## 2 Principle of SM Using a Novel Active Transmitter Detection Scheme with SSD
### The Novel Active Transmitter Detection Scheme
#### 2.3.2 Power-Based Statistical Method for Active Transmitter Detection  
The power-based statistical method for active transmitter detection is expressed in Equations (3) and (4):  
\[Z(t,j)=\left\langle\mathbf{a},\ \mathrm{sort}\Big{(}\big{|}\boldsymbol{x}_{t}^{j} \big{|}\Big{)}\right\rangle+b\times\Big{(}\mathrm{max}\Big{(}\boldsymbol{x}_{t }^{j}\Big{)}-\mathrm{min}\Big{(}\boldsymbol{x}_{t}^{j}\Big{)}\Big{)}+c\times \sigma\Big{(}\boldsymbol{x}_{t}^{j}\Big{)} \tag{3}\]  
where \(Z(t,j)\) is the power-based statistical function of the \(j\)-th symbol at the two-symbol-based time slot \(t\), where \(j\)\(\in\{1,2\}\) since two symbols form a signal group to provide SSD. Besides, \(\boldsymbol{x}_{t}^{j}\) is the received signal sample vector of the \(j\)-th symbol at the two-symbol-based time slot \(t\) with each vector length = oversampling factor, \(\mathrm{sort}\Big{(}\big{|}\boldsymbol{x}_{t}^{j}\big{|}\Big{)}\) is the absolute value of \(\boldsymbol{x}_{t}^{j}\) arranged in descending order, \(\mathbf{a}\) is the weight coefficient vector obtained via training with the reference of ideal active transmitter detection, \(\left\langle\cdot\right\rangle\) is the inner product, b and c are the weight coefficients also obtained via training, and \(\sigma\Big{(}\boldsymbol{x}_{t}^{j}\Big{)}\) is the standard deviation of \(\boldsymbol{x}_{t}^{j}\).  
When \(Z(t,1)\) and \(Z(t,2)\) are obtained, the spatial bits \(I(t)\) in a signal group can be recovered as:  
\[I(t)=\begin{cases}(0,1)\\ (1,0)\end{cases} \tag{4}\]  
Figure 2: Signal pre-distortion technique. (\(\mathbf{a}\)) Signal flow for pre-distortion. (\(\mathbf{b}\)) Example of the generation of a pre-distorted CAP waveform.  
The proposed power-based statistical method is based on the observation of received CAP signal characteristics and the channel gain conditions. In Equation (3), received signal power distribution is expressed statistically in three terms. The first inner product term gives a weighted symbol level since the contribution of each sample differs to represent a symbol. The second term indicates the range of sample levels in each symbol. The third term describes the amount of variation in the samples in each symbol. This received signal power distribution is then used to recover the spatial bits in Equation (4) with reference to the measured total channel gains. An example is given to illustrated how it works: when the values of \(Z(t,1)\) and \(Z(t,2)\) coincide with the reference total channel gains, i.e., \(Z(t,1)>Z(t,2)\) and \(\text{h}_{1}>\text{h}_{2}\), it indicates the first spatial bit is from \(\text{Tx}_{1}\) and the second is from \(\text{Tx}_{2}\). Thus, the pair of spatial bits (0,1) is recovered. Otherwise, when the values of \(Z(t,1)\) and \(Z(t,2)\) is opposite to the reference total channel gains, the pair of spatial bits (1,0) is recovered.  
This power-based method with three statistical terms enables a comprehensive description of the signal power distribution and thus enhances the robustness of power-based active transmitter detection under noisy channel conditions.