## 2 System Model
### AML Detection  
The JML detection approach described in (14) entails significant computational complexity because of the exhaustive search over \(|\mathcal{A}|^{4N_{S}}\) possible symbol combinations. An approximation detection method based on error propagation at the relay is proposed in [26] to alleviate this burden. This method applies a combination of the max-log and PEP approximations to simplify the original JML formulation. \[\hat{\mathbf{s}}_{S} \stackrel{{(a)}}{{=}}\underset{\mathbf{x}_{S}\in \mathcal{A}^{2N_{S}}}{\operatorname{argmax}}\mathbb{P}\{\mathbf{q}_{SD}| \mathbf{x}_{S}\}\times\underset{\mathbf{x}_{R}\in\mathcal{A}^{2N_{S}}}{ \operatorname{max}}\mathbb{P}\{\mathbf{x}_{R}|\mathbf{x}_{S}\}\mathbb{P}\{ \mathbf{q}_{RD}|\mathbf{x}_{R}\}\] \[=\underset{[\mathbf{x}_{S}^{T}\mathbf{x}_{R}^{T}]^{T}\in \mathcal{A}^{4N_{S}}}{\operatorname{argmax}}\mathbb{P}\{\mathbf{q}_{SD}| \mathbf{x}_{S}\}\mathbb{P}\{\mathbf{x}_{R}|\mathbf{x}_{S}\}\mathbb{P}\{ \mathbf{q}_{RD}|\mathbf{x}_{R}\}\] \[\stackrel{{(b)}}{{=}}\underset{[\mathbf{x}_{S}^{T} \mathbf{x}_{R}^{T}]^{T}\in\mathcal{A}^{4N_{S}}}{\operatorname{argmax}} \mathbb{P}\{\mathbf{q}_{SD}|\mathbf{x}_{S}\}\mathbb{P}\{\mathbf{x}_{S}\to \mathbf{x}_{R}\}\mathbb{P}\{\mathbf{q}_{RD}|\mathbf{x}_{R}\}. \tag{15}\]  
Step \((a)\) employs the max-log approximation to simplify the logarithmic sum of the detection metric. In step \((b)\), the conditional error probability is further approximated using the PEP, which enables a tractable analysis of the detection reliability of the relay. Term \(\mathbb{P}\{\mathbf{x}_{S}\to\mathbf{x}_{R}\}\) represents the PEP, which indicates the likelihood that the relay erroneously detects \(\mathbf{x}_{R}\) given that \(\mathbf{x}_{S}\) is originally transmitted, and it is also expressed as the weighted Hamming distance function [26]. \[\log(\mathbb{P}\{\mathbf{x}_{S}\to\mathbf{x}_{R}\})\leq-d_{w}(\mathbf{c}_{SR}( \mathbf{x}_{S}),\mathbf{c}_{SR}(\mathbf{x}_{R});\mathbf{w}_{SR}(\mathbf{x}_{S },\mathbf{\tilde{x}}_{R}),\mathbf{\tilde{w}}_{SR}(\mathbf{x}_{S},\mathbf{x}_{ R})). \tag{16}\]  
Since the PEP admits an upper bound that depends on the weighted Hamming distance between two quantized codewords \(\mathbf{c}_{ab}(\mathbf{x}_{a})\) and \(\mathbf{c}_{ab}(\mathbf{\tilde{x}}_{a})\), the detection rule in (15) can be reformulated in a more compact and computationally efficient form as  
\[\hat{\mathbf{x}}_{S}=\underset{[\mathbf{x}_{S}^{T}\mathbf{x}_{R}^{T}]^{T}\in \mathcal{A}^{4N_{S}}}{\operatorname{argmin}}\Bigg{(} \begin{bmatrix}\mathbf{q} _{SD}\\ \mathbf{q}_{RD}\\ \mathbf{c}_{SR}(\mathbf{x}_{S})\end{bmatrix} , \begin{bmatrix}\mathbf{c}_{SD}( \mathbf{x}_{S})\\ \mathbf{c}_{RD}(\mathbf{x}_{R})\\ \mathbf{c}_{SR}(\mathbf{x}_{R})\end{bmatrix} ; \begin{bmatrix}\mathbf{w}_{SD}( \mathbf{x}_{S})\\ \mathbf{w}_{RD}(\mathbf{x}_{R})\\ \mathbf{w}_{SR}(\mathbf{x}_{S},\mathbf{x}_{R})\end{bmatrix} , \begin{bmatrix} \mathbf{\tilde{w}}_{SD}(\mathbf{x}_{S})\\ \mathbf{\tilde{w}}_{RD}(\mathbf{x}_{R})\\ \mathbf{\tilde{w}}_{SR}(\mathbf{x}_{S},\mathbf{x}_{R})\end{bmatrix} \Bigg{)}. \tag{17}\]  
This unified representation consolidates three separate weighted Hamming distance terms into a single expression, simplifying implementation and reducing computational complexity. The detection strategy in (17) enables the interpretation of a DF relay system with an antenna configuration \((N_{S},N_{R},N_{D})\) as an equivalent MIMO system comprising \((N_{S}+N_{R})\) transmitting antennas and \((2N_{D}+N_{R})\) receiving antennas. This structural equivalence enables the low-complexity detection method proposed in [27] to be applied directly to (17). Consequently, AML achieves a substantial reduction in computational complexity compared to the original JML detection rule. A comparison of the detection schemes is given below:  
* ML detection: Optimal decision rule for point-to-point MIMO systems. It maximizes the likelihood probability based on the received signal and provides the lowest bit error rate under perfect channel state information.