# Efficient Low-Complexity Turbo-Hadamard Code in UAV Anti-Jamming Communications
## 4 Simulation Results  
In this section, simulation results are provided to illustrate the bit error rate (BER) and frame error rate (FER) performance of the proposed tail-biting turbo-Hadamard codes, the turbo-Hadamard codes reported in [20] and the concatenated zigzag Hadamard codes reported in [31]. Meanwhile, the performance of our proposed algorithm is analyzed with and without jamming. Since the concatenated codes were implemented on the FPGA platform in [23; 24], the computational complexity comparison of our proposed algorithm \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Calculation** & **The Proposed Code** & **Code in [20]** & **Code in [31]** \\ \hline Branch Metric & ADD:\(K(2^{r+1}-2)\) & ADD:\((K+\eta)(2^{r+1}-2)\) & ADD:\(3K(2^{r+1}-2)\) \\  & EXP:\((K+\eta)2^{r+1}\) & EXP:\(3K\cdot 2^{r+1}\) \\ \hline Full Branch & ADD:\(K(2^{r+1}-4)\) & ADD:\((K+\eta)(2^{r+1}-4)\) & - \\ Metric & ADD:\(K(2^{r+1}-4)\) & ADD:\((K+\eta)2^{r+1}\) & ADD:\(K(2^{r+1}-2)\) \\ \hline Forward & ADD:\(2K\cdot 2^{r+2}\) & ADD:\((K+\eta)2^{r}\) & DIV:\(K\) \\ Recursion & ADD:\(2K\cdot 2^{r+2}\) & ADD:\((K+\eta)2^{r}\) & ADD:\(K(2^{r+1}-2)\) \\ Recursion & ADD:\(2K\cdot 2^{r+2}\) & MULT:\(3(K+\eta)2^{r}\) & DIV:\(K\) \\  & & LOG:\(K\) \\ \hline LLR of & ADD:\(K(2^{r+2}-1)\) & ADD:\((K+\eta)(3\cdot 2^{r+1}+2^{r+1}-2r-10)\) & ADD:\(K(2^{r}+1-2)\) \\  & ADD:\(K(2^{r+2}-1)\) & MULT:\((K+\eta)(2^{r+1})\) & DIV:\(Kr\) \\ Bits & BITS & DIV:\((K+\eta)r\) & LOG:\(Kr\) \\  & LOG:\((K+\eta)r\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of decoder complexity *. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Code Parameters** & **The Proposed Code** & **Code in [20]** & **Code in [31]** \\ \hline k = 200, r = 4, \(\eta=2\) & 370,000 LUTs & 1,712,960 LUTs & 679,200 LUTs \\  & & 1.0 BRAMs & 1.0 BRAMs \\ \hline k = 200, r = 5, \(\eta=2\) & 626,000 LUTs & 2,805,376 LUTs & 1,402,400 LUTs \\  & & 1.0 BRAMs & 1.0 BRAMs \\ \hline k = 200, r = 4, \(\eta=3\) & 498,000 LUTs & 2,319,072 LUTs & 679,200 LUTs \\  & & 1.0 BRAMs & 1.0 BRAMs \\ \hline k = 200, r = 5, \(\eta=3\) & 754,000 LUTs & 3,416,896 LUTs & 1,402,400 LUTs \\  & & 1.0 BRAMs & 1.0 BRAMs \\ \hline \hline \end{tabular}
\end{table} 
Table 2: LUTs used in the decoder hardware implementation. and codes [23; 24] is illustrated from the perspective of implementation. Particularly, the trellis terminal state of turbo-Hadamard codes return to zeros; however, the trellis terminal state of our proposed algorithm is based on a cyclic state. Supposing that the capture process and synchronization process are perfect, we simulate a digital baseband FH system. The common FH system parameters are shown in the following Table 3.  
In Figure 6, we show the BER performance of the proposed tail-biting turbo-Hadamard codes and terminate-state-zeroturbo-Hadamard codes in [20] considering different numbers of component coders. We set the length of uncoded bits \(k=200\), the order of Hadamard matrix \(r=5\), the memory depth of convolutional code \(\eta=2\) and the number of component encoders \(M=2\) or \(M=3\). We observed that tail-biting turbo-Hadamard codes outperformed terminate state-zero turbo-Hadamard codes regardless of whether \(M\) equaled 2 or 3. This is because for each component coder, the starting state and terminating state were the same. However, only the terminating of the first component coder is a state of zero and the other is unknown in the terminate-state-zero turbo-Hadamard codes [20].The terminating state of every component coder in our proposed algorithm is determinant. This means that our proposed algorithm has more prior status information than the codes presented in [20]. Furthermore, our algorithm's performance with regard to \(M=3\) exceeded that regarding \(M=2\) in every turbo-Hadamard code.The reason for this is that the codes with three component codes have more constraints on the information bits than those with only two component codes.