# Efficient Low-Complexity Turbo-Hadamard Code in UAV Anti-Jamming Communications
## 4 Simulation Results
This result implies that when the value of \(r\) is small, the gain provided by Hadamard codes is insignificant, which leads to the fact that the performance gap between the two codes is small. To compare the anti-jamming performance against the order of Hadamard codes, the simulation setup in Figure 10 was \(r=5\), and other simulation conditions were similar to those presented in Figure 9. Therefore, the code rates of the tail-biting turbo-Hadamard codes and terminate-state-zero turbo-Hadamard codes were \(1/17.2\), and the code rate of concatenated zigzag Hadamard codes was \(1/16.6\). In this simulation, we repeated the \(1/6\) turbo codes 3 times, by which we obtained an equivalent of \(1/18\) overall codes. We can see that our proposed algorithm obviously outperformed the turbo-Hadamard codes [20] with and without jamming. This is because the gain brought by Hadamard codes is considerable when \(r=5\). When BER performance reaches \(\text{e}^{-5}\), the proposed codes are worsened by \(1.1\) dB under 20% jamming and \(2.5\) dB under 40% jamming compared with the conditions without jamming. It is worth noting that although there is little difference in the relative performance degradation between \(r=5\) and \(r=4\), the BER performance of \(r=5\) under the same SNR condition is much better than that of \(r=4\). Moreover, we observe that the scheme of the turbo codes transmitted repeatedly showed the same coding  
Figure 10: Error performance of turbo-Hadamard codes with \(r=5\) with no jamming (a), 20% (b), 40% (c) frequency bins to be jammed. gain as the normal turbo codes. However, repetitive coding can decentralize the jamming power to each decoded symbol, which results in a considerable anti-jamming performance. The decoding complexity performance of the proposed codes and other codes [23; 24] is shown in Figure 11. The parameters of codes are set as \(k=200\), \(r=4,5\) and \(\eta=2,3\). Since there are no multiplication or division operations in our proposed codes, the usage of LUTs is far less than that of terminate-state-zero turbo-Hadamard codes. We observed that when \(r=4\) changed into \(r=5\), the usage of LUTs increased by 1092 k in the terminate-state-zero codes, the usage of LUTs increased by 723 k in the sigzag-Hadamard codes, whereas the usage increased by 256 k in our proposed codes. We also noted that when \(\eta=2\) became \(\eta=3\), the usage of LUTs increased by 606 k in codes [23], whereas the usage increased by 128 k in our proposed codes. Moreover, the computational complexity of our proposed codes was much smaller than those of the terminate-state-zero codes. In the case of \(r=5,\eta=2\), the complexity of our codes was only 22.3% of that of terminate state-zero turbo-Hadamard codes, and 44.6% of that of zigzag-Hadamard codes. The transmitting efficiency of our proposed codes and other codes [23; 24] is shown in Figure 12. Since the tail-biting turbo-Hadamard method does not need to transmit extra bits for its terminating state, the actual coding efficiency is 100% in all simulation conditions. It was observed that when \(r=4,\eta=2\), the transmission efficiency was reduced by 7.4% in terminate-state-zero codes. In the condition of \(r=5,\eta=2\), the transmission efficiency was reduced by 9.1% in codes [23]. Furthermore, as the memory depth \(\eta\) increased, the transmission efficiency was also degraded in the algorithm reported in [23]. The reasons for this are that the canonical turbo-Hadamard codes need to transmit \(M\eta(2^{r}-r)\) bits for their termination. However, the efficiency of zigzag-Hadamard codes was 100%, as in the tail-biting turbo-Hadamard codes, although the performance regarding coding gains and the complexity of the tail-biting turbo-Hadamard codes was much better than that of the zigzag-Hadamard codes. Figure 6:0 Complexity comparison of different turbo-Hadamard codes.