# Low-Complexity GSM Detection Based on Maximum Ratio Combining
## 2 System Model  
Consider a GSM system with \(N_{t}\) transmit antennas and \(N_{t}\) receive antennas. The block diagram of GSM is shown in Figure 1. The source binary sequence is transmitted from \(N_{p}\) antennas, \(N_{p}<N_{t}\), and all activated antennas transmit the same M-QAM or M-PSK symbols. When choosing \(N_{p}\) active transmit antennas out of \(N_{t}\) ones, there are \(C_{N_{t}}^{N_{p}}\) possible transmit antenna combinations (TACs). Among these TACs, only \(N_{\text{C}}=2\left\lfloor\log_{2}C_{N_{t}}^{N_{p}}\right\rfloor\) TACs are chosen to convey information. In each time slot, the information bits with length of \(x_{1}+x_{2}\) are divided into two parts: the TACs selection bits and the modulation symbol bits, where \(x_{1}=\left\lfloor\log_{2}C_{N_{t}}^{N_{p}}\right\rfloor\) information bits are used to map the TAC indices and \(\kappa_{2}=\log_{2}M\) information bits are mapped the constellation symbol \(\mathbf{s}=\left[s_{1},s_{2},\cdots,s_{N_{p}}\right]^{T}\), where \(s_{1},s_{2},\cdots,s_{N_{p}}\in S\) denotes the modulation symbol. \(M\) and \(S\) represent the modulation order and the modulation symbol set, respectively. The transmitted signal vector \(\mathbf{x}\in\mathbb{C}^{N_{t}\times 1}\) with \(N_{p}\) non-zero elements can be expressed as \(\mathbf{x}=\left[\cdots,0,s_{1},0,\cdots,0,\cdots s_{2},0,\cdots,0,s_{N_{p}},0, \cdots\right]^{T}\). We only study the GSM system that transmits same symbol \(s\) in each active transmit antenna, that is, \(s_{1}=s_{2}=\cdots=s_{N_{p}}=s\). It is assumed that the \(i\)-th TACs is used to transmit modulation symbols under the condition that the channel is Rayleigh fading channel and the channel gain is constant within a symbol period. Then the received signal vector \(\mathbf{y}\in\mathbb{C}^{N_{t}\times 1}\) can be expressed as  
\[\mathbf{y}=\mathbf{H}\mathbf{x}+\mathbf{n}=\mathbf{H}_{\mathbf{i}}\mathbf{s}+\mathbf{n}, \tag{1}\]  
where \(\mathbf{s}=\left[s_{1},s_{2},\cdots,s_{N_{p}}\right]^{T}\in\mathbb{C}^{N_{p}\times 1}\), \(\mathbf{n}\in\mathbb{C}^{N_{t}\times 1}\) is an additive white Gaussian noise with mean 0 and variance \(\sigma^{2}\). \(\mathbf{H}\in\mathbb{C}^{N_{t}\times N_{t}}\) is the channel gain matrix, and each element of \(\mathbf{H}\) obeys complex Gaussian distribution with mean 0 and variance 1. \(\mathbf{H}_{I}=\left(\mathbf{h}_{i_{1}},\mathbf{h}_{i_{2}},\cdots\mathbf{h}_{i_{N_{p}}}\right)\) is the submatrix with \(N_{p}\) columns of \(\mathbf{H}\) corresponding to the TAC set \(I\). Since all activated antennas transmit the same symbol \(s\), the system model can be rewritten as  
\[\mathbf{y}==\sum_{k=1}^{N_{p}}\mathbf{h}_{i_{k}}s+\mathbf{n}=\mathbf{\bar{h}}_{i}s+\mathbf{n}, \tag{2}\]  
where \(i_{k}\) stands for the \(k\)-th element in TAC set \(I_{i}\), \(\mathbf{h}_{i_{k}}\) denotes the \(i_{k}\)-th column of the channel gain matrix \(\mathbf{H}\), \(\mathbf{\bar{h}}_{i}\) represents the sum of the column vectors of \(\mathbf{H}\) corresponding to the \(i\)-th TAC. For GSM system, ML detection algorithm given in [18] can be formulated as  
\[\left(\hat{I},\hat{s}\right)=\underset{I\in\mathcal{I},s\in\mathcal{S}}{\arg \min}\|\mathbf{y}-\mathbf{H}_{I}\mathbf{s}\|^{2}, \tag{3}\]  
where \(\mathcal{I}=\left\{I_{1},I_{2},\cdots,I_{N_{C}}\right\}\), \(I_{i}\) with \(i\in\left\{1,2,\cdots,N_{C}\right\}\) is defined as the set of \(N_{p}\) active transmit antennas in the \(i\)-th TAC, and \(\mathcal{S}=S^{N_{p}\times 1}\) represents the set of \(N_{p}\)-dimensional modulation symbol vectors. Figure 1: The block diagram of GSM system with \(N_{t}\) transmit antennas and \(N_{r}\) receive antennas. The ML detection can also be rewritten as  
\[\left(\hat{I},\hat{s}\right)=\underset{i\in I,s\in S}{\arg\min}\left\|\mathbf{y}- \bar{\mathbf{h}}_{i}\mathbf{s}\right\|^{2}. \tag{4}\]  
If only real-valued multiplications and divisions are considered, the computational complexity of ML detection algorithm is \(6N_{r}MN_{\text{C}}\) in the GSM system that transmits the same symbols.