# Waveform Design for Space-Time Coded MIMO Systems with High Secrecy Protection +
## 3 Secrecy-Enhancing LD-QO-STBC Scheme
### The Design of Artificial Interference  
From the investigation in the previous section, we note that the probability of having the null value of \(x_{i}\), i.e., \(x_{i}=0+j0\) is the maximum amongst all possible values in \(O_{b}\). We also note from (6) that whenever we have zero value of \(x_{i}\), there will be \(2s_{i}\) value in the same row in \(\mathbf{X}\). For example, consider the case when \(x_{1}=0\), which means \(s_{1}=-s_{3}\), because \(x_{1}=s_{1}+s_{3}\) as defined in (11). This implies that \(x_{3}=2s_{3}\), because \(x_{3}=s_{3}-s_{1}\), as in (11). We refer to these two elements that have the null value and \(2s_{i}\) as a pair. Investigation of (6) reveals that there are two pairs in each row; that is, the first and third elements form one pair, and the second and fourth elements form the other pair. We design the AI in order to satisfy the following condition. First, by adding AI, we escape the zero value transmission. For this, whenever we have a zero value of \(x_{i}\), we add AI that will change the transmitted symbol to a non-zero value. At the same time, we subtract the added value from the paired elements, which reduces the original amplitude. Second, the AI is added in such a way that the paired elements contribute to canceling each other's AI at the receiving end. For this, we multiply the channel gains in such a way that the added AI is canceled. To achieve the above design goal, we represent the received signal at Bob during the \(i\)-th time period, \(y_{\text{bi}}\), as follows:  
\[y_{\text{bi}}=\sum_{j=1}^{4}\left(\chi_{ij}+\omega_{ij}\right)h_{j}+n_{\text{ bi}}, \tag{17}\]  
where, \(\chi_{ij}\) is the element of \(\mathbf{X}\) in the \(i\)-th row and \(j\)-th column, \(\omega_{ij}\) is the AI term, and \(n_{\text{bi}}\) is the noise at [PERSON] during the \(i\)-th time period. Because \(\chi_{ij}\) and \(\chi_{i(j+2)},j=1,2\), always form a pair, we represent (17) as the following pairwise form:  
\[y_{\text{bi}}=\sum_{j=1}^{2}\left\{\left(\chi_{ij}+\omega_{ij}\right)h_{j}+ \left(\chi_{i(j+2)}+\omega_{i(j+2)}\right)h_{j+2}\right\}+n_{\text{bi}}. \tag{18}\]  
In order to satisfy the first condition, when \(\chi_{ij}=0\), we set \(\omega_{ij}\) to a non-zero value, so that it can increase the power of \(\chi_{ij}\). At the same time, we decrease the same amount of power from the paired element. In other words, we set \(\omega_{ij}=\omega_{i(j+2)}=\chi_{i(j+2)}/2\), in the case \(\chi_{ij}=0\). This pairwise power reallocation will contribute to decreasing the difference of power allocations across the transmit antennae. Even though AI is added, it is a kind of reallocation of power, and thus there will be no additional average power allocation. In addition, thanks to the following solution to the second condition, this AI will be perfectly eliminated. The second condition requires that AI should be canceled at the legitimate receiver. For this, we represent (18) by separating the signal and AI terms, that is  
\[y_{\text{bi}}=\sum_{j=1}^{2}\left(\chi_{ij}h_{j}+\chi_{i(j+2)}h_{j+2}\right)+ \sum_{j=1}^{2}\left(\omega_{ij}h_{j}+\omega_{i(j+2)}h_{j+2}\right)+n_{\text{ bi}}. \tag{19}\]  
Because the AI term in (19) is \(\sum_{j=1}^{2}\left(\omega_{ij}h_{j}+\omega_{i(j+2)}h_{j+2}\right)\), we need to make it zero, i.e.,  
\[\sum_{j=1}^{2}\left(\omega_{ij}h_{j}+\omega_{i(j+2)}h_{j+2}\right)=0. \tag{20}\]  
A simple solution to (20) can be letting \(\omega_{ij}\) and \(\omega_{i(j+2)}\) contain \(-h_{(j+2)}\) and \(h_{j}\) terms, respectively. By combining the above two conditions at the same time, \(\omega_{ij}\) can be defined as follows:  
\[ \begin{array}{ll}\omega_{ij}=\frac{\chi_{ij(j+2)}h_{j+2}}{2},&\omega_{i(j+2)}= -\frac{\chi_{ij(j+2)}h_{j}}{2},&\text{if $j=1,2$, and $\chi_{ij}=0$},\\ \omega_{ij}=\frac{\chi_{i(j-2)}h_{j-2}}{2},&\omega_{i(j-2)}=-\frac{\chi_{ii(j-2) }h_{j}}{2},&\text{if $j=3,4$, and $\chi_{ij}=0$},\\ \omega_{ij}=0,&\chi_{ij}\
eq 0,\ \chi_{i(j\pm 2)}\
eq 0.\end{array}