## 2 Turbo Coded OFDM with MIMO (2 \(\times\) 2)
The received signal after removal of cyclic prefix and N-DFT operation at time slot 1 and time slot 2 are modeled as follows:  
\[\mathrm{Y}_{\mathrm{1c(n)}}=\mathrm{DFT}[\mathrm{h}_{\mathrm{11}}\otimes \left(\mathrm{IDFT}(\mathrm{c}_{1})\right)+\mathrm{h}_{\mathrm{12}}\otimes \left(\mathrm{IDFT}(\mathrm{c}_{2})\right)+\mathrm{v}_{\mathrm{1(n)}}] \tag{1}\]  
\[\mathrm{Y}_{\mathrm{1c(n)}}=\mathrm{H}_{\mathrm{11}}\times\mathrm{C}_{1}+ \mathrm{H}_{\mathrm{12}}\times\mathrm{C}_{2}+\mathrm{V}_{\mathrm{1(n)}} \tag{2}\]  
\[\mathrm{Y}_{\mathrm{2c(n)}}=\mathrm{DFT}[\mathrm{h}_{\mathrm{21}}\otimes \left(\mathrm{IDFT}(\mathrm{c}_{1})\right)+\mathrm{h}_{\mathrm{22}}\otimes \left(\mathrm{IDFT}(\mathrm{c}_{2})\right)+\mathrm{v}_{\mathrm{2(n)}}] \tag{3}\]  
\[\mathrm{Y}_{\mathrm{2c(n)}}=\mathrm{H}_{\mathrm{21}}\times\mathrm{C}_{1}+ \mathrm{H}_{\mathrm{22}}\times\mathrm{C}_{2}+\mathrm{V}_{\mathrm{2(n)}} \tag{4}\]  
\[\mathrm{Y}_{\mathrm{1c(n+1)}}=\mathrm{DFT}[\mathrm{h}_{\mathrm{11}}\otimes \left(\mathrm{IDFT}(-\mathrm{c}_{2}^{*})\right)+\mathrm{h}_{\mathrm{12}} \otimes\left(\mathrm{IDFT}(\mathrm{c}_{1}^{*})\right)+\mathrm{v}_{\mathrm{1(n+1)}}] \tag{5}\]  
\[\mathrm{Y}_{\mathrm{1c(n+1)}}=-\mathrm{H}_{\mathrm{11}}\times\mathrm{C}_{2}^{*} +\mathrm{H}_{\mathrm{12}}\times\mathrm{C}_{1}^{*}+\mathrm{V}_{\mathrm{1(n+1)}} \tag{6}\]  
\[\mathrm{Y}_{\mathrm{2c(n+1)}}=\mathrm{DFT}[\mathrm{h}_{\mathrm{21}}\otimes \left(\mathrm{IDFT}(-\mathrm{c}_{2}^{*})\right)+\mathrm{h}_{\mathrm{22}} \otimes\left(\mathrm{IDFT}(\mathrm{c}_{1}^{*})\right)+\mathrm{v}_{\mathrm{2(n+1) }}] \tag{7}\]  
\[\mathrm{Y}_{\mathrm{2c(n+1)}}=-\mathrm{H}_{\mathrm{21}}\times\mathrm{C}_{2}^{*} +\mathrm{H}_{\mathrm{22}}\times\mathrm{C}_{1}^{*}+\mathrm{V}_{\mathrm{2(n+1)}} \tag{8}\]  
where the operator \"\(\otimes\)\" denotes the circular convolution. \(\mathrm{C}_{1}\) and \(\mathrm{C}_{2}\) are the corresponding frequency domain representation of \(\mathrm{c}_{1}\) and \(\mathrm{c}_{2}\) respectively. The subscript \(\mathrm{c}\) of \(\mathrm{Y}_{\mathrm{1c(n)}}\), \(\mathrm{Y}_{\mathrm{2c(n)}}\), \(\mathrm{Y}_{\mathrm{1c(n+1)}}\) and \(\mathrm{Y}_{\mathrm{2c(n+1)}}\) represents the combined received signal at receive antenna1 \(\mathrm{R}_{\mathrm{x}1}\), receive antenna2 \(\mathrm{R}_{\mathrm{x}2}\) in time slot-1 and time slot-2 respectively. The terms such as \(\mathrm{v}_  
impulse response from transmit antenna2 T\({}_{\chi 2}\) to the received antenna1 R\({}_{\chi 1}\) and h\({}_{22}\) represents channel impulse response from transmit antenna2 T\({}_{\chi 2}\) to the received antenna2 R\({}_{\chi 2}\). Subsequently H\({}_{11}\), H\({}_{12}\), H\({}_{21}\) and H\({}_{22}\) are diagonal matrix whose diagonal entries are DFTs of the corresponding channel impulse response. Now the Alamouti STBC decoding in frequency domain is given as follows:  
\[\hat{\mathfrak{c}}_{1}=\mathrm{H}_{11}^{*}Y_{1\mathrm{c(n)}}+\mathrm{H}_{12}Y_{ 1\mathrm{c(n+1)}}^{*}+\mathrm{H}_{21}^{*}Y_{2\mathrm{c(n)}}+\mathrm{H}_{22}Y_{ 2\mathrm{c(n+1)}}^{*} \tag{9}\]  
\[\mathrm{Y}_{\mathrm{1c(n)}}=\mathrm{DFT}[\mathrm{h}_{\mathrm{11}}\otimes \left(\mathrm{IDFT}(\mathrm{c}_{1})\right)+\mathrm{h}_{\mathrm{12}}\otimes \left(\mathrm{IDFT}(\mathrm{c}_{2})\right)+\mathrm{v}_{\mathrm{1(n)}}] \tag{1}\]0  
The estimated signals i.e., \(\hat{\mathfrak{c}}_{1}\) and \(\hat{\mathfrak{c}}_{2}\) are combined to get the signal \(\hat{\mathfrak{c}}\). The signal \(\hat{\mathfrak{c}}\) is given as input to the de-mapper that performs the corresponding demodulation process to obtain the estimated sequence of soft bits \(\hat{\mathfrak{f}}\). Finally, joint turbo decoding is performed to get final estimation of information sequences \(\hat{\mathfrak{m}}\). The details of joint turbo decoding will be discussed in the subsequent section of this manuscript.