# Design of Chaotic Interleaver Based on Duffing Map for Turbo Code
## 4 Empirical Tests in Software
### Test Results  
To compare turbo code performances, four data lengths are considered, and the same encoder and decoder are maintained, substituting only the interleaver. The communication system shown in Figure 4 is simulated, and the tests assess the performance of BER versus the SNR. The tests have the objective of showing changes in the performance of the turbo code of each interleaver when the block length is modified. For each interleaver, a vector is obtained that contains the indexes applied to the encoder and decoder in the turbo code simulation. Since the main focus is a performance analysis on bits of short lengths, the results of the interleavers for shorter lengths are compared. Figure 5 shows these results for the 512-bit length of the random, S-random, QPP interleavers, and chaotic Lozi map, as well as the proposed chaotic interleaver. The comparison in Figure 5 shows that for a 512-bit length, the performance of the chaotic Duffing interleaver reaches an SNR of 2 dB, while BER has a curve similar to that of a random interleaver. For an SNR of 2 dB, the BER of the random interleaver reaches the value of \(4.8787\times 10^{-6}\), the result obtained by the S-random interleaver is a BER of \(1.6262\times 10^{-5}\), the result of the Lozi chaotic interleaver is a BER of \(5.0414\times 10^{-5}\), and of the proposed chaotic interleaver reached is a BER of \(6.5050\times 10^{-6}\). These four cases present an excellent performance. For the QPP interleaver, the result obtained for an SNR of 1 dB is a BER of \(0.0032\), while the result obtained for an SNR of 1.5 dB is a BER of \(4.3909\times 10^{-5}\), achieving very poor and excellent performances, respectively. In both cases, BER is smaller than for other interleavers. Figure 6 shows the comparison of the results for a 256-bit length. Figure 5: BER \(\
u/s\)\(E_{b}/N_{0}\) comparison for 512 bits. The comparison in Figure 6 shows that for a 256-bit length, the performance of the chaotic Duffing interleaver reaches an SNR of 2.5 dB, while BER exhibits a curve similar to that of the Lozi chaotic interleaver. For an SNR of 2 dB, the BER of the random interleaver reaches the value of \(9.7575\times 10^{-5}\), the result of the S-random interleaver is a BER of \(2.9272\times 10^{-5}\), and of the QPP interleaver is a BER of \(6.5050\times 10^{-6}\). These three cases present an excellent performance. The Lozi chaotic interleaver achieved a BER of \(2.3093\times 10^{-4}\), and the proposed chaotic interleaver reached a BER of \(2.0442\times 10^{-4}\). Therefore, both have good performance. Figure 7 shows the comparison of the results for a 128-bit length. Figure 6: BER \(\
u/s\)\(E_{b}/N_{0}\) comparison for 256 bits. Figure 7: BER \(\
u/s\)\(E_{b}/N_{0}\) comparison for 128 bits. The comparison in Figure 7 shows that for a 128-bit length, the performance of the chaotic Duffing reaches an SNR of 3 dB, while BER exhibits a curve similar to that of the random interleaver until 2.5 dB, with a slightly lower BER. For an SNR of 2.5 dB, the BER of the random interleaver reaches the value of \(2.471\times 10^{-4}\), and the result obtained by the Lozi chaotic interleaver is a BER of \(2.8622\times 10^{-4}\); in both cases, the performance is good. The result obtained by the S-random interleaver is a BER of \(2.6020\times 10^{-5}\), the result of the QPP interleaver is a BER of \(4.5535\times 10^{-5}\), and of the proposed chaotic interleaver is a BER of \(6.5050\times 10^{-5}\). In these three cases, the performance is excellent. Figure 8 shows the comparison of the results for a 64-bit length. The main interest of this study is to evaluate the performance of the turbo code in terms of an interleaver for short lengths. Table 3 below shows the quality state of BER according to Table 1 for the performance of the 64-bit turbo code.