# Wireless Covert Communication with Polarization Dirty Constellation
## 3 Basis of Our Scheme  
### Binary Hypothesis Testing at [PERSON]  
According to the system model shown in Figure 1, the performance elements of the wireless covert channels mainly include two aspects: undetectability and reliability. In communication, [PERSON] totally transmits \(n\) symbols to [PERSON]. We denote the finite block as \(m[i]\) (\(i\in[1,n]\)), which consist of normal information \(m_{c}[i]\) and secret information \(m_{s}[i]\). [PERSON]'s task is to collect the messages \(m[i]\) transmitted by [PERSON] and detect whether covert communication exists or not. Let the AWGN between [PERSON] and [PERSON] as \(r[i]\sim(0,\sigma_{w}^{2})\). The signal received by [PERSON] can be expressed as  
\[y_{w}[i]=m[i]+r[i] \tag{1}\]  
Figure 2: The schematic diagram of wireless covert communication with dirty constellation.  
[PERSON]'s task is to detect whether covert communication exists. We define two hypotheses, \(H_{0}\) and \(H_{1}\), to distinguish these two cases:  
\[\begin{cases}H_{0}:m[i]=m_{c}[i]\\ H_{1}:m[i]=m_{c}[i]+m_{s}[i]\end{cases} \tag{2}\]  
Let \(H_{0}\) denotes that covert communication exists, \(H_{1}\) denotes covert communication does not exist. In covert communication, the ultimate goal of [PERSON] is to minimize the total error rate (i.e., \(\xi\)). Let \(T\) and \(F\) denote binary decisions that infer whether covert communication exists or not. The false positive rate and miss detection rate can be expressed as  
\[\begin{cases}P_{F}=P_{r}(T|H_{0})\\ P_{M}=P_{r}(F|H_{1})\end{cases} \tag{3}\]  
Assuming [PERSON] performs the optimal detection. Following [PERSON]'s inequality [22; 23], we can obtain  
\[P_{F}+P_{M}\geq 1-\sqrt{\frac{1}{2}D(P_{0}||P_{1})} \tag{4}\]  
where relative entropy \(D(P_{0}||P_{1})\) (also called K-L divergence) is defined as follows:  
\[D(P_{0}||P_{1})=\int_{n}p_{0}(x)\ln\frac{p_{0}(x)}{p_{1}(x)}dx \tag{5}\]  
where \(n\) is the value range of \(x\). Let \(p_{0}(x)\sim(0,\sigma_{u})\sim(0,P+\sigma_{\omega}^{2})\) denotes the distribution of sequence \(P_{0}\) detected by [PERSON], and \(p_{1}(x)\sim(0,\sigma_{\omega})\) denotes the distribution of AWGN \(P_{1}\) collected by [PERSON]. Let \(P\) denote the transmit power. We can equivalently express Equation (5) as  
\[\begin{split}& D(P_{0}||P_{1})=\int_{n}\frac{1}{\sqrt{2\pi} \sigma_{\omega}}e^{-\frac{x^{2}}{\sigma_{\omega}^{2}}}\cdot\ln(\frac{\sigma_{ s}}{\sigma_{\omega}}\cdot e^{\frac{x^{2}}{\sigma_{\omega}^{2}}-\frac{x^{2}}{ \sigma_{\omega}^{2}}})dx\\ &=\int_{n}\frac{1}{\sqrt{2\pi}\sigma_{\omega}}e^{-\frac{x^{2}}{ \sigma_{\omega}^{2}}}\cdot(\ln(\frac{\sqrt{P+\sigma_{\omega}^{2}}}{\sigma_{ \omega}})+\frac{x^{2}\cdot P}{(P+\sigma_{\omega}^{2})\cdot\sigma_{\omega}^{2} })dx\\ &=\frac{1}{2}(\ln\frac{P+\sigma_{\omega}^{2}}{\sigma_{\omega}^{2} }-\frac{P}{P+\sigma_{\omega}^{2}})\end{split} \tag{6}\]  
It can be obtained that the K-L divergence deteriorates with the increase of transmit power \(P\). We can ameliorate the value of K-L divergence by constraining the transmit power. Except for KL-divergence, we can also calculate the distance between distributions with the KS-Test (also called Kolmogorov-Smirnov statistic). The KS-distance [24] is defined as follows:  
\[D_{KS}=\max\lvert F_{1}(x)-F_{0}(x)\rvert \tag{7}\]  
where \(F_{1}(x)=\frac{\sum_{j=1}^{x}h_{1}(j)}{n}\), \(F_{0}(x)=\frac{\sum_{j=1}^{x}h_{0}(j)}{n}\). Let \(F_{1}(x)\) and \(F_{0}(x)\) denote the cumulative distribution functions of \(P_{1}\) and \(P_{0}\). \(P_{1}\) and \(P_{0}\)power of covert communication. However, the BLER (Block Error Rate) would deteriorate with low transmit power. We can ameliorate the reliability of covert communication by encoding the covert messages with polar code.