# MIMO Self-Heterodyne OFDM Using Band Selection Technique
## 2 The FSTBC MIMO Self-Het OFDM System
\begin{bmatrix}&x_{1}&-\,\,x_{2}^{*}\\ &x_{2}&x_{1}^{*}\\ &x_{3}&-\,\,x_{4}^{*}\\ &x_{3}&\end{bmatrix} . \tag{8}\]  
Let \(\alpha_{c}\,\triangleq\,\,\text{H}_{11}+\text{H}_{12}+\text{H}_{13}+\text{H }_{14}\,\,\) be the sum channel gains to the 1 st receiving antenna from the four transmitting antennas and \(\beta_{c}\,\triangleq\,\,\text{H}_{21}+\text{H}_{22}+\text{H}_{23}+\text{H}_ {24}\) is the sum of channel gains of the 2 nd receiving antenna from the four transmitting antennas. Then, the received signal at each receiving antenna is given by:  
\[\mathbf{Y}\,=\,\,\mathbf{H}_{\text{eq}}\,\,\mathbf{x}+\mathbf{Z}, \tag{9}\]where \(\mathbf{Z}\) is the AWGN components and \(\mathbf{H}_{\text{eq}}\) is the channel MIMO which represented in a matrix form as:  
\[\mathbf{H}_{\text{eq}}\ \triangleq\ \left[ \begin{array}{cccc}\alpha_{c}^{*}H_{11}& \alpha_{c}^{*}H_{12}&\alpha_{c}^{*}H_{13}&\alpha_{c}^{*}H_{14}\\ \beta_{c}^{*}H_{21}&\beta_{c}^{*}H_{22}&\beta_{c}^{*}H_{23}&\beta_{c}^{*}H_{24} \\ \alpha_{c}H_{12}^{*}&-\alpha_{c}H_{11}^{*}&\alpha_{c}H_{14}^{*}&-\alpha_{c}H_{1 3}^{*}\\ \beta_{c}H_{22}^{*}&-\beta_{c}H_{21}^{*}&\beta_{c}H_{24}^{*}&-\beta_{c}H_{23}^{* }\\ \end{array} \right], \tag{10}\]  
where \(\mathbf{H}_{ra}\) is the channel gain between the \(a\)th transmitting antenna and \(r\)th receiving antennas and (\(*\)) is the conjugate operation. The channel is assumed to be constant during two successive symbol intervals [22]. At the receiver side, no local carrier oscillator or recovering of carrier phase is required. A low pass filter (LPF) with a cut-off frequency equal to \(f_{S}+f_{s}\) and filters of DC components are used to eliminate the high frequency signals and the DC components. Then, the CP is removed, and the fast Fourier transform (FFT) is performed after serial-to-parallel (S/P) conversion. Thus, the received signal can be expanded as:  
\[\left[ \begin{array}{c}y_{11}\\ y_{21}\\ y_{12}^{*}\\ y_{22}^{*}\\ \end{array} \right]\ =\ \left[ \begin{array}{cccc}\alpha_{c}^{*}H_{11}& \alpha_{c}^{*}H_{12}&\alpha_{c}^{*}H_{13}&\alpha_{c}^{*}H_{14}\\ \beta_{c}^{*}H_{21}&\beta_{c}^{*}H_{22}&\beta_{c}^{*}H_{23}&\beta_{c}^{*}H_{24 }\\ \alpha_{c}H_{12}^{*}&-\alpha_{c}H_{11}^{*}&\alpha_{c}H_{14}^{*}&-\alpha_{c}H_{1 3}^{*}\\ \beta_{c}H_{22}^{*}&-\beta_{c}H_{21}^{*}&\beta_{c}H_{24}^{*}&-\beta_{c}H_{23}^{* }\\ \end{array} \right]\left[ \begin{array}{c}x_{1}\\ x_{2}\\ x_{3}\\ x_{4}\\ \end{array} \right]\ +\ \left[ \begin{array}{c}z_{11}\\ z_{21}\\ z_{22}^{*}\\ \end{array} \right], \tag{11}\]  
where \(\mathbf{y}_{11}\) and \(\mathbf{y}_{21}\) are the received signals, in time domain, at the 1 st receiving antenna during the 1 st time slot and the second time slot successively. \(\mathbf{y}_{21}\) and \(\mathbf{y}_{22}\) are the received signals at the 2 nd receiving antennas during the 1 st and 2 nd time slots respectively. As well, \(\mathbf{z}_{11}\) and \(\mathbf{z}_{21}\) are the AWGN noise components at the 1 st receiving antenna during the 1 st time slot and the second time slot successively. \(\mathbf{z}_{21}\) and \(\mathbf{z}_{22}\) are the AWGN noise components at the 2 nd receiving antennas during the 1 st and 2 nd time slots respectively [22].