# Iterative Analog-Digital Multi-User Equalizer for Wideband Millimeter Wave Massive MIMO Systems
## 4 Analog-Digital Receiver Design
### Iterative Analog-Digital Equalizer  
We assume at the receiver, a hybrid iterative \(S\)-length block space-frequency decoder, as shown in Figure 2. At the \(k\)th subcarrier, the corresponding concatenated received signal of all users, \(\tilde{\mathfrak{c}}_{k}^{(i)}=\left[\tilde{c}_{u,k}^{(i)}\right]_{1\leq u\leq U}\in \mathbb{C}^{U}\), is  
\[\tilde{\mathfrak{c}}_{k}^{(i)}=\mathbf{W}_{d,k}^{(i)}{(\mathbf{W}_{a}^{(i)})}^ {H}\mathbf{y}_{k}-\mathbf{B}_{d,k}^{(i)}\hat{\mathfrak{c}}_{k}^{(i-1)}, \tag{11}\]  
where \(\mathbf{W}_{a}^{(i)}\in\mathbb{C}^{N_{rx}\times N_{rx}^{RF}}\) denotes the analog part of feed-forward matrix for the \(i\)th iteration, and \(\mathbf{W}_{d,k}^{(i)}\in\mathbb{C}^{U\times N_{rx}^{RF}}\) and \(\mathbf{B}_{d,k}^{(i)}\in\mathbb{C}^{U\times U}\) denote the digital part of the feed-forward and feedback matrices, computed at the \(i\)th iteration at the \(k\)th subcarrier, respectively. The vector \(\hat{\mathfrak{c}}_{k}^{(i)}=\left[\hat{\mathfrak{c}}_{u,k}^{(i)}\right]_{1 \leq u\leq U}\in\mathbb{C}^{U}\), where the block \(\left\{\hat{\mathfrak{c}}_{u,k}^{(i)}\right\}_{k=1}^{S}\) is the FFT of the block of time domain estimates conditioned to the detector output for user \(u\) and iteration \(\left\{\hat{\mathfrak{s}}_{u,t}^{(i)}\right\}_{t=1}^{S}\). The vector \(\hat{\mathfrak{s}}_{k}^{(i)}=\left[\hat{\mathfrak{s}}_{u,t}^{(i)}\right]_{1 \leq u\leq U}\), where \(\hat{\mathfrak{s}}_{u,t}^{(i)}\) is the hard decision associated with the data symbols of user \(u\) at iteration \(i\). The received signal is first processed through the analog phase shifters, with \(\left|\mathbf{W}_{a}(n,l)\right|_{2}=N_{rx}^{-1}\), and then follows the baseband processing composed by \(N_{rx}^{RF}\) RF chains. The digital baseband processing includes a feedback closed-loop that employs a forward path and a feedback path for each subcarrier. In the forward path, the signal first passes through a linear filter \(\mathbf{W}_{d,k}^{(i)}\) used at subcarrier k and then follows SC-FDMA decoding and data demodulation. The data recovered from the forward path are  
Figure 2: Proposed receiver structure. modulated, and SC-FDMA encoded in the feedback path, and then, the data pass through the feedback filter \(\mathbf{B}_{d,k}^{(i)}\) employed at subcarrier k.  
From the central limit theorem the entries of vector, \(\mathbf{c}_{k}=\left[c_{u,k}\right]_{1\leq u\leq U}\in\mathbb{C}^{U}\), \(k\in\{1,\ldots,S\}\) are Gaussian distributed, then as the input-output relationship between variables \(\mathbf{c}_{k}\) and \(\hat{\mathbf{c}}_{k}^{(i)}\), is memoryless, thus by the [PERSON] theorem [33] is approximately given by  
\[\hat{\mathbf{c}}_{k}^{(i)}\approx\mathbf{\Psi}^{(i)}\mathbf{c}_{k}+\hat{ \mathbf{e}}_{k}^{(i)},\quad k\in\{1,\ldots,S\}, \tag{12}\]  
where \(\hat{\mathbf{e}}_{k}^{(i)}\) is a zero mean error vector uncorrelated with \(\mathbf{c}_{k},k\in\{1,\ldots,S\}\), 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 [22]. The coefficients of each block can be estimated at the receiver, as discussed in [10]. The error between the estimated signal before the S-IFFT \(\overline{\mathbf{c}}_{k}^{(i)}\) given by (11) and the transmit signal after the S-FFT \(\mathbf{c}_{k}\) may be expressed by  
\[\overline{\mathbf{c}}_{k}^{(i)}=\overline{\mathbf{c}}_{k}^{(i)}-\mathbf{c}_{k }=\left(\mathbf{W}_{ \begin{subarray}{c}\text{\emph{add}},k}^{(i)}\mathbf{H}_ {k}-\mathbf{I}_{U}-\mathbf{B}_{d,k}^{(i)}\overline{\mathbf{\Psi}}^{(i-1)} \right)\mathbf{c}_{k}-\mathbf{B}_{d,k}^{(i)}\hat{\mathbf{e}}_{k}^{(i-1)}+ \mathbf{W}_{\begin{subarray}{c}\text{\emph{add}},k}^{(i)}\mathbf{n}_{k}\quad, \\ 1\end{subarray}\end{subarray}