# Low-Complexity GSM Detection Based on Maximum Ratio Combining
## 4 MRC-MP Detection Algorithm  
In this section, a maximum ratio combining detection algorithms based on the iterative idea of the orthogonal matching pursuit, termed as the MRC-MP algorithm, is proposed. The MRC-MP algorithm can be divided into three stages. In the first stage, the TACs are constructed combining with the iteration idea of the OMP algorithm. By multiple iterations, new TACs with fewer antenna combinations are obtained. In the second stage, the transmit symbol is quantitated. In the third stage, the TAC and the modulation symbol are estimated jointly. The second and third stages are same as that of the MRC-RS algorithm. The specific implementation steps are as follows. \begin{table}
\begin{tabular}{l l} \hline \hline
1: & Input : \(\mathbf{y}\), \(\mathbf{H}\), \(N_{r}\), \(N_{p}\). \\ \multicolumn{3}{l}{Calculate the MRC values of \(N_{\text{C}}\) TACs \(k_{i}=\frac{\left\|\left(\hat{\mathbf{h}}\right)^{h}y\right\|^{2}}{\left\|\hat\mathbf{{h}}\right\|^{2}}\),} \\
2: & and obtain the weight vector \(\mathbf{k}=\left[k_{1},k_{2},\cdots,k_{N_{\text{C}}}\right]^{\text{T}}\), \\ \multicolumn{3}{l}{where \(\bar{\mathbf{h}}_{i}=\mathbf{h}_{i(1)}+\mathbf{h}_{i(2)}+\cdots+\mathbf{h}_{i(N_{p})}=\sum \limits_{l=1}^{N_{p}}\mathbf{h}_{i(l)}\).} \\
3: & Sort \(N_{\text{C}}\) TACs according to weight value : \(\left[w_{1},w_{2},\cdots,w_{N_{\text{C}}}\right]=\text{arg}\;sort(\mathbf{k})\). \\ \multicolumn{3}{l}{Obtain the \(L\) new TACs, calculate the MRC values of \(L\) TACs \(r_{j}=\frac{\left\|\left(\hat{\mathbf{h}}\right)^{h}y\right\|^{2}}{\left\|\hat\mathbf{{h}}\right\|^{2}}\),} \\
4: & get the weight vector \(\mathbf{r}=\left[r_{1},r_{2},\cdots,r_{L}\right]^{T}\), \\ \multicolumn{3}{l}{and sort \(\left[n_{1},n_{2},\cdots,n_{L}\right]=\text{arg}sort(\mathbf{r})\), where \(\bar{\mathbf{h}}_{j}=\mathbf{h}_{j(1)}+\mathbf{h}_{j(2)}+\cdots+\mathbf{h}_{j(N_{p})}=\sum \limits_{l=1}^{N_{p}}\mathbf{h}_{j(l)}\).} \\
5: & Reserve only \(round(L/2)\) TACs for estimating the solution. \\
6: & Obtain the quantitated modulation symbol \(s_{\alpha}\) using Equations (11)-(15) for M-QAM \\
7: & constellation and Equations (11), (12), (17) and (18) for M-PSK constellation, respectively. \\
7: & The TAC and the modulation symbol are estimated using ML algorithm. \\
8: & Output the transmit antenna combination and modulation symbol \(\left(I,s\right)\). \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Proposed MRC-RS detector. Step 1: Initialize the required variables, the iteration counter and residual are \(\beta=1\) and \(\mathbf{r}_{1}=y\), respectively. The active antenna index set is \(\psi=\left\{\psi_{1},\psi_{2},\cdots,\psi_{N_{p}}\right\}\), and \(\psi\) is an empty set. Step 2: In the \(\beta\)-th iteration, the MRC value \(k_{\beta}^{l}\) between \(\mathbf{r}_{\beta}\) and \(\mathbf{h}_{l}\) is calculated as  
\[k_{\beta}^{l}=\frac{\left\|\mathbf{h}_{l}{}^{H}\mathbf{r}_{\beta}\right\|^{2}}{\left\| \mathbf{h}_{l}\right\|^{2}}, \tag{20}\]  
where \(\mathbf{h}_{l}\) is the \(l\)-th column of the channel gain matrix \(\mathbf{H}\), \(l=1,2,\cdots,N_{l}\). Step 3: Calculate the average value of \(N_{l}\) MRC values in the \(\beta\)-th iteration and find out the qualified antenna indices. \[\eta_{avr}=\frac{\sum\limits_{l=1}^{N_{l}}k_{\beta}^{l}}{N_{l}}, \tag{21}\]  
where \(\eta_{avr}\) denotes the average value of \(N_{l}\) MRC values. All \(N_{l}\) MRC values are compared with \(\eta_{avr}\). If the MRC value is greater than the average value \(\eta_{avr}\), that is, \(k_{\beta}^{l}>\eta_{avr}\), the antenna index corresponding to the MRC value is added into the set \(\psi_{\beta}\),  
\[\psi_{\beta}=\{l\}, \tag{22}\]  
where \(l\in\{1,2,\cdots,N_{t}\}\). Step 4: The signal is estimated using the least square method in the \(\beta\)-th iteration.