# Efficient Massive MIMO Detection for M-QAM Symbols
## 3 Detector Based on Iterated BOXDCD with Negative Diagonal Loading Regularization
### Combination of Box-Constrained Minimization with the DCD Algorithm
The algorithm's worst-case complexity is bounded by \(K_{t}(2N_{u}+M_{b}-1)+N_{u}\) shift-accumulation operations with given \(N_{u}\) and \(M_{b}\). In practice, however, there should be several successful iterations in each pass. The average complexity of the BOXDCD algorithm is close to \(2K_{t}N_{u}\). For dealing with a \(K_{t}\times K_{t}\) system, the real part and the imaginary part are processed sequentially. The whole average computational complexity of the BOXDCD is approximately \(4K_{t}N_{u}\). An iterative method can obtain a more accurate solution to Equation (9). Here, applying an iteration with negative diagonal loading regularization yields the following solution:  
\[\mathbf{\hat{s}}^{(v)}=\arg\min_{\mathbf{s}\in\mathcal{U}_{A}}\left\{\mathbf{s}^ {H}\mathbf{R}\mathbf{s}-2\Re(\mathbf{y}+\mu\mathbf{\hat{s}}^{(v-1)})^{H}\mathbf{ s}-\mu\mathbf{s}^{H}\mathbf{s}\right\} \tag{10}\]  
where \(\mathbf{\hat{s}}^{(-1)}=\mathbf{0}\), \(\mathbf{\hat{s}}^{(v-1)}=\mathbf{\hat{s}}^{(v-1)}\). The solution to (10) can be regarded as the solution to the equation  
\[(\mathbf{R}-\mu\mathbf{I})\mathbf{\hat{s}}^{(v)} = \mathbf{y}+\mu\mathbf{\hat{s}}^{(v-1)} \tag{11}\] \[\implies\mathbf{\hat{R}}\mathbf{\hat{s}}^{(v)} = \mathbf{y}^{(v)}.\]  
In each iteration, Equation (11) can be solved using the BOXDCD algorithm. Consequently, we obtain the iterated BOXDCD with a negative diagonal loading (INL-BOXDCD) detector for M-QAM symbols, which is summarized in Algorithm 2, and the individual optimization steps are given below. **Step 1:**: The BOXDCD algorithm is applied to solve equation \(\mathbf{y}=\mathbf{\hat{R}}\mathbf{s}\). \(\mathbf{s}\) is initialized as \(\mathbf{s}_{0}\) and \(\mathbf{\hat{R}}=\mathbf{R}-\mu\mathbf{I}\), where regularization parameter \(\mu=\eta\sigma^{2}\). **Step 2:**: The soft output \(\mathbf{\hat{s}}_{0}\) obtained from step 1 is demodulated and mapped into the M-QAM constellation to obtain \(\mathbf{\hat{s}}^{(0)}\). **Step 3:**: For each iteration, the vector \(\mathbf{y}\) is updated as \(\mathbf{y}^{(v)}=\mathbf{y}+\mu\mathbf{\hat{s}}^{(v-1)}\). Then, the BOXDCD is applied with \(\mathbf{\hat{s}}^{(v-1)}\) and the updated \(\mathbf{y}\) to obtain the solution vector \(\mathbf{\hat{s}}^{(v)}\). **Step 4:**: \(\mathbf{\hat{s}}^{(V)}\) is mapped into the M-QAM constellation to obtain \(\mathbf{\hat{s}}\). For a given \(N_{u}\), the INL-BOXDCD detector has an average computational complexity of \(4K_{t}N_{u}(1+V)\) additions for a \(K_{t}\times K_{t}\) MIMO system. ```
Step Operation Input: \(\mathbf{s}_{0}\), \(\mathbf{R}\), \(\mathbf{y}\), \(\mathbf{\hat{c}}\), \(N_{u}\), \(M_{b}\), \(\mu\) Output: \(\mathbf{\hat{s}}\) Initialization \(\mathbf{s}=\mathbf{s}_{0}\), \(\mathbf{\hat{R}}=\mathbf{R}-\mu\mathbf{I}\)  Use \(\text{BOXDCD}(\mathbf{s},\mathbf{\hat{R}},\mathbf{y},N_{u},M_{b})\) to solve \(\mathbf{y}=\mathbf{\hat{R}}\mathbf{s}\) to obtain \(\mathbf{\hat{s}}_{0}\). Map \(\mathbf{\hat{s}}_{0}\) into the M-QAM constellation to obtain \(\mathbf{\hat{s}}^{(0)}\). for\(v=1,2,3\cdots,V\) \(\mathbf{y}^{(v)}=\mathbf{y}+\mu\mathbf{\hat{s}}^{(v-1)}\)  Use \(\text{BOXDCD}(\mathbf{\hat{s}}^{(v-1)},\mathbf{\hat{R}},\mathbf{y}^{(v)},N_{ u},M_{b})\) to solve \(\mathbf{y}^{(v)}=\mathbf{\hat{R}}\mathbf{s}^{(v)}\) to obtain \(\mathbf{\hat{s}}^{(v)}\). endfor  Map \(\mathbf{\hat{s}}^{(V)}\) into the M-QAM constellation to obtain \(\mathbf{\hat{s}}\). ```  
**Algorithm 2**Proposed INL-BOXDCD detector.