# Wireless Covert Communication with Polarization Dirty Constellation
## 3 Basis of Our Scheme
### Dirty Constellation Polarization  
We always denote a binary discrete memoryless channel (B-DMC) \(W\) as \(W:X\to Y\) with input alphabet \(X\) and output alphabet \(Y\). The input alphabet \(X\) belongs to set\(\{0,1\}\), the output alphabet \(Y\) may be arbitrary. To measure the channel condition, [PERSON] [21] mainly used two parameters: the symmetric capacity  
\[I(W)\triangleq\sum_{y\in Y}\sum_{x\in X}\frac{1}{2}W(y|x)\,\log\frac{W(y|x)}{ \frac{1}{2}W(y|0)+\frac{1}{2}W(y|1)}, \tag{9}\]  
and the Bhattacharyya parameter as  
\[Z(W)\triangleq\sum_{y\in Y}\sqrt{W(y|0)W(y|1)}. \tag{10}\]  
When the input alphabet is just 1 or 0, \(Z(W)\) is an upper bound of the error probability of the maximum-likelihood (ML) decision. It is easy to see that \(Z(W)\) and \(I(W)\) both take values in [0,1]. Intuitively, we will expect \(I(W)\approx 1\) iff \(Z(W)\approx 0\) or \(I(W)\approx 0\) iff \(Z(W)\approx 1\). In [25], [PERSON] proposed that the ensemble average probability of decoding error based on ML decoding satisfies  
\[\bar{P}_{e}\leq\exp\{-N[E_{0}(\rho,Q)-\rho R]\}, \tag{11}\]  
where  
\[E_{0}(\rho,Q)=-\ln\sum_{j=0}^{J-1}\left[\sum_{k=0}^{K-1}Q(k)P(j|k)^{1/(1+\rho)} \right]^{1+\rho}. \tag{12}\]  
We denote the transition probability of discrete memoryless channel as \(P(j|k)\), positive integer as \(N\), and positive number as \(R\). In the ensemble of \((N,R)\) block codes, every code is independently chosen with the probability assignment \(Q(k)\). Assuming \(\rho=1\), we can transform Equation (12) as  
\[E_{0}(1,Q)=-\ln\sum_{j=0}^{J-1}\left[\sum_{k=0}^{K-1}Q(k)P(j|k)^{1/2}\right]^{ 2}. \tag{13}\]  
For binary symmetric channel, \(E_{0}(1,Q)\) is maximized over \(Q\) by \(Q(0)=Q(1)=1/2\). For this \(Q\), we have  
\[ \begin{split}& E_{0}(1,Q)=\ln 2-2\ln\Bigl{[}p^{1/2}+\left(1-p \right)^{1/2}\Bigr{]}\\ &=\ln\frac{2}{\Bigl{(}p^{1/2}+\left(1-p\right)^{1/2}\Bigr{)}^{2} }\\ &=\ln\frac{2}{1+Z(W)}.\end{split}  \tag{14}\]  
The Equation (11) can be converted to  
\[NR\geq\ln(\bar{P}_{e})+N\cdot\ln\frac{2}{1+Z(W)}\sim\log(\bar{P}_{e})+N\cdot \log\frac{2}{1+Z(W)}. \tag{15}\]  
\(R\) in nats (that is, using \(\ln\)) is the number of binary digits entering the encoder per transmitted per second. Thus, we use \(R\) in bits (that is, using \(\log\)2) to interpret Equation (15),which is the number of binary digits entering the encoder per transmitted channel digit. It is known that \(I(W)=C\geq R\), so we have  
\[I(W)\geq\log\frac{2}{1+Z(W)}. \tag{16}\]  
For any B-DMC, we have  
\[\left\{ \begin{aligned} I(W)&\geq\log\frac{2}{1+Z(W)}\\ I(W)&\leq\sqrt{1-Z(W)^{2}}.\end{aligned} \right. \tag{17}\]  
When \(W\) is a symmetric channel, the symmetric capacity of \(W\) is equal to the Shannon capacity. Referring to Equation (17), it can be obtained that \(I(W)\) approaching 1 when \(Z(W)\approx 0\). It shows that there exists a sequence of \((2^{NR},N)\) codes with the maximum probability of error \(P_{e}\to 0\) when we encode the covert information in B-DMC. The polarization code can realize \(P_{e}\to 0\) by polarizing the channel. Channel polarization is an operation consisting of channel combining and channel splitting. With the length of block \(N\) trending to infinity, the symmetric capacity \(I(W)\) tends to be two-polar. Part of the symmetric capacity tends to be 1, the rest tends to be 0. If we polarize the covert channel in B-DMC, as the length of covert information blocks \(N\) becomes large, part of the covert information can be transmitted with the Shannon capacity. We can improve the undetectability by reducing the power of the covert signal without deteriorating the reliability of covert communication. If the communication channel is AWGN (Additive White Gaussian Noise) channel, the transmit process of covert information can be expressed as \(X_{1}^{N}\to S_{1}^{N}\to Y_{1}^{N}\). The signal \(Y_{1}^{N}\) received by [PERSON] can be expressed as \(y_{i}=s_{i}+n_{i}\), \(1\leq i\leq N\). We denote the encoded covert information as \(S_{1}^{N}\).