# Covert Anti-Jamming Communication Based on Gaussian Coded Modulation
## 5 Implementation Problems
### Modulation Method  
In our proposed system, we used Gaussian samples. This has a noise-mimicking property that is highly likely to remain undetected by the watchful [PERSON]. This undetectable characteristic of the Gaussian code is bolstered with the time-frequency modulation, which is not available with traditional digital modulation constellations, such as quadrature phase-shift keying (QPSK) and quadrature amplitude modulation (QAM). We can recall that in Section 4, the proposed method showed higher undetectability than the traditional method, especially in the scenario in which [PERSON] is the most threatening. The proposed system has to be implemented in signal processing processors that have a finite dynamic range and a finite number of precision levels. Hence, truncation and quantization must be considered when implementing the Gaussian constellation. For several decades, optimal quantization methods have been studied in signal processing and machine learning communities. Well-known quantization methods include Lloyd's algorithm I [36]. It determines a partition with respect to \(k\) mean points or quantization points and then updates these points by calculating the center of each partition. Generalized vector quantization [37] is a relaxation of Lloyd's algorithm. It combines Lloyd's algorithm with a global optimization method to avoid local minimums. There are several other heuristic algorithms for determining optimal quantization, such as competitive learning vector quantization [38]. Figure 7: Per-symbol detection probability of proposed Gaussian and conventional binary symbols: **(a)** false alarm probability of \(0.1\) (\(\alpha=1.645\)), **(b)** false alarm probability of \(0.01\) (\(\alpha=2.576\)). The most important parameter for these methods is the number of quantization points, which determines the truncation limits and quantization precision. If the number of quantization points is large, the undetectability performance of the quantized Gaussian signal becomes closer to that of an ideal Gaussian signal. As the number of quantization points increases, a more precise analog-to-digital converter is required. The quantized Gaussian constellation methods allow us to transmit and receive an approximation of the proposed Gaussian-coded signal by bounding the maximum magnitude with finite values. For example, [PERSON] [38] designed a 500-point quantized constellation that closely approximates the ideal Gaussian signal. For this constellation, the maximum magnitude of a sample is bounded by \(3\sqrt{E_{s}}\), where \(E_{s}\) is the average energy of the signal constellation. This result shows that the proposed method with quantization can be achieved in modern radios. For example, the commercial mobile radio platform of Qualcomm, called Snapdragon 855, includes a long-term evolution modem that supports 256-QAM [39]. When 256-QAM is used, the maximum magnitude of a sample is bounded by \(1.6\sqrt{E_{s}}\). Thus, the dynamic range required for a quantized Gaussian constellation is not significantly far from the dynamic range of the specifications of commercial hardware. If the quantized Gaussian signal is used instead of the true Gaussian signal, performance degradation might occur due to the gap between the two distributions. However, the difference will be negligible if the quantization precision is sufficiently high. In Table 1, the results of the Anderson-Darling test for samples from a quantized and truncated Gaussian distribution are listed. If the quantized Gaussian distribution is close to the real Gaussian distribution, the test returns 0; otherwise, it returns 1. In the numerical simulation, 1024 realizations of 8-, 10-, 12-, 14-, and 16-bit quantized distributions were considered; moreover, a [-3, 3]-truncated standard normal distribution was tested. The results of the tests were averaged from 10,000 simulations per simulation point.