# Reduced Complexity Detection in MIMO Systems with SC-FDE Modulations and Iterative DFE Receivers
## 3 Multiuser Detection  
The detection of the different signals that are received at a given antenna \(r\) relies on an iterative scheme based on the IB-DFE concept [19]. This method takes advantage of the symbol's detection on the previous iteration to compute a more updated estimate of the transmitted data symbols associated with each link from the \(p\)-th antenna. In the \(i\)-th iteration, the set of estimated symbols \(\mathbf{\hat{s}}_{n}\) is the hard decisions of the time-domain detector output \(\{\mathbf{\hat{s}}_{n}\}=\mathrm{IDFT}\{\mathbf{\hat{S}}_{k}\}\), where \(\mathbf{\hat{S}}_{k}\) is a column vector of size \(P\times 1\) and is given by:  
\[\mathbf{\hat{S}}_{k}=\mathbf{F}_{k}^{T}\mathbf{Y}_{k}^{Q}-\mathbf{B}_{k}^{T} \mathbf{\hat{S}}_{k}. \tag{13}\]  
As part of the iterative DFE process, \(\mathbf{F}_{k}^{T}\) and \(\mathbf{B}_{k}^{T}\) are the feedforward and feedback coefficient matrices of size \(P\times R\) and \(P\times P\), respectively, expressed as:  
\[\mathbf{F}_{k}^{T}= \begin{bmatrix}F_{k,p}^{(1)}&\dots&F_{k,p}^{(R)}\\ \vdots&\ddots&\vdots\\ F_{k,p}^{(1)}&\dots&F_{k,P}^{(R)}\end{bmatrix}  \tag{14}\]  
and:  
\[\mathbf{B}_{k}^{T}= \begin{bmatrix}B_{k,p}^{(1)}&\dots&B_{k,p}^{(P)}\\ \vdots&\ddots&\vdots\\ B_{k,P}^{(1)}&\dots&B_{k,P}^{(P)}\end{bmatrix} . \tag{15}\]  
\(\mathbf{\hat{S}}_{k}\) is a \(P\times 1\)-sized column vector given by \(\mathbf{\hat{S}}_{k}=\left[\bar{S}_{k,1},\dots,\bar{S}_{k,p-1},\bar{S}_{k,p}, \dots,\bar{S}_{k,P}\right]^{T}\), where block \(\{\bar{S}_{k,p}\}\) is the DFT of the block of time-domain average values conditioned to the detector output \(\{\bar{s}_{n,p}\}\). In order to obtain the optimum coefficients \(\mathbf{F}_{k}\) and \(\mathbf{B}_{k}\), which at a given iteration define the state of the detector, the Minimum Mean Squared Error (MMSE) criterion is followed. At each subcarrier \(k\) and each data stream \(p\) the MSE on the frequency-domain samples \(\tilde{S}_{k,p}\) corresponds to:  
\[ \begin{split}\Theta_{k,p}&=\mathbb{E}\left[\left| \tilde{S}_{k,p}-S_{k,p}\right|^{2}\right]\\ &=\mathbb{E}\left[\left|\mathbf{F}_{k,p}^{T}\mathbf{Y}_{k}^{Q}- \mathbf{B}_{k,p}^{T}\tilde{\mathbf{S}}_{k,p}-S_{k,p}\right|^{2}\right],\end{split}  \tag{16}\]  
where its minimization is conditioned to:  
\[\gamma_{p}=\frac{1}{N}\sum_{k=0}^{N-1}\sum_{r=1}^{R}F_{k,p}^{(r)}H_{k,p}^{eq( r)}=1, \tag{17}\]  
with \(\gamma_{p}\) indicating the average overall channel frequency response. Applying the gradient for the Lagrange function as:  
\[J=\Theta_{k,p}+\lambda\left(\gamma_{p}-1\right), \tag{18}\]  
the optimum coefficients \(\mathbf{F}_{k}\) and \(\mathbf{B}_{k}\) are given by:  
\[\mathbf{F}_{k}=\kappa\left(\mathbf{H}_{k}^{H}\left(\mathbf{I}_{P}-\mathbf{P}^ {2}\right)\mathbf{H}_{k}+\mathbf{D}_{k}\right)^{-1}\mathbf{H}_{k}^{H} \tag{19}\]  
and:  
\[\mathbf{B}_{k}=\alpha\mathbf{H}_{k}\mathbf{F}_{k}-\mathbf{I}_{P}, \tag{20}\]  
(see the details in [5]), with \(\mathbf{D}_{k}\) corresponding to a diagonal matrix defined as:  
\[\mathbf{D}_{k}=\text{diag}\left(\left\{\frac{\left|\mathbf{\omega}\right|^{2} \sigma_{N}^{(r)^{2}}+\sigma_{D}^{(r)^{2}}(k)}{\left|\mathbf{\omega}\right|^{2 }\sigma_{S}^{2}};r=1,...,R\right\}\right) \tag{21}\]  
and \(\kappa\) selected to ensure that \(\gamma_{p}=1\), in order to have a normalized FDE with \(\mathbb{E}\left[\tilde{s}_{n,p}\right]=s_{n,p}\). In addition, \(\sigma_{N}^{(r)^{2}}\) and \(\sigma_{S}^{2}\) represent the variance of the real and imaginary parts of the channel noise and data sample components, respectively. This method for the multiuser detection requires inverting matrices of size \(R\times R\), which can be inefficient, namely in BS cooperation schemes or massive MIMO systems with fewer users. Since in these cases, the relation between transmitting and receiving antennas is \(R\gg P\), then a method of inverting matrices of size \(P\times P\) instead of \(R\times R\) is desirable.