# Improving Passband Characteristics in Chebyshev Sharpened Comb Decimation Filters
## 4 Results
### Comparisons
#### 4.1.2 Comparison with Method in [11]  
The authors in [11] proposed four Chebyshev sharpened combs and the corresponding optimum compensator designs to improve the passband while keeping the number of adders as low as possible. We compare our designs, taking narrowband and wideband cases.  
* Chebyshev sharpening polynomial \(p(x)=1-2^{9}x^{2}+2^{15}x^{4}\), \(x=H(z,\,1,\,32)\) The passband edge equals \(\omega_{p}=0.226\pi/M\) where \(M=32\). The design in [11] requires \(N\) = 3 adders resulting in \(\delta_{p}=0.02\) dB. In our design, we chose the narrowband compensator \(C_{1}\). The results of the PSO design are \(C_{110}=0.3543\) and \(\delta_{1}0=0.0043\) dB, while the results of the SPT design are shown in Table 11.  
The proposed design offers better compensation requiring more adders. As shown in the first row of Table 11, the proposed design exhibits better compensation at the expense of one additional adder. Unlike [11], the proposed design's flexibility offers further advantages and improved passband compensation by slightly increasing the adders.  
* Chebyshev sharpening polynomial \(p(x)=-1+27(2^{4}x-2^{10}x^{2}+2^{14}x^{3})\), \(x=H(z,\,1,\,32)\)  
\begin{table}
\begin{tabular}{c c c} \hline \(C_{31}\) & \(C_{32}\) & \(\delta_{3}\) [dB] & \(N_{3}\) \\ \hline \(2^{-1}\)-\(2^{-13}\) & \(2^{-1}\)-\(2^{-4}\) & **0.0101** & **9** \\ \(2^{-1}\)-\(2^{-13}\)-\(2^{-16}\) & \(2^{-1}\)-\(2^{-4}\) & **0.0100** & **11** \\ \(2^{-1}\)-\(2^{-11}\)-\(2^{-13}\) & \(2^{-1}\)-\(2^{-4}+2^{-10}\) & **0.0098** & **12** \\ \hline \end{tabular}
\end{table}
Table 10: SPT compensator \(C_{3}\) coefficients, passband deviations and the number of adders.  
\begin{table}
\begin{tabular}{c c c} \hline \(C_{11}\) & \(\delta_{1}\) [dB] & \(N_{1}\) \\ \hline \(2^{-1}\)-\(2^{-3}\) & **0.0169** & 4 \\ \(2^{-1}\)-\(2^{-3}\)-\(2^{-6}\) & **0.0064** & 5 \\ \(2^{-1}\)-\(2^{-3}\)-\(2^{-6}\)-\(2^{-8}\) & **0.0045** & 6 \\ \(2^{-1}\)-\(2^{-3}\)-\(2^{-6}\)-\(2^{-8}\)-\(2^{-11}\) & **0.0043** & 7 \\ \hline \end{tabular}
\end{table}
Table 11: SPT compensator \(C_{1}\) coefficient, passband deviations and the number of adders.  
\begin{table}
\begin{tabular}{c c c} \hline \(C_{11}\) & \(\delta_{1}\) [dB] & \(N_{1}\) \\ \hline \(2^{-1}\)-\(2^{-5}\) & **0.0056** & **4** \\ \(2^{-1}\) + \(2^{-5}\)-\(5\)-\(2^{-8}\) & **0.0042** & **5** \\ \(2^{-1}\) + \(2^{-5}\)-\(5\)-\(2^{-7}\) + \(2^{-9}\) & **0.0036** & **6** \\ \(2^{-1}\) + \(2^{-5}\)-\(2^{-7}\) + \(2^{-9}\)-\(2^{-11}\) & **0.0035** & **7** \\ \hline \end{tabular}
\end{table}
Table 9: SPT compensator \(C_{1}\) coefficients, passband deviation and the number of adders.  
The passband edge equals \(\omega_{p}=0.483\pi/M\) and \(M=32\). The optimum design in [11] requires \(N=7\)**adders**, resulting in \(\delta_{p}=0.12\)**dB**. Since \(C_{3}\) requires fewer adders than \(C_{2}\), we apply the wideband compensator \(C_{3}\) in our design. The results of the optimal PSO design are \(C_{310}=0.8888\), \(C_{320}=1.8668\) and \(\delta_{30}=0.0554\) dB, while the SPT design results are given in Table 12.  
The proposed design in the first row of Table 12 exhibits much better compensation, requiring two additional adders. Unlike [11], other results in Table 12 offer a trade-off between improving compensation and slightly increasing the number of adders. Additionally, all compensators in [11] need the normalization factor.