# Improving Passband Characteristics in Chebyshev Sharpened Comb Decimation Filters
## 4 Results
### Comparisons
#### 4.1.1 Comparison with Method in [10]  
In Table 1[10], the authors proposed four compensated Chebyshev sharpened combs. In continuation, we then present a comparison of all the cases.  
* Chebyshev sharpening polynomial \(p(x)=1-2^{9}x+2^{15}x^{2}\), \(x=H(z,\,1,\,32)\)  
The passband edge is \(\omega_{p}=0.226\pi/M\) where \(M=32\). The design in [10] requires \(\mathbf{N=9\) adders, resulting in \(\mathbf{\delta_{p}=0.06}\) dB. We used the narrowband compensator \(C_{1}\). The PSO design results in \(C_{110}=0.3549\) and \(\delta o=0.0043\) dB. The SPT design results are shown in Table 7.  
The results in Table 7 highlight the efficiency of the proposed designs (denoted in bold), which outperform those in [10], requiring fewer adders and providing better compensation.  
* Chebyshev Sharpening Polynomial \(p(x)=1-2^{10}x+2^{17}x^{2}\), \(x=H(z,\,1,\,32)\)  
The passband edge is \(\omega_{p}=0.164\pi/M\) where \(M=32\). The design in [10] requires \(\mathbf{N=8}\) adders, resulting in \(\mathbf{\delta_{p}=0.02}\) dB. In the proposed design, we choose the compensator \(C_{1}\). The optimum PSO design gives \(C_{110}=0.3443\) and \(\delta o=0.0012\) dB. The results of the SPT design are shown in Table 8.  
The proposed designs (denoted in bold) confidently present superior compensation, requiring fewer adders.  
* Chebyshev sharpening polynomial \(p(x)=-1+27(2^{6}x-2^{14}x^{2}+2^{20}x^{3})\), \(x=H(z,\,1,\,32)\)  
The passband frequency equals \(\omega_{p}=0.187\pi/M\) and \(M=32\) resulting in \(\mathbf{N=13}\) adders and \(\mathbf{\delta_{p}=0.05}\) dB in the design [10]. We applied the compensator \(C_{1}\) and the PSO design, obtaining \(C_{110}=0.5248\) and \(\delta o=0.0034\) dB. The results of the SPT design are shown in Table 9.  
\begin{table}
\begin{tabular}{c c c} \hline \(\mathbf{C_{11}}\) & \(\mathbf{\delta_{1}\) [dB]} & \(\mathbf{N_{1}}\) \\ \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 7: SPT compensator \(C_{1}\) coefficients, passband deviation and the number of adders.  
\begin{table}
\begin{tabular}{c c c} \hline \(\mathbf{C_{11}}\) & \(\mathbf{\delta_{1}\) [dB]} & \(\mathbf{N_{1}}\) \\ \hline \(2^{-1}\)-\(2^{-3}\) & **0.0157** & **4** \\ \(2^{-1}\)-\(2^{-3}\)-\(2^{-5}\) & **0.0015** & **5** \\ \(2^{-1}\)-\(2^{-3}\)-\(2^{-5}\)-\(2^{-11}\) & **0.0012** & **6** \\ \hline \end{tabular}
\end{table}
Table 8: SPT compensator \(C_{1}\) coefficients, passband deviation and the number of adders.  
Once again, the proposed design shown in Table 9 reassures us with its significantly improved compensation and fewer adders.
* Chebyshev sharpening polynomial \(p(x)=-1+27(2^{4}x-2^{10}x^{2}+2^{14}x^{3})\), \(x=H(z,\,1,\,32)\) The passband edge equals \(\omega_{p}=0.354\pi/M\) and \(M=32\). The design in [10] resulted in \(N=13\) adders and \(\delta_{p}=0.05\) dB. In our design, we chose the wideband compensator \(C_{3}\). The PSO design results in \(C_{310}=0.4998\), \(C_{320}=0.4300\) and \(\delta_{o}=0.0023\) dB, while the results of the SPT design are shown in Table 10.  
In Table 10, presented in bold, the proposed design demonstrates better compensation and fewer adders in all cases.  
Additionally, all compensators in [10] need the normalization factor.