## 3 Two-Source Asymmetric Turbo-Coded Cooperative (ATCC) Scheme Based on CMI
### Two-Source Asymmetric Turbo-Coded Cooperative Spatial Modulation (ATCC-SM) Scheme Based on CMI
Likewise, the transmission matrix at \(\mathsf{S_{2}}\) is:  
\[\mathbf{V^{S_{2}}_{m}}=\!\!\bigg{[}\big{(}\mathbf{v^{S_{2}}_{1,m}}\big{)}^{ \mathsf{T}},\,(\mathbf{v^{S_{2}}_{2,m}}\big{)}^{\mathsf{T}},\,\ldots,\,( \mathbf{v^{S_{2}}_{N_{\mathrm{T}},m}})^{\mathsf{T}}\bigg{]}^{\mathsf{T}}, \tag{24}\]  
which is similar to \(\mathbf{V^{S_{1}}_{m}}\) given in Equation (23). Next, the matrix \(\mathbf{V^{S_{2}}_{m}}\) is transmitted towards \(\mathsf{S_{1}}\) and the destination node over the slow Rayleigh fading MIMO channel. During the first time slot, the received matrices \(\mathbf{Z}_{\mathsf{S_{1}},\mathsf{S_{2}}}\), \(\mathbf{Z}_{\mathsf{S_{2}},\mathsf{S_{1}}}\), \(\mathbf{Z}_{\mathsf{S_{1}},\mathrm{D}}\) and \(\mathbf{Z}_{\mathsf{S_{2}},\mathrm{D}}\) are mathematically modelled as:  
\[\mathbf{Z}_{\mathsf{S_{1}},\mathsf{S_{2}}}=\ \mathbf{H^{\prime}_{S_{1}, \mathsf{S_{2}}}}\mathbf{V^{S_{1}}_{m}}+\mathbf{N}_{\mathsf{S_{1}},\mathsf{S_{2 }}},\qquad\mathbf{Z}_{\mathsf{S_{2}},\mathsf{S_{1}}}=\ \mathbf{H^{\prime}_{S_{2}, \mathsf{S_{1}}}}\mathbf{V^{S_{2}}_{m}}+\mathbf{N}_{\mathsf{S_{2}},\mathsf{S_{ 1}}} \tag{25}\]  
\[\mathbf{Z}_{\mathsf{S_{1}},\mathrm{D}}=\ \mathbf{H^{\prime}_{S_{1}, \mathrm{D}}}\mathbf{V^{S_{1}}_{m}}+\mathbf{N}_{\mathsf{S_{1}},\mathrm{D}}, \qquad\mathbf{Z}_{\mathsf{S_{2}},\mathrm{D}}=\ \mathbf{H^{\prime}_{S_{2}, \mathrm{D}}}\mathbf{V^{S_{2}}_{m}}+\mathbf{N}_{\mathsf{S_{2}},\mathrm{D}}, \tag{26}\]  
where \(\mathbf{H^{\prime}_{S_{1},\mathsf{S_{2}}}}\), \(\mathbf{H^{\prime}_{S_{2},\mathsf{S_{1}}}}\), \(\mathbf{H^{\prime}_{S_{1},\mathrm{D}}}\), and \(\mathbf{H^{\prime}_{S_{2},\mathrm{D}}}\) represent the channel matrices of \(\mathsf{S_{1}}\)-\(\mathsf{S_{2}}\), \(\mathsf{S_{2}}\)-\(\mathsf{S_{1}}\), \(\mathsf{S_{1}}\) to destination, and \(\mathsf{S_{2}}\) to destination, respectively, and are defined similarly to \(\mathbf{H}\) in Equation (13). \(\mathbf{N}_{\mathsf{S_{1}},\mathsf{S_{2}}}\), \(\mathbf{N}_{\mathsf{S_{2}},\mathsf{S_{1}}}\), \(\mathbf{N}_{\mathsf{S_{1}},\mathrm{D}}\), and \(\mathbf{N}_{\mathsf{S_{2}},\mathrm{D}}\) denote the complex additive Gaussian noise of \(\mathsf{S_{1}}\)-\(\mathsf{S_{2}}\), \(\mathsf{S_{2}}\)-\(\mathsf{S_{1}}\), \(\mathsf{S_{1}}\) to destination, and \(\mathsf{S_{2}}\) to destination, respectively, and are defined similarly to \(\mathbf{N}\) in Equation (15). The matrices \(\mathbf{Z}_{\mathsf{S_{1}},\mathsf{S_{2}}}\) and \(\mathbf{Z}_{\mathsf{S_{2}},\mathsf{S_{1}}}\) are fed into the VBLAST detector and the RSC-1 decoder to produce the estimated sequences \(\mathbf{\hat{a}^{\prime 1}_{1}}\), \(\mathbf{\hat{a}^{\prime 2}_{1}}\),..., \(\mathbf{\hat{a}^{\prime N_{\mathrm{T}}}_{1}}\), and \(\mathbf{\hat{a}^{\prime 1}_{2}}\), \(\mathbf{\hat{a}^{\prime 2}_{2}}\),..., \(\mathbf{\hat{a}^{\prime N_{\mathrm{T}}}_{2}}\), respectively. These sequences are delivered to the CMI, the RSC-2 encoder and the M-PSK/QAM modulator to get \(1\times\) (L\({}_{1}\)/2) dimensional  
Figure 7: Two-source ATCC-VBLAST scheme with the CMI. modulated parity sequences \(\overline{\mathbf{V}}_{1,\text{m}}^{\text{S}_{2}}\), \(\overline{\mathbf{V}}_{2,\text{m}}^{\text{S}_{2}}\),... \(\overline{\mathbf{V}}_{\text{N}_{\text{T},\text{m}}}^{\text{S}_{2}}\) and \(\overline{\mathbf{V}}_{1,\text{m}}^{\text{S}_{1}}\), \(\overline{\mathbf{V}}_{2,\text{m}}^{\text{S}_{1}}\),... \(\overline{\mathbf{V}}_{\text{N}_{\text{T},\text{m}}}^{\text{S}_{1}}\), respectively. After that, the newly re-arranged matrices  
\[\mathbf{v^{S_{1}}_{l_{1}r_{1}}}(\mathbf{m}_{1})\ =\ [0,\ \ldots,\ 0,\ \mathbf{v^{S_{1}}_{r_{1}}}( \mathbf{m}_{1})\,\ 0,\ \ldots,\ 0]^{\mathrm{T}},\ \ \ \ \mathbf{v^{S_{2}}_{l_{2}r_{2}}}(\mathbf{m}_{2})\ =\ [0,\ \ldots,\ 0,\ \mathbf{v^{S_{2}}_{r_{2}}}( \mathbf{m}_{2}),\ 0,\ldots,\ 0]^{\mathrm{T}} \tag{16}\]0  
from \(\text{S}_{2}\) and \(\text{S}_{1}\) are transmitted towards the common destination node.