# A Wireless Covert Channel Based on Dirty Constellation with Phase Drift
## 3 Phase Drift-Based Dirty Constellation  
The developed dirty constellation with the phase drift is based on the DCM [33]. The DCM solution was used to create a watermark in audio files. The covert information is represented in monotonic phase changes of the selected signal harmonics. The \(m\)th harmonic selected for steganography is expressed by the following formula [33]  
\[y_{k}(t)=A_{m}\exp\mathrm{j}(2\pi f_{m}t+\varphi_{x}+\Delta\chi_{m}), \tag{1}\]  
where \(A_{m}\) and \(f_{m}\) represent amplitude and frequency of the mth signal, respectively, \(\varphi_{x}\) is an initial phase and \(\Delta\chi_{m}\) is the preset phase drift carrying covert information. In the DCM detector, the signal is subjected to the phase angle scanning procedure, which results in finding the maximum of the virtual fringe module \(\chi_{V\mathrm{max}}\). As shown in [33], the DCM demonstrates good steganographic properties. In our approach, we adopted the DCM for creating the dirty constellation of the steganographic channel. For further considerations, we assume that the QAM signal (without the use of the OFDM) is the carrier (cover) of the hidden transmission. The proposed method uses a covert symbol represented by the \(K\geq 2\) phase drifts of the successive cover constellation points. Therefore, the \(P\) symbols of the QAM can carry \(P/K\) covert symbols. For the \(N\)-ary QAM, the single constellation point is defined in the complex form:  
\[x_{n}=A_{n}\exp(\mathrm{j}\varphi_{n})\quad\text{for}\quad n=1,2,\ldots,N, \tag{2}\]  
where \(A_{n}\) and \(\varphi_{n}\) are the amplitude and phase of the \(n\)th constellation point. A single \(M\)-ary covert message adopts the form of a \(K\)-component complex vector:  
\[y_{k}=A_{m}\exp\mathrm{j}\left(\varphi_{x}+k\varphi_{m}\right)\quad\text{for }\quad k=1,2,\ldots,K, \tag{3}\]  
Figure 1: Dirty constellation: (**a**) assigning bits to constellation points; (**b**) chosen constellation point. where \(A_{m}\) and \(\varphi_{m}=\Delta\chi_{m}\) are the centroid distance and phase drift step for the proposed dirty constellation, respectively, and \(\varphi_{x}\) is a random initial phase, which does not change when the covert symbol is in progress. Steganographic information is created by combining (adding) two successive \(K\) cover symbols with successive covert message symbols (i.e., vectors \(y_{k}\)). An example of creating hidden information for a single covert symbol and \(K=3\) is presented in Figure 2.  
Possible phase increments correspond to the \(M\)-ary DCM. As a result, we receive new constellation points in the following form:  
\[y_{l}=A_{l}\exp(\mathrm{j}\theta_{l}), \tag{4}\]  
where \(A_{l}\) and \(\theta_{l}\) mean the amplitude and phase of the new point in the constellation relative to the origin of the \(lQ\) coordinate system, respectively, which is depicted in Figure 3.  
The parameters \(A_{l}\) and \(\theta_{l}\) are limited by the following relationships:  
\[A_{n}-A_{m}\leq A_{l}\leq A_{n}+A_{m}, \tag{5}\]  
\[-|\arctan(A_{m}/A_{n})|\leq\theta_{l}\leq|\arctan(A_{m}/A_{n})|. \tag{6}\]  
Equations (5) and (6) mean that the maximum distortions of module \(A_{n}\) and phase \(\varphi_{n}\) of carrier symbols might be \(\pm A_{l}\) and \(\pm\theta_{l}\), respectively. These values describe the degradation level of the original signal by the DCM. On the one hand, it handicaps the  
Figure 3: Additional \(y_{k}\) and carrier \(x_{n}\) constellation points. \[x_{n}=A_{n}\exp(\mathrm{j}\varphi_{n})\quad\text{for}\quad n=1,2,\ldots,N, \tag{2}\]0 Example of constellation with drift phase: (**a**) vector \(y_{k}\) for \(K=3\); (**b**) quadrature amplitude modulation (QAM) cover (transmitted overt sequence of symbols: \(1+\mathrm{j}\), \(1-\mathrm{j}\), \(1-\mathrm{j}\)) and resultant constellation with drift phase applied for preset vector \(y_{k}\). recreation of the cover information. On the other hand, it shows the possibility of detecting stenographic transmission.