# MIMO Self-Heterodyne OFDM Using Band Selection Technique
## 2 The FSTBC MIMO Self-Het OFDM System  
The system model of the FSTBC MIMO self-het OFDM consists of four antennas at the transmitter and two antennas at the receiver as shown in Figure 1. The H \(\in\) C\({}^{A_{t}}\)\(\times\)\(A_{r}\) is channel response, where \(A_{t}\) is the number of transmitting antennas and \(A_{r}\) is the number of receiving antennas. The information subcarriers are divided into \(A_{s}\) and \(A_{g}\) which represent the OFDM number of subcarriers used to transmit the information symbols and the number of subcarriers that are zero-padded in each self-het OFDM transmitter respectively. It is shown in Figure 1, that the proposed system transmits four OFDM symbols on two successive symbol intervals using four transmitting antennas and two receiving antennas. The proposed system increases the data rate to double that proposed by the conventional STBC MIMO self-het OFDM system. This is in addition to the increase in the diversity gain due to the increase in the number of transmitting antennas. The proposed system performs multiplexing in the spatial time and frequency domain, where the first and second symbol intervals are space-time encoded in the lower band of sub-carriers (zeros are embedded in the upper band), while the third and fourth symbols are space-time encoded in the upper band (zeros are embedded in the lower band). Thus, each pair of successive symbols can be received and separated at the receiver depending on the sub-band they were sent on (i.e., frequency multiplexing on two sub-bands). The transmitted data symbols X are modulated using QPSK modulation. Then the modulated data symbols are divided into four code matrices. These code matrices are  
Figure 1: FSTBC self-het OFDM Proposed System Model. transmitted by \(A_{t}\) transmitting antennas over \(A_{s}\) sub-carriers. The code matrices are successively defined as:  
\[d_{1}[L] = \text{X}[1+4(L-1)],\,\text{where}\,\,L=\,\,1,2,\,\,\ldots,\,\,\frac{N} {4}. \tag{1}\] \[d_{2}[L] = \text{X}[2+4(L-1)],\,\text{where}\,\,L=\,\,1,2,\,\,\ldots,\,\,\frac{N }{4}.\] (2) \[d_{3}[L] = \text{X}[3+4(L-1)],\,\text{where}\,\,L=\,\,1,2,\,\,\ldots,\,\,\frac {N}{4}.\] (3) \[d_{4}[L] = \text{X}[4+4(L-1)],\,\text{where}\,\,L=\,\,1,2,\,\,\ldots,\,\,\frac {N}{4}. \tag{4}\]  
where \(N\) is the number of the transmitted data symbols and \(d_{a}\) is code matrices for each transmitter antenna (\(a=1,\,2,\,\ldots,\,A_{t}\)) [22]. The modulated data symbols are applied to the inverse fast Fourier transform (IFFT) stage. The IFFT input vectors for each transmitter antenna (\(a\)) are represented as (size (\(1\times\frac{N}{2}\)):  
\[\text{X}_{a}[K]= \begin{cases}A&K=0\\ 0&K=1,\,\cdots,\,\frac{N}{4}-2\\ d_{a}&K=\frac{N}{4}-1,\,\cdots,\,\frac{N}{2}-1\end{cases} ,\,\,a\,\,=\,1,\,2 \tag{5}\]  
\[\text{X}_{a}[K]= \begin{cases}A&K=0\\ d_{a}&K=1,\,\cdots,\,\frac{N}{4}\\ 0&K=\frac{N}{4}+1,\,\cdots,\,\frac{N}{2}-1\end{cases} \,\,\,a\,\,=\,\,3,\,4, \tag{6}\]  
where \(K=\frac{N}{2}\), is the number of subcarriers and also the size of IFFT/FFT (\(K=A_{g}+A_{s}\)). The OFDM signal in the time domain \(x_{a}(\text{t})\) is generated as:  
\[x_{a}(\text{t})=\mathcal{R}\,\left\{\sum_{\text{K}=0}^{\frac{N}{2}-1}\,\, \text{X}_{a}[\text{K}]\,\,\text{e}^{\text{j}2\pi\,\Delta\zeta(\,\text{K}\,)t} \right\},\,a=1,\cdots,\,4, \tag{7}\]  
where \(\Delta\zeta\) is OFDM subcarrier frequency spacing [22]. Then, the parallel-to-series (P/S) conversion is applied to the output of the IFFT stage and after that the cyclic prefix (CP) is added to eliminate the inter symbol interference (ISI). A digital-to-analogue-converter (DAC) is then used to remove the high frequencies. Then, the transmitter encodes the information signal by the space time encoder. Let \(\mathbf{x}=[x_{1},x_{2},x_{3},x_{4}]\) be the information symbols vector, in time domain, which is transmitted on each of the OFDM subcarrier, then the code-word matrix which is generated by Alamouti code is represented in a matrix form as:  
\[\mathbf{X}_{c}=