# Covert Anti-Jamming Communication Based on Gaussian Coded Modulation
## 4 Undetectability Analysis
### Undetectability under an Exact Noise Level  
Now, let us focus on Case 2. In this case, [PERSON] is assumed to have perfect information on the noise level. The simplest solution for this testing strategy is to use an energy detector of which the goal is to determine whether the signal is present by comparing the energy of the received symbol. In the energy detection, the test statistic of Equation (26) becomes the squared sum of symbols. As we assumed, [PERSON] does not know the exact time-frequency band of the target signals. Then, [PERSON] has two strategies for testing. The first is obtaining a single test statistic by integrating the whole energy over a suspected time-frequency band. It is obvious that the test statistic depends on both the SNR and the suspected region, regardless of the coding and modulation method of the target signal. Then the test result depends only on how accurate the suspected region is. Rather than this trivial case, we focus on the other strategy: dividing the suspected region with as fine a grid as [PERSON] can produce, and applying per-symbol energy detection to determine the signal existence according to each time-frequency slot. By this strategy, [PERSON] can evaluate the slot-by-slot possibility of signal presence. Based on this possibility, [PERSON] can jointly estimate the suspected region and probability of signal presence. The performance of this joint detection of possibility and region depends on the per-symbol test. From this aspect, we derive the true positive of the per-symbol hypothesis test (i.e., probability of detection). We compare the probability for the proposed method and that for the conventional binary method, which follows Lemma 2. **Lemma 2**. : _Let [PERSON], using an energy-detecting strategy, know the exact noise level. If [PERSON] has a false alarm probability \(P_{F}\), then the per-symbol detection probabilities of the proposed signal \(P_{D,Gaussian}\) and BPSK-modulated signal \(P_{D,Binary}\) are_  
\[ \begin{array}{l}P_{D,Binary}=Q\big{(}\alpha-\sqrt{\beta}\big{)}+Q\big{(} \alpha+\sqrt{\beta}\big{)},\\ P_{D,Gaussian}=2Q\bigg{(}\frac{\alpha}{\sqrt{1+\beta}}\bigg{)},\end{array}  \tag{27}\]  
_where \(\alpha=Q^{-1}(P_{F}/2)/\sigma_{n}\), \(\beta=\sigma_{c}^{2}/\sigma_{n}^{2}\), and the \(Q\)-function is defined by_  
\[Q(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty}e^{-t^{2}/2}dt. \tag{28}\]  
Proof of Lemma 2.: In Case 2, [PERSON] has information about the wiretap channel noise level. Without the loss of generality, the hypothesis test in Equation (26) is simplified into a per-symbol hypothesis test, as shown below:  
\[\left\{ \begin{array}{rl}H^{\prime}{}_{0}:d=n\\ H^{\prime}{}_{1}:d=c+n,\end{array} \right. \tag{29}\]  
where \(d\), \(c\), and \(n\) are the corresponding single samples of \(\mathbf{d}\), \(\mathbf{c}\), and \(\mathbf{n}\). We determine a threshold \(\alpha\) and use the following likelihood ratio test to determine which hypothesis is true:  
\[\left|d\right|_{\frac{H_{\alpha}}{H_{\alpha}}}^{\frac{H_{1}}{H_{\alpha}}} \alpha\sigma_{n}. \tag{30}\]  
The threshold \(\alpha\) is selected according to the desired false alarm probability \(P_{F}\). Because the noise is Gaussian, \(\alpha\) can be calculated using the \(Q\)-function:  
\[\alpha=Q^{-1}\bigg{(}\frac{P_{F}}{2}\bigg{)}. \tag{31}\]  
We can now compare the probability of detection for the traditional binary modulation and proposed modulation methods. The probability of detection can be calculated as the \(\alpha\sigma_{n}\)-tail probability of the distribution of \(d\) given that \(H^{\prime}{}_{1}\) is true, through a process similar to that used in [13,14].