# Covert Channel Based on Quasi-Orthogonal Coding
## 3 Proposed Model
### Cover Signal Sorting  
In this section, it will be shown on a selected example that by sorting the IQ samples of the cover signal in an appropriate way, its spectral characteristics can be changed, so that the influence of the residual signal on the covert signal \(x_{2}\) is reduced (Figure 2).  
The impact of cover IQ sample sorting will be tested in the frequency domain using fast Fourier transform (FFT) analysis on the example of QAM amplitude-phase modulation. Sorting is performed for a sequence (block) of random IQ samples of signal \(x_{1}\) consisting of \(N=64\) samples of 64 order QAM modulation (\(M=64\)). The probability of each symbol  
Figure 2: Covert signal transceiver system.  
(from 0 to 63) is equally likely. The averaged FFT spectrum of such a signal does not have a clear main peak. We then sort the samples according to the phase increment defined as the angle \(angle(x_{1})\). The signal sorted in this way is denoted \(x_{1\_sorted}\). Since the phases of the QAM constellation points vary from \(-\pi\) to \(\pi\), a \(x_{1\_sorted}\) signal close to a sine wave (Figure 3a) with a period equal to \(N\) samples will be obtained. The FFT spectrum of the signal thus formed has one strong spectral line for the fundamental frequency (\(f_{0}\)), the normalized value of which is \(f_{0}=2/N\). The same will be true if \(N\) is a multiple of the modulation order \(M\). By reordering the \(x_{1\_sorted}\) samples, we can obtain shifted spectral lines on the frequency \(f_{n}=nf_{0}\), where \(f_{0}\) is a fundamental frequency, \(n\) is a power of 2 and satisfies the condition \(-0.5\leq f_{n}\leq 0.5\). The above actions can be written using matrix operations. If we have a set of sorted (according to phase increase) samples of the N-QAM \(x_{1\_sorted}\) signal with indices [1:N], the \(i\) indices of the sorted signal with the fundamental frequency \(\mathrm{k}\cdot f_{0}\) (\(\mathrm{k}=1,\ 2,\ 4\ldots\frac{N}{2}\)) are obtained according to the formula:  
\[i=\text{reordered\_index\_for}\ x_{1\_sorted}=\text{parallel\_to\_serial} \left\{\begin{bmatrix}1:\mathrm{k}:\mathrm{N}\\ 2:\mathrm{k}:\mathrm{N}\\.\\.\\.\\ \mathrm{k}:\mathrm{k}:\mathrm{N}\end{bmatrix}\right\} \tag{9}\]  
For example, if we have a sorted (according to the phase increment) set of samples of a 64 QAM signal labelled \(x_{1\_sorted}\) with base frequency \(f_{0}\) and corresponding indexes from 1 to 64, then the indices of the signal with frequency \(16\cdot f_{0}\) and \(32\cdot f_{0}\) (which corresponds to the normalized frequency equal to \({}^{1}/_{4}\) and \({}^{1}/_{2}\)) are as follows:  
\[i=\text{parallel\_to\_serial}\left\{\begin{bmatrix}1,\ 17,33,49\\ 2,\ 18,\ 34,\ 50\\ 3,\ 19,\ 35,\ 51\\.\\.\\ 16,\ 32,\ 48,\ 64\end{bmatrix}^{\mathrm{T}}\right\}\Rightarrow \tag{10}\]  
and for \(k=32\) (\(32\cdot f_{0}\))  
\[i=\text{parallel\_to\_serial}\left\{\begin{bmatrix}1,\ 33\\ 2,\ 34\\.\\.\\.\\ 32,64\end{bmatrix}^{\mathrm{T}}\right\}=\ [1,\ 33,\ 2,\ 34,\ \ldots,\ 32,\ 64] \tag{11}\]  
In Figure 3, for 64-QAM modulation, for a sequence of 64 random IQ samples, the time and frequency spectrum waveforms are shown for sequences sorted with fundamental frequencies of \(f_{0}\), \(16\cdot f_{0}\) and \(32\cdot f_{0}\), respectively.  
According to Figure 3, for 64 random 64 QAM symbols we can, as a result of sorting, control (to a certain extent) the position of the spectral lines. This enables the parameters of the covert signal to be selected for optimal reception. On the basis of the presented method, the reception of the covert signal in the form of FSK and PSK modulation and the different modulation values of the cover and covert signal will be presented.