## 4 Turbo Coded OFDM with MIMO Antennas in Cooperation
\(\mathbf{N}_{S-D}\) is the complex Gaussian noise matrix whose entries are modelled as I.I.D complex Gaussian random variables with zero mean and variance \(N_{0}/2\) per dimension. The channel entries remain static during the transmission of one complete Alamouti STBC. Therefore, the channel qualifies to be called as fast Rayleigh fading channel. At the same time, in first time slot the signal received at the relay-R is modeled as  
\[\mathbf{Y}_{S-R}=\mathbf{H}_{S-R}\mathbf{X}_{s}+\mathbf{N}_{S-R} \tag{27}\]  
where \(\mathbf{X}_{s}\) is the Alamouti space time code word \(\mathbf{H}_{S-R}\) and \(\mathbf{N}_{S-R}\) channel matrix and Gaussian noise matrix defined similarly as \(\mathbf{H}_{S-D}\) and \(\mathbf{N}_{S-D}\) respectively. The received signal \(\mathbf{Y}_{S-R}\) is decoded using Alamouti decoding [19]. The signal is passed through OFDM decoder and corresponding soft demodulator to estimate the original information sequence \(\hat{i}_{1}\). The estimated information sequence \(\hat{i}_{1}\) may or may not be same as original information sequence \(\hat{i}_{1}\). It depends upon the source-S to relay-R channel. The estimated information sequence \(\hat{i}_{1}\) is interleaved by CMI that permutes the estimated information sequence \(\pi(\hat{i}_{1})\) in an optimum way. The relay-R encodes the interleaved estimated information sequence \(\pi(\hat{i}_{1})\) using RSC-2 encoder. The RSC-2 encoded sequences of parity bits \(\mathbf{p}_{2}\) are mapped into digital modulator to obtain the complex digital modulated symbols \(\mathbf{b}\). The RSC-2 encoded symbols \(\mathbf{b}\) is delivered to OFDM block which takes N point IDFT and appends a cyclic prefix to form RSC-2 encoded OFDM symbol \(\overline{\mathbf{X}}_{k}\). The resulted RSC-2 encoded OFDM symbol \(\overline{\mathbf{X}}_{k}\) is again given as input to Alamouti STBC which results in two parallel sequences \(\overline{\mathbf{X}}_{k}^{i}\) and \(\overline{\mathbf{X}}_{k}^{i+1}\). These two Alamouti code words are propagated towards  
Figure 5: Distributed turbo coded OFDM with MIMO (\(2\times 2\)) antennas. destination-D according to Alamouti encoding scheme [20]. The signal received at the destination-D during the second time slot is modelled as follows as:  
\[\mathbf{Y}_{R-D}=\mathbf{H}_{R-D}\mathbf{X}_{R}+\mathbf{N}_{R-D} \tag{28}\]  
\[\mathbf{X}_{R}=\left[ \begin{array}{cc}\mathbf{\overline{X}}_{k}^{i}&-\mathbf{ \overline{X}}_{k}^{(i+1)*}\\ \mathbf{\overline{X}}_{k}^{(i+1)}&\mathbf{\overline{X}}_{k}^{i*}\end{array} \right]\]  
where \(\mathbf{H}_{R-D}\) and \(\mathbf{N}_{R-D}\) are channel matrix and Gaussian noise matrix respectively and are defined similarly as \(\mathbf{H}_{S-D}\) and \(\mathbf{N}_{S-D}\). The received signals \(\mathbf{Y}_{S-D}\) and \(\mathbf{Y}_{R-D}\) are delivered to Alamouti STBC decoder and OFDM demodulator in their respected time slots. OFDM demodulator takes N-point DFT and removes the cyclic prefix to form the complex symbols. These complex symbols are digitally demodulated by SISO demodulator produces log likelihood ratios (LLRs) for each of the received coded bits. These soft bit LLRs for first and second time slot are given an input to joint iterative turbo decoder, which iterates for a desired number of iterations and pass the updated LLRs to the slicer. The term joint here refers to the fact the decoding of the signals \(\mathbf{Y}_{S-D}\) and \(\mathbf{Y}_{R-D}\) are performed together rather than decoding each of the received signal separately. The joint iterative turbo decoding for the proposed scheme is comprehensively discussed in the subsequent section of the manuscript. In the end the slicer performs the hard decoding operation on each received soft bit to recover the original transmitted sequence of information bits \(\hat{t}_{1}\).