## 4 Joint Iterative ATC Decoder  
Another elegant feature for the ATC-coded cooperative architecture is the joint iterative ATC decoding employed at the destination. The joint ATC decoder consists of two different soft input soft output (SISO) decoders, as given in the literature [33]. For simplicity, the diagram of the joint iterative log maximum a posterior (log-MAP) [34,35] ATC decoder of two-source ATCC-SM in Section 3.1 is shown in Figure 8.  
Figure 8: Joint iterative ATC decoder.  
In Figure 8, the LLRs \(\gamma_{\text{S}_{1},\text{D}}(\mathbf{v}^{\text{S}_{1}})\) and \(\gamma_{\text{S}_{2},\text{D}}(\mathbf{v}^{\text{S}_{2}})\) are de-multiplexed into pairs of \(\gamma(\mathbf{q}_{1}^{\prime})\) and \(\gamma(\mathbf{p}_{1}^{\prime})\), and pairs of \(\gamma(\mathbf{q}_{2}^{\prime})\) and \(\gamma(\mathbf{p}_{2}^{\prime})\) by the de-multiplexer, where \(\gamma(\mathbf{q}_{1}^{\prime})\) and \(\gamma(\mathbf{q}_{2}^{\prime})\) denote the LLRs of systematic bits \(\mathbf{q}_{1}^{\prime}\) and \(\mathbf{q}_{2}^{\prime}\), whereas \(\gamma(\mathbf{p}_{1}^{\prime})\) and \(\gamma(\mathbf{p}_{2}^{\prime})\) represent the LLRs of the parity bits \(\mathbf{p}_{1}^{\prime}\) and \(\mathbf{p}_{2}^{\prime}\), respectively.  
During the first time slot, the LLRs \(\gamma(\mathbf{q}_{1}^{\prime})\) and \(\gamma(\mathbf{p}_{1}^{\prime})\) are fed into SISO decoder-1 to generate extrinsic information \(\gamma_{\text{EXT1},2}(\mathbf{q}_{1}^{\prime})\). Similarly, \(\gamma(\mathbf{q}_{2}^{\prime})\), together with \(\gamma(\mathbf{p}_{2}^{\prime})\), is given as an input to SISO decoder-1 that produces extrinsic information \(\gamma_{\text{EXT1},2}(\mathbf{q}_{2}^{\prime})\). These LLRs \(\gamma(\mathbf{q}_{1}^{\prime})\), \(\gamma(\mathbf{q}_{2}^{\prime})\), \(\gamma_{\text{EXT1},2}(\mathbf{q}_{1}^{\prime})\) and \(\gamma_{\text{EXT1},2}(\mathbf{q}_{2}^{\prime})\) are interleaved into \(\gamma^{1}(\mathbf{q}_{1}^{\prime})\), \(\gamma^{1}(\mathbf{q}_{2}^{\prime})\), \(\gamma^{1}_{\text{EXT1},2}(\mathbf{q}_{1}^{\prime})\) and \(\gamma^{1}_{\text{EXT1},2}(\mathbf{q}_{2}^{\prime})\), respectively, by passing through the CMI. Next, \(\gamma^{1}(\mathbf{q}_{1}^{\prime})\) and \(\gamma^{1}_{\text{EXT1},2}(\mathbf{q}_{1}^{\prime})\), together with parity LLRs \(\overline{\gamma}_{\text{S}_{2},\text{D}}(\overline{\mathbf{v}}^{\text{S}_{ 2}})\), are delivered to SISO decoder-2. During the second time slot, the extrinsic information \(\gamma^{1}_{\text{EXT2},1}(\mathbf{q}_{1}^{\prime})\) is generated. Similarly, SISO decoder-2 accepts \(\gamma^{1}(\mathbf{q}_{2}^{\prime})\), \(\gamma^{1}_{\text{EXT1},2}(\mathbf{q}_{2}^{\prime})\) and \(\overline{\gamma}_{\text{S}_{1},\text{D}}(\overline{\mathbf{v}}^{\text{S}_{ 1}})\) to produce extrinsic information \(\gamma^{1}_{\text{EXT2},1}(\mathbf{q}_{2}^{\prime})\). Afterwards, \(\gamma^{1}_{\text{EXT2},1}(\mathbf{q}_{1}^{\prime})\) and \(\gamma^{1}_{\text{EXT2},1}(\mathbf{q}_{2}^{\prime})\) are passed through the CMI\({}^{-1}\) to generate the de-interleaved sequences \(\gamma_{\text{EXT2},1}(\mathbf{q}_{1}^{\prime})\) and \(\gamma_{\text{EXT2},1}(\mathbf{q}_{2}^{\prime})\), which are given back to SISO decoder-1 for the next iteration, as shown in Figure 8. After reaching a specific number of iterations, the extrinsic information \(\gamma^{1}_{\text{EXT1},2}(\mathbf{q}_{1}^{\prime})\) and \(\gamma^{1}_{\text{EXT2},1}(\mathbf{q}_{1}^{\prime})\) are summed up and delivered to the de-interleaver and slicer that finally estimates sequence \(\hat{\mathbf{q}}_{1}^{\prime}\). Similarly, the estimation \(\hat{\mathbf{q}}_{2}^{\prime}\) is generated when the sum of \(\gamma^{1}_{\text{EXT1},2}(\mathbf{q}_{2}^{\prime})\) and \(\gamma^{1}_{\text{EXT2},1}(\mathbf{q}_{2}^{\prime})\) is considered as the input of the de-interleaver and the slicer.