# Chaotic Pulse-Shaping Filter Based on Root-Raised-Cosine Division
## 3 RRC-Divided Chaotic Filter
### RRC-Division Scheme  
Since the RRC waveform attenuates symmetrically with \(t=0\) as the center, taking the first zero crossing time \(T_{c}\) as a division point (one positive and one negative), the waveform can be divided into three parts. \(T_{c}\) can be calculated by:  
\[\frac{1}{1-\left(\frac{4\beta T_{c}}{T_{s}}\right)^{2}}\bigg{[}\cos\frac{(1+ \beta)\pi T_{c}}{T_{s}}+\frac{T_{s}}{4\beta T_{c}}\sin\frac{(1-\beta)\pi T_{c }}{T_{s}}\bigg{]}=0, \tag{6}\]  
which is a function of the roll-off factor \(\beta\) and symbol time \(T_{s}\). Since Equation (6) has infinite solutions, there is no exact analytic solution for zero crossing time. Using fsolve in MATLAB to solve the nonlinear equation, a positive \(T_{c}\) with various \(\beta\) and \(T_{s}\) can be obtained, as shown in Figure 1.  
We can see that when \(T_{s}\) is constant, \(T_{c}\) decreases linearly with the growth of \(\beta\). Thus, \(T_{c}\) is a linear function of \(T_{s}\). The intercept is \(T_{s}\), and the slope of the function is \(-0.25\beta\). Thus, \(T_{c}\) can be derived as:  
\[T_{c}=-0.25\beta T_{s}+T_{s}. \tag{7}\]  
Since RRC is symmetric at \(t=0\), taking \(-T_{c}\) and \(T_{c}\) as division points, the waveform of RRC can be divided into three parts. By multiplying the first part by two and then combining it with the second part, we can obtain an asymmetric basis function \(p(t)\). By combining the second part with the third part by two, another asymmetric basis function \(g(t)\) is derived.  
From the points discussed above, the impulse responses of \(p(t)\) and \(g(t)\) are obtained as:  
\[p(t)=\left\{\begin{aligned} & l2h_{rrc}(t),&& t<-T_{c},\\ & h_{rrc}(t),&&-T_{c}\leq t\leq T_{c},\end{aligned}\right. \tag{8}\]  
\[g(t)=p(-t)=\left\{\begin{aligned} & h_{rrc}(t),&& -T_{c}\leq t\leq T_{c},\\ & 2h_{rrc}(t),&& t>T_{c},\end{aligned}\right. \tag{9}\]  
where \(h_{rrc}\) is shown in Equation (5).  
Figure 2 gives the basis function of Equations (8) and (9) with different \(\beta\). It can be seen that a large value of \(\beta\) results in fast sidelobe decay rates. Figure 3 compares the results of convolution of those two pulses and a theoretical RC pulse. Figure 3a shows the waveforms of two pulses in the time domain, and Figure 3b shows the corresponding frequency domain responses. Due to the approximation error of the division point, the frequency response is slightly jitter compared with the standard RC waveform. However, the ability to provide a zero crossing at the optimal sampling point of other pulse intervals is presented.  
All the pulses satisfying the vestigial sideband criterion, namely, that the pulse spectrum has odd symmetry about the corresponding ideally bandlimited spectrum band edge, will be ISI-free. There are an infinite number of such pulses corresponding to different vestigial sidebands. The proposed division method is applicable to all those pulses with the implementation of zero crossing points.  
Figure 3: Comparison of convolution of \(p(t)\) and \(g(t)\) and raised-cosine (RC) pulse in (**a**) time domain and (**b**) frequency domain for \(\beta=0.3,0.5,1\) and \(T_{s}=1\).  
Figure 2: Basis function of (**a**): \(p(t)\) and (**b**): \(g(t)\) for \(\beta=0.3,0.5,1\) and \(T_{s}=1\).