# Estimating the Performance of MIMO SC-FDE Systems Using SISO Measurements
## 2 System Characterisation  
The scenario considered in this paper is a point-to-point connection in a SISO system with a highly frequency selective channel.  
The system is modelled assuming a data block, of size \(N\), transmitted in the frequency-domain and expresses as \(\{X_{k};n=0,1,\ldots,N-1\}\), which is the Discrete Fourier Transform (DFT) of the time-domain as a data block \(\{x_{n};n=0,1,\ldots,N-1\}\), where data symbols \(x_{n}\) are selected according to the Quadrature Phase-Shift keying (QPSK) rule with Gray mapping. The received signal in the frequency-domain is denoted as \(\{Y_{k};n=0,1,\ldots,N-1\}\) whose Inverse Discrete Fourier Transform (IDFT) corresponds to \(\{y_{n};n=0,1,\ldots,N-1\}\) samples in the time-domain. The received signal can also be expressed by Equation (1):  
\[Y_{k}=H_{k}X_{k}+N_{k}, \tag{1}\]  
where the frequency-domain block of the transmitted block \(x_{n}\) is represented by \(X_{k}\), the \(N_{k}\) and \(H_{k}\) represent the channel noise and the channel for the \(k^{th}\) subcarrier, respectively.  
The iterative FDE receiver considered is the Iterative Block Decision Feedback Equalization (IB-DFE) [19] presented in Figure 2. This IB-DFE receiver has two filters the feedforward filter, \(F_{k}\), which equalises the channel, and the feedback filter, \(B_{k}\), which is responsible for removing the Inter-Symbol Interference (ISI), using the previous iteration values.  
For the \(i^{th}\) iteration and \(k^{th}\) subcarrier, the estimated signal, \(\widetilde{X}_{k}^{(i)}\), is given by:  
\[\widetilde{X}_{k}^{(i)}=F_{k}Y_{k}-B_{k}^{(i)}\overline{X}_{k}^{(i-1)}, \tag{2}\]  
which, for the first iteration, can be simplified to  
\[\widetilde{X}_{k}^{(1)}=F_{k}Y_{k}. \tag{3}\]  
The \"soft decision\" of the previous iteration is expressed by \(\overline{X}_{k}^{(i-1)}\), which is the DFT of the time-domain block \(\overline{\pi}_{k}^{(i-1)}\), whose average value, according to [20], is given by:  
\[\overline{x}_{n}=\tanh\left(\frac{L_{n}^{Re}}{2}\right)+j\tanh\left(\frac{L_{ n}^{Im}}{2}\right), \tag{4}\]  
Figure 2: IB-DFE receiver structure.  
where  
\[L_{n}^{Re}=\frac{2}{\sigma_{n}^{2}}Re\{\hat{x}_{n}\}, \tag{5}\]  
\[L_{n}^{Im}=\frac{2}{\sigma_{n}^{2}}Im\{\hat{x}_{n}\}, \tag{6}\]  
and  
\[\sigma_{n}^{2}=\frac{1}{2N}\sum_{n^{\prime}=0}^{N-1}\left|\hat{x}_{n}-x_{n^{ \prime}}\right|^{2}\simeq\frac{1}{2N}\sum_{n^{\prime}=0}^{N-1}\left|\hat{x}_{ n^{\prime}}-\hat{x}_{n^{\prime}}\right|^{2}. \tag{7}\]  
Coefficients \(F_{k}\) and \(B_{k}\) are chosen so as to minimise the Mean Square Error (MSE) and, consequently, to minimise the BER. They can be given by [21; 22]:  
\[F_{k}^{(i)}=\frac{H_{k}^{*}}{\frac{\sigma_{n}^{2}}{\sigma_{n}^{2}}+(1-(\rho^{(i -1)})^{2})\left|H_{k}\right|^{2}}, \tag{8}\]  
\[B_{k}^{(i)}=\rho^{(i-1)}(F_{k}^{(i)}H_{k}-1), \tag{9}\]  
where \(\rho\) represents the correlation coefficient and is defined by:  
\[\rho^{(i)}=\frac{\mathbb{E}[\hat{x}_{k}^{(i)}x_{k}^{*}]}{\mathbb{E}[\left|x_{ k}\right|^{2}]}. \tag{10}\]  
Considering QPSK modulation, \(\rho\) can be approximately given by:  
\[\rho_{p}\approx\frac{1}{2N}\sum_{n=0}^{N-1}\left(\left|\rho_{n}^{Re}\right|+ \left|\rho_{n}^{Im}\right|\right), \tag{11}\]  
where  
\[\rho_{n}^{Re(i)}=\tanh\left(\frac{\left|L_{n}^{Re(i)}\right|}{2}\right), \tag{12}\]  
and  
\[\rho_{n}^{Im(i)}=\tanh\left(\frac{\left|L_{n}^{Im(i)}\right|}{2}\right). \tag{13}\]