# Covert Anti-Jamming Communication Based on Gaussian Coded Modulation
## 5 Implementation Problems
### Computational Complexity and Power Consumption  
The computational complexity is another important consideration for real-time implementation. The receivers of the proposed system include the SSR algorithm for solving the SJE problem and removing it from the received vector. The additional computation of the SSR algorithm requires a certain amount of processing time and power consumption. On the one hand, for the GSJE algorithm, the dominant complexity term is that of the least-squares method for obtaining nonzero coefficients. The computational complexity of the least-squares or the equivalent matrix inversion method is \(O(NK)\) per iteration, because the Cholesky decomposition of \(\mathbf{A}_{\mathrm{S}}^{H}\mathbf{A}_{\mathrm{S}}\) can be precalculated for all possible support sets, \(\mathrm{S}\). The amount of overall computations for \(K\) number of iterations is \(O(NK^{2})\). Conversely, the BSJE algorithm has to calculate an inversion of the matrix \(\left(\mathbf{A}\mathbf{A}^{H}+\sigma^{2}\mathbf{A}\mathbf{A}^{H}\right)^{-1}\) for each iteration, resulting in \(O(MN)\) computations, where \(\mathbf{A}\in\mathbb{C}^{M\times N}\) and \(M<N\). We now discuss the complexity of two other conventional covert anti-jamming methods. The simplest conventional method is the DSSS, which is compared with the proposed method in Section 4. The receiver for DSSS requires \(O(N)\) computations to perform a matched filtering of length \(2N\). The other conventional method, FHSS-type cognitive radio [41; 42], also requires \(O(N)\) computations for spectrum sensing of \(N_{F}\) subbands within \(O(N_{T})\) hops. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Quantization Level [bits]** & **16** & **14** & **12** & **10** & **8** \\ \hline Average Return & 0.0557 & 0.0583 & 0.0624 & 0.0661 & 0.1969 \\ \hline \end{tabular}
\end{table} 
Table 1: Average results of the Anderson–Darling test for quantized Gaussian distribution. The results show that the complexity of the SSR algorithm is marginally higher than that of these conventional methods. However, owing to the recent advances in software and hardware solutions, this complexity increase does not pose any limitation to real-time implementation. We aim to show this in the following section. Let the block size be \(N=2^{9}\) for transmitting a message of length \(l_{m}=16\). Then, the computational complexity of the GSJE algorithm is approximately \(2^{23}\approx 8\times 10^{6}\) floating-point operations per second (FLOPS), assuming the worst case where \(K=N/4\). We can use serial processing with modern digital signal processing chips to compute the algorithm. For example, a single core of the TMS320C6678 digital signal processor offers a few tens of giga-FLOPS. When we employ these chips, we can process a few thousand blocks in real time, while consuming 1 W [43]. This implies that the throughput that the proposed transceiver offers can easily support communication speeds on the order of kilobits per second(kbps), while offering covert communication with strong anti-jamming protection. We can increase the speed of the SSR algorithm. To achieve this, we can utilize recent advances in research on parallel computation for matrix inversion and multithread processing on GPUs. Examples include Gauss-Jordan-based implementations [44] and squared Givens rotation-based implementations [45]. [PERSON] et al. [44] claimed that the time complexity of the matrix inversion decreased up to \(O(N)\) and [PERSON] et al. [45] showed that their CPU/GPU-combined implementations achieved more than 20 times higher throughput than CPU-only implementations. These examples suggest the possibility of a real-time implementation on the order of megabits per second based on a GPU or a cluster of GPUs. However, the challenge of power consumption remains. For example, the GPU cores used in [45] consume a maximum of 49 W. These additional power requirements might not be bearable for certain power-critical applications, such as a mobile radio device.