(i)},s.t.\sum_{k=1}^{ N_{c}}\text{diag}(\mathbf{W}_{d,k}^{(i)}(\mathbf{W}_{a})^{H}\mathbf{H}_{k})=N_{c} \mathbf{I}_{U},\mathbf{W}_{a}\in\mathcal{W}_{a}, \tag{11}\]  
where \(\mathbf{H}_{k}=[\mathbf{H}_{1,k}\mathbf{f}_{a,1},\ldots,\mathbf{H}_{1,k} \mathbf{f}_{a,1}]^{T}\in\mathbb{C}^{N_{rx}\times U}\) is the equivalent channel, and \(\mathcal{W}_{a}\) denotes the set of feasible analog coefficients. Considering that the entries of \(\mathbf{c}_{k}=[c_{1,k^{\prime}}\ldots,c_{U,k}]^{T}\in\mathbb{C}^{U}\) are approximately Gaussian distributed, it can be proven that the input-output relationship between \(\mathbf{c}_{k}\) and \(\mathbf{c}_{k}^{(i)}\), \(k\in\{1,\ldots,N_{c}\}\) is [41]:  
\[\mathbf{c}_{k}^{(i)}\approx\mathbf{\Psi}^{(i)}\mathbf{c}_{k}+\mathbf{\epsilon }_{k}^{(i)},\quad k\in\{1,\ldots,N_{c}\}, \tag{12}\]  
where \(\mathbf{\epsilon}_{k}^{(i)}\in\mathbb{C}^{U}\) is the zero mean error, uncorrelated with \(\mathbf{c}_{k}\). Considering (7) and (12) the \(\text{MSE}_{k}^{(i)}\) can be expressed as:  
\[ \begin{array}{rl}\text{MSE}_{k}^{(i)}=&\|\mathbf{W}_{d,k}^{(i)}(\mathbf{W}_ {a})^{H}\mathbf{H}_{k}-\mathbf{B}_{d,k}^{(i)}\mathbf{\Psi}^{(i-1)}-\mathbf{I} _{U}\|_{F}^{2}\sigma_{u}^{2}\\ &+\|\mathbf{B}_{d,k}^{(i)}\left(\mathbf{I}_{U}-\left|\mathbf{\Psi}^{(i-1)} \right|_{2}\right)^{1/2}\|_{F}^{2}\sigma_{u}^{2}+\|\mathbf{W}_{d,k}^{(i)}( \mathbf{W}_{a})^{H}\|_{F}^{2}\sigma_{n}^{2}.\end{array}  \tag{13}\]  
where \(\sigma_{u}^{2}=\left(2\pi h\right)^{2}\sigma_{s}^{2}\) and \(\mathbf{\Psi}^{(i)}\in\mathbb{C}^{U\times U}\) is a diagonal matrix whose \(u\)th element gives a blockwise reliability measure of the estimates of \(u\)th block, associated to the \(i\)th iteration [32]. The bit-error-rate(BER) for iteration \(i\), at receiver side, can be estimated by the semi-analytic BER approximation [42], given by:  
\[\text{BER}^{(i)}=\frac{\alpha}{\text{U}N_{c}}\sum_{u=1}^{U}\sum_{k=1}^{N_{c}} \mathbb{Q}\left(\sqrt{\beta\Big{(}\text{MSE}_{k,u}^{(i)}\Big{)}^{-1}}\right) \tag{14}\]  
where \(\alpha=4(1-1/\sqrt{M})/\log_{2}[M]\), \(\beta=3/(M-1)\), for a \(M\)-QAM constellation, and \(Q(. )\) denotes the Q-function. \(\text{MSE}_{k,u}^{(i)}\) is the mean square error on samples \(\overline{c}_{k,u}^{(i)}\), with \(\text{MSE}_{k}^{(i)}=\sum_{u=1}^{U}\text{MSE}_{k,u^{\prime}}^{(i)}\) at iteration \(i\). First, let us calculate the digital feedback matrix, \(\mathbf{B}_{d,k}^{(i)}\), through the following optimization problem:  
\[\mathbf{B}_{d,k}^{(i)}\Big{[}\mathbf{W}_{a},\mathbf{W}_{d,k}^{(i)}\Big{]}= \text{argmin}\sum_{k=1}^{N_{c}}\text{MSE}_{k}^{(i)}, \tag{15}\]  
where \(\mathbf{B}_{d,k}^{(i)}\) appears as a function of \(\mathbf{W}_{a}\) and \(\mathbf{W}_{d,k}^{(i)}\). The problem (15) does not have any constraint (and it is convex), since \(\mathbf{B}_{d,k}^{(i)}\) is independent of the constraints of (11).