# Efficient Low-Complexity Turbo-Hadamard Code in UAV Anti-Jamming Communications
## 2 System Model  
As shown in Figure 1, we consider an FH communication system having one single-antenna transmitter and one single-antenna receiver, but in which the system performance is degraded by jamming methods, such as single-tone jamming, multiple tone jamming, and partial band jamming, etc. At the transmission side, extra parity bits are added via an encoder to improve the reliability of the communication system. In light of the high performance and low complexity, we adopt turbo-Hadamard as the encoding scheme. Therefore, the \(N\) encoded bits \(\mathbf{c}=(c_{1},c_{2},\cdots,c_{N})\) can be represented as  
\[\mathbf{c}=\text{TH}(\mathbf{u}), \tag{1}\]  
where \(\text{TH}(\cdot)\) represents the turbo-Hadamard encoder function, which will be discussed in detail in Section 3, and \(\mathbf{u}=(u_{1},u_{2},\cdots,u_{W})\) denotes \(W\) uncoded bits to be transmitted. For simplicity but without the loss of generality, suppose that binary phase shift keying (BPSK) is adopted. It is worth noting that other modulation schemes are also suitable in this scenario. The transmitted symbols are then modulated by \(G\) frequency bins according to the frequency hopping pattern. To combat complex electromagnetic jamming in the UAV communication system, the number of frequency bins can be as large as hundreds or thousands. The transmitted signal \(\mathbf{x}_{n},n=1,\cdots,N\), is thus given by  
\[\mathbf{x}_{n}=\mathbf{c}_{n}e^{2\pi f_{n}+\theta_{n}}, \tag{2}\]  
where \(f_{n}\) and \(\theta_{n}\) denote the frequency and phase, respectively, for the \(n\)th encoded bit. Considering the effect caused by jamming, we assume that \(\phi\) (\(0\leq\phi\leq\Phi\)) frequency bins are influenced by suppressive jamming in this paper. The jammed frequency bins can be represented as a set \(\mathcal{H}=\{\hat{f}_{j},j=1,2,...,\phi\}\), where \(\hat{f}_{j}\) denotes the \(j\)th jammed frequency bin. Consequently, the received signal \(\mathbf{y}_{n},n=1,\cdots,N\), can be written as  
\[\mathbf{y}_{n}=\left\{ \begin{array}{ll}\mathbf{x}_{n}+\omega+i,&f_{n}\in\mathcal{H },\\ \mathbf{x}_{n}+\omega,&f_{n}\
otin\mathcal{H},\end{array} \right. \tag{3}\]  
where \(\omega\) denotes the additive Gaussian white noise (AWGN) with a power spectral density of \(\omega_{0}\), and \(i\) represents the suppressive jamming, with a power spectral density of \(i_{0}\). Usually, the strength of the implemented jamming strength is much larger than that of the UAV communication signal and noise. As a result, we assume that the received signals are superimposed on additional noise with high power once the frequency bin is jammed, whereas the signals on the other frequency bins are treated as normal. Since there is an obvious strength gap between signals with and without jamming, the receiver can easily find out which frequency bins are jammed based on the received power. Figure 1: System block diagram. Therefore, the receiver can set the power of signal on the jammed frequency bins to zero by employing notch filter on account of the detection of the whole spectrum. As it is not possible to know which frequency bins are jammed before communication, the transmitted encoded bits are interleaved as randomly as possible before modulation in order to lower the impact of jamming. Each bit is modulated onto every frequency bin in the same probability due to the above interleaving operation. Therefore, the probability of received signals in the jammed frequency bins is \(\phi/\Phi\), whereas the probability of normal received signals is \(1-\phi/\Phi\). Hence, the received signal \(\hat{y}_{n},n=1,\cdots,N\), can be represented as  
\[\hat{y}_{n}=\left\{ \begin{array}{ll}0,&\text{probability }\phi/\Phi,\\ \mathbf{x}_{n}+\omega,&\text{probability }1-\phi/\Phi.\end{array} \right. \tag{4}\]  
Admittedly, the performance of the BPSK/FH demodulator depends heavily on both a reliable capture progress and a precise synchronization progress.