# Parallel Complex Quadrature Spatial Modulation
## 3 Parallel Complex Quadrature Spatial Modulation  
In the PCQSM, the set of transmit antennas is divided into \(P\) equal-size subgroups and the system performs CQSM in each group, using the same two signal symbols, \(s_{a}\) and \(s_{b}\). Each subgroup has \(n_{t}=N_{t}/P\) transmit antennas that are indexed independently. As such, the spectral efficiency of PCQSM is \(\log_{2}(n_{T}^{2P}M^{2})\) bits/s/Hz. A block representation of the PCQSM system model is given in Figure 1. At the receiver side, the received signal vector is expressed as follows,  
\[\mathbf{y}=\frac{1}{\sqrt{P}}\left(\sum_{i=1}^{P}\mathbf{h}_{i}s_{a}+\sum_{i= 1}^{P}\mathbf{h}_{k_{i}}s_{b}\right)+\mathbf{n}, \tag{6}\]  
where \(\mathbf{h}_{i}\in(\mathbf{h}_{i_{1}},\mathbf{h}_{i_{2}},\cdots,\mathbf{h}_{l _{P}})\) and \(\mathbf{h}_{k_{i}}\in(\mathbf{h}_{k_{i}},\mathbf{h}_{k_{2}},\cdots,\mathbf{h} _{k_{P}})\) represent the channel vectors that transmit \(s_{a}\) and \(s_{b}\), respectively, in the \(i\)-th group. The antenna indices for the \(i\)-th group range from \((i-1)n_{T}+1\) to \(in_{T}\), such that \(1\leq i\leq P\). Similar to the CQSM, \(s_{a}\) and \(s_{b}\) belong to distinct constellations of size \(M\). The first constellation is a conventional QAM/PSK constellation and the second is a rotated version of the first. The results of the rotation angle optimization for the PCQSM are shown in Figure 2 for several system configurations with \(P=2\) and \(P=4\). The angle that minimizes the the BER is considered as the optimal angle and is used in subsequent simulations for the given system configuration. Employing the maximum-likelihood (ML) detection at the receiver side, the estimated transmitted symbols and antenna indices are given by  
\[\left[\hat{\mathbf{l}},\hat{\mathbf{k}},\hat{s}_{a},\hat{s}_{b}\right]= \arg\min_{\mathbf{l},\mathbf{k},s_{a},s_{b}}\|\mathbf{y}-\sum_{i=1}^{P} \mathbf{h}_{i}s_{a}-\sum_{i=1}^{P}\mathbf{h}_{k_{i}}s_{b}\|^{2} \tag{7}\] \[= \arg\min_{\mathbf{l},\mathbf{k},s_{a},s_{b}}\|\mathbf{y}-\mathbf{ g}\|^{2}\] \[= \arg\min_{\mathbf{l},\mathbf{k},s_{a},s_{b}}\|\mathbf{g}\|^{2}-2 \Re\{\mathbf{y}^{H}\mathbf{g}\},\]  
where  
\[\mathbf{g}\triangleq\sum_{i=1}^{P}\mathbf{h}_{i}s_{a}+\sum_{i=1}^{P}\mathbf{h} _{k_{i}}s_{b}. \tag{8}\]  
Figure 1: A block representation of parallel complex quadrature spatial modulation (PCQSM) with \(P\) antenna groups.  
The sets \(\mathbf{I}=\{l_{1},l_{2},\cdots,l_{P}\}\) and \(\mathbf{k}=\{k_{1},k_{2},\cdots,k_{P}\}\) are the spatial indices from which \(s_{a}\) and \(s_{b}\) are transmitted in each group, respectively.  
Assume a PCQSM system that achieves 12 bits/s/Hz with \(N_{t}=8\), \(N_{r}=4\), \(M=4\), and \(P=2\). Accordingly, the first and second group of the antenna set are \(\{1,2,3,4\}\) and \(\{5,6,7,8\}\), respectively. Let the information stream at one channel use be [0 1 1 0 1 0 1 1 0 0 1 0], which is split into  
\[m_{a} =[0\ 1],\qquad m_{b}=[1\ 0],\] \[p_{1} =[1\ 0\ 1\ 1],\quad p_{2}=[0\ 0\ 1\ 1].\]  
The first part \(m_{a}\) modulates \(s_{a_{2}}=-1+j\) from the QPSK constellation. The second part \(m_{b}\) modulates \(s_{b_{3}}\), where \(s_{b_{2}}=s_{a_{2}}e^{\mathrm{i}\theta_{opt}}\) and \(s_{a_{3}}=1-j\). Using the same optimization method of [22], the optimal angle, obtained through extensive simulations, is \(\theta_{opt}=43^{\circ}\) for the \(8\times 4\) PCQSM system. Therefore, \(s_{b_{3}}=-0.2767-j1.3869\). Based on the \(p_{1}\) part, the transmit antennas of index 3 and 4 in the first group are activated to transmit \(s_{a_{2}}\) and \(s_{b_{3}}\), respectively. Similarly, the transmit antennas of index 5 and 8 in the second group are used to transmit the signal symbols. The signal symbols are normalized before transmission to achieve a unity transmission power. The received vector is then given by  
\[\mathbf{y}=\frac{1}{\sqrt{2}}\big{(}s_{a_{2}}(\mathbf{h}_{3}+\mathbf{h}_{5})+ s_{b_{3}}(\mathbf{h}_{4}+\mathbf{h}_{8})\big{)}+\mathbf{n}. \tag{9}\]