# Chaotic Pulse-Shaping Filter Based on Root-Raised-Cosine Division
## 4 Wireless Communication System Using Chaotic Filter
### Threshold Calculation
#### 4.3.1 BER Performance Analysis  
* AWGN channel  
For an AWGN channel (\(L=1\)), \(\tau=0\) and \(\gamma=0\). The simulation results of CBWCS [21], the proposed method, conventional RRC filter, and the theoretical BER of BPSK under an AWGN channel are presented in Figure 7. In this simulation, for fair comparison, the threshold \(\theta(n)\) was 0, and any channel equalization methods were not implemented at receiver. In all of the following simulations, the oversampling rate \(N_{o}\) was 10, and the tap number of filter \(N_{f}\) was 6. More than 100,000 trials were performed on average. From Figure 7, since the proposed shaping filter is designed without ISI, the chaotic waveform can achieve the same BER performance as an RRC filter. CBWCS has the worst BER performance because of ISI caused by the chaotic filter itself. * Multi-path channel  
For a multi-path channel, we assessed the performance under two conditions: a two-path channel (\(L=2\)), \(\tau_{0}=0\), \(\tau_{1}=1\), \(\gamma=0.6\), and a three-path channel (\(L=3\)), \(\tau_{0}=0\), \(\tau_{1}=1\), \(\tau_{2}=2\), \(\gamma=0.6\). First, we discuss the case with \(\theta(n)=0\) and no channel equalization. The BER performance comparison between CBWCS, the proposed method, and the RRC filter is shown in Figure 8. The dotted lines and solid lines indicate the performance under a two-path channel and a three-path channel, respectively. From Figure 8, it can be seen that ISI caused by multi-path propagation leads to worse BER performance compared with the AWGN channel. The RRC filter and the proposed method have similar BER performance, while CBWCS not only has ISI itself but is also effected by ISI due to multi-path propagation, so it has the worst BER performance. When BER = \(10^{-3}\), the proposed method outperforms CBWCS by more than 2 dB under a two-path channel. Next, we implemented the calculated threshold in symbol detection. Figure 9 shows the comparison results of CBWCS, the proposed method, and the RRC filter with the optimal threshold (Equation (21)). For the conventional RRC-based communication system, a minimum mean square error (MMSE) equalizer is used for channel equalization. For BER = \(10^{-3}\), the proposed method has about 0.2 dB and 0.25 dB gain under two-path and three-path channels compared with CBWCS, respectively, because the ISI, which is not completely eliminated, effects the accuracy of CBWCS. The gap between the chaotic  
Figure 7: Bit error rate (BER) comparison under additive white Gaussian noise (AWGN) wireless channel. waveform without channel equalization and the traditional non-chaotic waveform using linear equalization shows that the threshold can be implemented without amplifying the channel noise to combat multi-path effects, because the Lyapunov exponent of the chaotic waveform stays constant after passing through the wireless channel. Figure 10 shows BER performance with the suboptimal threshold (Equation (22)). Similar to Figure 9, the performance of the two-path channel generally outperforms that of the three-path channel. When BER is \(10^{-3}\), the BER of the suboptimal threshold is about 1.7 dB worse than the optimal threshold. All these analyses are based on the premise that the state of the channel, including decay and delay, are constant. Thus, for all frames, a perfect channel knowledge assumption is employed for threshold calculation. However, in an actual environment, the channel state of each frame will change. We assume that the channel parameter \(\tau_{L}\) is invariant in one frame and uniformly distributed in the range of \([0.3,0.9]\) in different frames of both the  
Figure 8: BER comparison under multi-path wireless channel with \(\theta(n)=0\). Figure 9: BER comparison under static multi-path wireless channel with optimal threshold. two-path channel and three-path channel. In this simulation, 1024 bits in one frame are divided into two parts: 896 data bits and 128 pilot bits. Pilot bits are used to carry out the least squares (LS) algorithm for channel estimation.