# Waveform Design for Space-Time Coded MIMO Systems with High Secrecy Protection +
## 3 Secrecy-Enhancing LD-QO-STBC Scheme  
### Waveform Analysis for LD-QO-STBC  
In order to design an efficient waveform for secrecy enhancement, we first investigate the signal constellation of the LD-QO-STBC scheme. By letting combinations of modulation symbols be used in (6) as follows:  
\[\begin{array}{ll}x_{1}=s_{1}+s_{3},&x_{2}=s_{2}+s_{4},\\ x_{3}=s_{3}-s_{1},&x_{4}=s_{4}-s_{2},\end{array} \tag{11}\]  
the encoding matrix \(\mathbf{X}\) in (6) can be represented as:  
\[\mathbf{X}=\left[\begin{array}{cccc}x_{1}&x_{2}&x_{3}&x_{4}\\ -x_{2}^{*}&x_{1}^{*}&-x_{4}^{*}&x_{3}^{*}\\ x_{3}&x_{4}&x_{1}&x_{2}\\ -x_{4}^{*}&x_{3}^{*}&-x_{2}^{*}&x_{1}^{*}\end{array}\right]. \tag{12}\]  
Because each element of \(\mathbf{X}\) is a linear combination of any possible values of \(s_{i}\) and \(s_{j}\), the waveforms of \(s_{i}\) and \(x_{i}\) will be different. In order to investigate the signal distributions, we denote constellation symbol sets used for \(s_{i}\) and \(x_{i}\) as \(a\in O_{a}\) and \(b\in O_{b}\), respectively. We note that for any Gray-coded QAM constellation symbol mapping, \(s_{i}\), \(-s_{i}\), \(s_{i}^{*}\), and \(-s_{i}^{*}\) are all in the same set \(O_{a}\), due to its symmetric constellation. Likewise, elements of \(\mathbf{X}\) that are either \(x_{i}\), \(-x_{i}\), \(x_{i}^{*}\), or \(-x_{i}^{*}\), are in the same set \(O_{b}\). Nevertheless, it is clear that \(O_{b}\) forms a different set from \(O_{a}\).  
Figure 2 demonstrates an example of waveform transformation from \(O_{a}\) to \(O_{b}\) when 4-QAM is used for \(a\). In the figure, we denote \(a_{i}\) and \(b_{i}\) as one of the complex constellation symbols included in \(O_{a}\) and \(O_{b}\), respectively. In addition, we derive the probability density function (PDF) of \(b\), and show that it is not a uniform distribution. For any \(M\)-ary QAM, we can assume that without loss of generality, the PDF of \(a\) follows a uniform distribution. That is,  
\[f_{a}(a)=1/M,\;\;a\in O_{a}. \tag{13}\]  
By assuming the independency of the real and imaginary parts of \(a\),  
\[f_{a^{R}}(a^{R})=1/\sqrt{M},\;\;f_{a^{I}}(a^{I})=1/\sqrt{M},\;\;a\in O_{a}, \tag{14}\]  
where \(a^{R}\) and \(a^{I}\) are the real and imaginary parts of a complex value \(a\), respectively.  
For any symbol \(b\), we note that it is the sum of two independent random variables, so its PDF can be found by first taking the convolution of \(f_{b^{R}}(b^{R})\) and \(f_{b^{I}}(b^{I})\) in their real and imaginary parts, respectively. Therefore,  
\[\begin{array}{ll}f_{b^{R}}(b^{R})&=f_{a^{R}}(a^{R})*f_{a^{R}}(a^{R}),\\ f_{b^{I}}(b^{I})&=f_{a^{I}}(a^{I})*f_{a^{I}}(a^{I}),\;\;a\in O_{a},\end{array} \tag{15}\]  
and finally using the independency of real and imaginary parts,  
\[f_{b}(b)=f_{b^{R}}(b^{R})f_{b^{I}}(b^{I}),\;\;b\in O_{b}. \tag{16}\]  
Figure 2 compares the PDFs, \(f_{a}(a)\) and \(f_{b}(b)\) when 4-QAM is used for \(O_{a}\). We find that the number of elements in \(O_{b}\) is greater than that in \(O_{a}\), and that there is a null value \(0+j0\) in \(O_{b}\), which is denoted by \(b_{1}\) in the example of 4-QAM of Figure 2. Whatever \(M\)-ary QAM is used for \(O_{a}\), this null value exists. We also find that the probability of having the null value of \(b\) is the maximum among the other possible values, i.e., \(f_{b}(b_{1})\) is the maximum amongst, and Figure 2 also demonstrates this. These findings will  
Figure 2: Signal constellations and PDFs for \(a\in O_{a}\) and \(b\in O_{b}\) used for \(s_{i}\) and \(x_{i}\), respectively.  
be applicable to higher-order \(M\)-ary QAM. The resultant \(O_{b}\) from the LD-QO-STBC scheme will incur zero-crossing amongst non-zero modulation symbol values.