# Covert Anti-Jamming Communication Based on Gaussian Coded Modulation
## 4 Undetectability Analysis
### Undetectability under an Unknown Noise Level  
In the first case, [PERSON] is assumed to have no information on the noise level (Case 1). In the following section, we provide a simple analysis and example showing that [PERSON] cannot distinguish the proposed signal, although she has a chance of detecting conventional signals. Let us recall the channel model depicted in Figure 6 to determine the eavesdropping problem. Using the same vectorizing process as the legitimate channel model as presented in Equations (3) and (4), [PERSON] obtains  
\[\mathbf{z}=\mathbf{H^{{}^{\prime}}x}+\mathbf{w^{{}^{\prime}}}, \tag{24}\]  
where \(\mathbf{H^{{}^{\prime}}}\) is a combined wiretap channel matrix and \(\mathbf{w^{{}^{\prime}}}\) is a CSCG noise vector, which are derived using the same method as those of a legitimate channel, as discussed in Section 3.2. Consider that [PERSON] monitors the communication channel and obtains a measurement vector assuming an exact carrier frequency, phase, and symbol duration. Moreover, consider a perfect channel state information and flat fading at [PERSON], which is a threat scenario for a WCS. Then, the receiving model is simplified to  
\[\mathbf{z}=h^{\prime}\mathbf{x}+\mathbf{w^{{}^{\prime}}}, \tag{25}\]  
where \(h^{\prime}\) is the channel coefficient satisfying \(\mathbf{H^{{}^{\prime}}}=h^{\prime}\mathbf{I}\). In this scenario, [PERSON] has to determine whether there is a signal using the hypothesis test:  
\[\left\{ \begin{array}{l}H_{0}:\mathbf{d}=\mathbf{n}\\ H_{1}:\mathbf{d}=\mathbf{c}+\mathbf{n},\end{array} \right. \tag{26}\]  
where \(\mathbf{c}=\left[ \begin{array}{cc}real(\mathbf{x})^{T}&imag(\mathbf{x})^{T }\end{array} \right]^{T}\) is the real-valued codeword of which the entries are zero-mean Gaussian random variables with variance \(\sigma_{c}^{2}\), \(\mathbf{n}=\left[ \begin{array}{cc}real(\mathbf{w^{{}^{\prime}}}/h^{\prime}) ^{T}&imag(\mathbf{w^{{}^{\prime}}}/h^{\prime})^{T}\end{array} \right]^{T}\) is the AWGN whose entries are zero-mean Gaussian random variables with variance \(\sigma_{n}^{2}\), and \(\mathbf{d}=\left[ \begin{array}{cc}real(\mathbf{z}/h^{\prime})^{T}&imag( \mathbf{z}/h^{\prime})^{T}\end{array} \right]^{T}\) is the channel-compensated measurement vector. In Case 1, [PERSON] has to determine which hypothesis is true based on the PDF of \(\mathbf{d}\). Let the variance of \(\mathbf{d}\) be \(\sigma_{\mathbf{d}}^{2}\). To test the two hypotheses, the only thing [PERSON] can do is compare how close the sample distribution is to the distribution of Gaussian noise. This test is well known as the normality test. First, assume that [PERSON] demonstrates perfection while calculating the sample distribution. Perfect here implies that [PERSON] can collect an infinite number of samples each with infinite resolution; thus, the histogram distribution of the perfect [PERSON] is identical to the distribution according to the true hypothesis. **Lemma 1**. : _Assume that [PERSON] is perfect while calculating the sample distribution. Let [PERSON] not know the noise level. Then, she cannot detect the existence of the proposed signal, but she can detect finite-field modulated signals._  
Proof of Lemma 1.: [PERSON] does not know the noise level. The addition of the Gaussian signal sample to the Gaussian noise sample produces a Gaussian sample. Thus, what [PERSON] observes are Gaussian samples; [PERSON] can only treat it as Gaussian noise. When [PERSON] observes finite-level signal samples, a perfect [PERSON] will notice that the observed signal deviates from the Gaussian samples. Thus, [PERSON] can detect the presence of an ongoing communication signal. \(\square\)  
It is worth noting that Lemma 1 is satisfied in the ideal case. Namely, it holds when the degree of freedom of the codebook is infinite and [PERSON] can correct an infinite number of samples. However, both conditions cannot be perfect in practice. The degree of freedom of a fixed codebook is limited by the size of the codebook.