# A Wireless Covert Channel Based on Dirty Constellation with Phase Drift
## 3 Phase Drift-Based Dirty Constellation
The proposed approach allows us to better hide the steganographic channel than the method based on the dirty constellation shown in [30]. This is possible thanks to the random initial phase \(\varphi_{x}\), the adaptive selection of the centroid distance \(A_{m}\) and the phase drift step \(\varphi_{m}\). In the general case, the parameters \(A_{m}\) and \(\varphi_{m}\) could also be random within defined ranges. An example of the phase drift-based dirty constellation for long bit strings is presented in \[x_{n}=A_{n}\exp(\mathrm{j}\varphi_{n})\quad\text{for}\quad n=1,2,\ldots,N, \tag{2}\]1  
The transceiver (transmitter-receiver) system is shown in \[x_{n}=A_{n}\exp(\mathrm{j}\varphi_{n})\quad\text{for}\quad n=1,2,\ldots,N, \tag{2}\]2 This approach is similar to the framework of wireless covert channel proposed in [31] (Figure 4, p. 5). In the transmitter system, the fixed phase drift is added to the cover constellation obtained at the output of the QAM modulator. The DCM mapper is used to assign vector \(y_{k}\). After putting together vector \(y_{k}\) with carrier constellation points \(x_{n}\), the transmission adopts the steganographic form. In the receiver system, if necessary, the detection of the overt signal (cover) is performed using a traditional QAM demodulator. By contrast, information from the covert channel is received by using the so-called DCM demapper. In the first instance, this system  
\[x_{n}=A_{n}\exp(\mathrm{j}\varphi_{n})\quad\text{for}\quad n=1,2,\ldots,N, \tag{2}\]3 Constellations of carrier and the covert channels applied with: (**a**) fixed initial phase \(\varphi_{x}\); (**b**) random \(\varphi_{x}\); and (**c**) random \(\varphi_{x}\) and \(A_{m}\). \[x_{n}=A_{n}\exp(\mathrm{j}\varphi_{n})\quad\text{for}\quad n=1,2,\ldots,N, \tag{2}\]4 Receiver and transmitter (framework) of phase drift-based covert channel. determines the residual signal, which is the difference between the constellation points of the carrier and the received signal, \(\hat{y}_{k}\). In more detail, it is an assigned complex vector \(y_{k}\) with error \(e_{k}\) caused by the signal transmission through the radio channel  
\[x_{n}=A_{n}\exp(\mathrm{j}\varphi_{n})\quad\text{for}\quad n=1,2,\ldots,N, \tag{2}\]5  
where \(\hat{A}_{m}\) and \(\hat{\varphi}_{m}\) are the amplitude and phase of the covert symbols in the received signal. The DCM demodulator performs the phase angle scanning procedure. In this case, based on the knowledge of the possible angle \(\varphi_{m}\) values used in the steganographic transmitter, it calculates the \(U_{m}\) values for each \(\varphi_{m}\) as follows  
\[x_{n}=A_{n}\exp(\mathrm{j}\varphi_{n})\quad\text{for}\quad n=1,2,\ldots,N, \tag{2}\]6  
The maximum value of the module \(U_{m}\) for a given \(\varphi_{m}\)\((m=1,2,\ldots,M)\) corresponds to the assigned symbol of the covert information. It is worth noting that according to Equation (8), the value \(K\) can be treated as the number of averages performed for a single covert symbol. By providing coherent averaging [34] of the hidden subsymbols, we obtain an increase in the detection gain of covert transmissions according to [33]. The multiplexing use of the hidden symbols may increase the effectiveness of its correct detection. In a similar way, the oversampling application of various credit-related datasets significantly improves the performance of a credit default prediction model presented in [47].