## 2 Convert Information Mapping Aided GSDM
### The GSDM System Model  
In order to take both transmission rate and security performance into consideration, one of functions of GSM, receiver subset selection (RRS), is introduced to SDM where multiple eavesdroppers exist. Based on RSS, in such a case that Bob is equipped with \(N_{r}\) receivers and only \(N_{u}\) of them are activated, the number of total single-antenna receivers combinations is \(C_{N}^{N_{u}}\). For the sake of convenience for information mapping, the number of receivers combinations chosen to use should be the power of two. Hence, there are \(f_{1}=2^{\left\lfloor\log_{2}C_{N_{r}}^{N_{u}}\right\rfloor}\) combinations allowed for modulation and \(k_{1}=\log_{2}f_{1}\) bits are delivered by RRS. Apart from information carried by indices of receiver subsets, \(N_{u}\) receivers are activated and each of them is able to deliver single \(M\)-ary amplitude-phase modulation (APM) symbol. Therefore, other \(k_{2}=N_{u}\times\log_{2}M\) bits are transmitted by conventional APM symbols though \(N_{u}\) activated receivers. As a result, the GSDM system is capable to convey \(k=k_{1}+k_{2}\) bits a time so the transmission rate is improved efficiently. For simplicity, assume that Alice is equipped with uniform linear array (ULA), where the phase center is the geometric center of array. Then the channel vector between Alice and the receiver located in direction of angle \(\theta\) is expressed as  
\[\mathbf{h}^{H}(\theta)=\left[e^{-j\left(\frac{N_{r}-1}{2}\right)\frac{2\pi}{ \lambda}d\cos\theta},e^{-j\left(\frac{N_{r}-1}{2}-1\right)\frac{2\pi}{\lambda} d\cos\theta},\ldots,e^{j\left(\frac{N_{r}-1}{2}\right)\frac{2\pi}{\lambda}d\cos \theta}\right], \tag{1}\]  
where \(\lambda\) denotes the signal wavelength and \(d\leq\lambda/2\) is the antennas spacing at Alice. Thus, the channel matrix between Alice and Bob can be expressed as  
\[\mathbf{H}(\Theta_{B})=\left[\mathbf{h}(\theta_{1}),\mathbf{h}(\theta_{2}), \ldots,\mathbf{h}(\theta_{N_{r}})\right]^{H}, \tag{2}\]  
where \(\mathbf{H}(\Theta_{B})\in\mathbb{C}^{N_{r}\times N_{r}}\), \(\Theta_{B}=\left\{\theta_{1},\theta_{2},\ldots,\theta_{N_{r}}\right\}\) and \(\mathbf{h}(\theta_{i}),i=1,2,\ldots,N_{r}\) represent the set of directional angles at Bob and the channel vector between Alice and \(i\)-th receiver at Bob, respectively. For the purpose of transmission security, the beamforming vector should provide the maximum power to Bob. Then the corresponding beamforming vector \(w_{i}\) to \(i\)-th receiver at Bob is given by  
\[\mathbf{w}_{i}=\mathbf{h}(\theta_{i})/N_{t}. \tag{3}\]  
Figure 1: The system of generalized spatial and direction modulation. When taking all receivers at Bob into consideration, the beamforming matrix \(\mathbf{W}\in\mathbb{C}^{N_{t}\times N_{t}}\) of the channel can be represented by  
\[\mathbf{W}=[\mathbf{w}_{1},\mathbf{w}_{2},\ldots,\mathbf{w}_{N_{t}}]. \tag{4}\]  
As aforementioned, \(k=k_{1}+k_{2}\) bits can be modulated at a time. More specifically, symbol vector \(\mathbf{x}_{l,s}\) produced by [PERSON] can be formulated as  
\[\mathbf{x}_{l,s}=\mathbf{I}_{l}\mathbf{b}_{s}, \tag{5}\]  
where \(\mathbf{I}_{l}(l=1,2,\ldots,2^{k_{1}})\) denotes matrix which consists of \(N_{u}\) column vector chosen from identity matrix \(\mathbf{I}_{N_{t}\times N_{t}}\), as \(N_{u}\) activated receivers. In addition, \(\mathbf{b}_{s}(s=1,2,\ldots,2^{k_{2}})\) is a column vector built up by \(N_{u}\) symbols selected from conventional APM symbols. It is noteworthy that the symbol vector \(\mathbf{x}_{l,s}\) involves information of activated receiver subset in conjunction with APM symbols which both are able to influence the performance. For the sake of security enhancement, an interference matrix is proposed here. The interference matrix is known to Bob but unknown to Eve.