# Improving Passband Characteristics in Chebyshev Sharpened Comb Decimation Filters
## 3 Passband Compensation of Chebyshev Sharpened Combs from [8; 10; 11]
### Chebyshev Sharpened Comb Filters [10; 11]
#### 3.3.2 Wideband Compensation  
The passband frequency \(\omega_{p}=\pi/2M\), while the decimation factor \(M\) and the Chebyshev sharpening polynomial are the same as in Section 3.2.1. We will consider the wideband compensators \(C_{2}\) and \(C_{3}\).  
* Compensator \(C_{2}\)  
The optimum values of the compensator parameters obtained using the PSO method are \(C_{21\text{o}}=0.6337\) and \(C_{22\text{o}}=0.6264\), while the passband deviation equals \(\delta_{\text{o}}=0.0171\) dB. To obtain a multiplierless design, the optimum parameter values are presented in the SPT form and the results are shown in Table 5. It can be observed that with \(N_{2}=15\) we can obtain almost optimum compensation.  
Figure 5 contrasts the magnitude responses in passband of the non-compensated, optimum compensated and the SPT compensated (\(N_{2}=12\)) Chebyshev sharpened comb filter [11].  
\begin{table}
\begin{tabular}{c c c} \hline \(C_{\mathbf{11}}\) & \(\mathbf{\delta_{1}}\) [dB] & \(N_{\mathbf{1}}\) \\ \hline \(2^{0}\)-\(2^{-2}\) & 0.0299 & 4 \\ \(2^{0}\)-\(2^{-2}\)-\(2^{-5}\) & 0.0125 & 6 \\ \(2^{0}\)-\(2^{-2}\)-\(2^{-5}\)-\(2^{-8}\) & 0.0115 & 7 \\ \(2^{0}\)-\(2^{-2}\)-\(2^{-5}\)-\(2^{-8}\) + \(2^{-11}\) & **0.0110** & **8** \\ \hline \end{tabular}
\end{table}
Table 4: SPT compensator \(C_{1}\) coefficients, passband deviation and the number of adders.  
Figure 4: Compensation of Chebyshev sharpened filter [11], \(M=20\), with the narrowband compensator \(C_{1}\). Passband magnitude responses of non-compensated, optimal compensated and SPT compensated \(N_{1}=4\) filters.  
\begin{table}
\begin{tabular}{c c c c} \hline \(C_{\mathbf{21}}\) & \(C_{\mathbf{22}}\) & \(\mathbf{\delta_{2}}\) [dB] & \(N_{\mathbf{2}}\) \\ \hline \(2^{-1}+2^{-3}\) & \(2^{-1}+2^{-3}\) & 0.0485 & 11 \\ \(2^{-1}+2^{-3}+2^{-7}\) & \(2^{-1}+2^{-3}\) & 0.0227 & 12 \\ \(2^{-1}+2^{-3}+2^{-7}\) & \(2^{-1}+2^{-3}+2^{-8}\) & 0.00177 & 13 \\ \(2^{-1}+2^{-3}+2^{-7}+2^{-11}\) & \(2^{-1}+2^{-3}+2^{-9}\) & 0.0174 & 14 \\ \(2^{-1}+2^{-3}+2^{-7}+2^{-10}\) & \(2^{-1}+2^{-3}+2^{-10}+2^{-12}\) & **0.0172** & **15** \\ \hline \end{tabular}
\end{table}
Table 5: SPT compensator \(C_{2}\) coefficients, passband deviations and the number of adders.  
* Compensator \(C_{3}\) The optimum parameters are obtained using PSO, resulting in C\({}_{210}\) = 0.5946, C\({}_{220}\) = 0.8939 and \(\delta_{20}\) = 0.0302 dB. The SPT parameter values are given in Table 6, along with the corresponding passband deviations and the number of compensator adders.  
As observed in Table 6, the approximate optimal optimization can be obtained with \(N_{3}\) = 14.  
Figure 6 compares the passband magnitude responses of the Chebyshev comb filter [11], non-compensated, compensated with the optimal \(C_{3}\) filter and compensated with the SPT \(C_{3}\) filter with \(N_{3}\) = 10.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \(C_{31}\) & \(C_{32}\) & \(\delta_{3}\) [dB] & \(N\) \\ \hline \(2^{-1}\)-\(2^{-4}\) & \(2^{0}\)-\(2^{-4}\) & 0.0597 & 9 \\ \(2^{-1}+2^{-4}\)-\(2^{-6}\) & \(2^{0}\)-\(2^{-4}\) & 0.0404 & 10 \\ \(2^{-1}+2^{-3}+2^{-6}\) & \(2^{0}\)-\(2^{-3}\)-\(2^{-6}\) & 0.0373 & 9 \\ \(2^{-1}+2^{-3}\)-\(2^{-5}+2^{-9}\) & \(2^{0}\)-\(2^{-3}+2^{-6}\) & 0.0317 & 13 \\ \(2^{-1}+2^{-3}\)-\(2^{-5}+2^{-11}\) & \(2^{0}\)-\(2^{-3}+2^{-6}+2^{-8}\) & **0.0304** & **14** \\ \hline \hline \end{tabular}
\end{table}
Table 6: SPT compensator \(C_{3}\) coefficients, passband deviations and the number of adders.  
Figure 5: Compensation of Chebyshev sharpened filter [11], \(M\) = 20, with the narrowband compensator \(C_{1}\). Passband magnitude responses of non-compensated, optimal compensated and SPT compensated, \(N_{1}\) = 4, filters.  
Figure 6: Compensation with the wideband compensator \(C_{3}\). Passband magnitude responses of non-compensated, optimal compensated and SPT compensated, \(N_{3}\) = 10, Chebyshev sharpening comb filters [11] with \(M\) = 20.