# A Nonlinear Optimization Design Algorithm for Nearly Linear-Phase 2D IIR Digital Filters
## 6 Experimental Results
### 2D Highpass Filter  
In this example, it is required to design a 2D highpass filter with the ideal filter specifications shown in Figure 2, where \(\theta_{i}=w_{i}T\). In addition to the given design specifications, we are including the transition band in our design, which aligns with the filter obtained in [8]. To assess the performance of our filter and the one proposed by [8], we have calculated the maximum transfer function magnitude in the transition band \(H_{\text{max}}^{t}\) for both filters, as presented in Table 1.  
In this example, the max. iterations = 140, \(\epsilon=1\times 10^{-7}\), \(\varepsilon_{s}=0.1\), \(\Gamma_{pb}=0.01\), \(\Gamma_{tb}=1.5\), \(\Gamma_{sb}=0.1\), \(H_{0}=1\). We implement the proposed algorithm for various values of \(\Gamma_{g}\) starting from 0.1 to 4.0. Using the quality factors defined in Section 5, the filter with \(\Gamma_{g}=1.65\) is the most optimal filter among the filters obtained. Figure 3 shows the magnitude response of the designed filter using the proposed method, and the filter presented in [8]. Figure 4 and Figure 5 show the group delays \(\tau_{1}\) and \(\tau_{2}\) of the obtained 2D  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Proposed Method** & **Method in [8]** \\ \hline Orders & (4,4) & (4,4) \\ \(H_{\text{min}}\) & 0.437 & 0.262 \\ \(H_{\text{max}}\) & 1.338 & 1.066 \\ \(Q_{h}\) & 50.76 & 60.54 \\ \(H_{\text{max}}^{t}\) & 0.664 & 0.849 \\ \(H_{\text{max}}^{s}\) & 0.389 & 0.426 \\ \(Q_{s}\) & 8.104 & \(-2.598\) \\ \(\tau_{\text{min}}\) & 0.962 & 0.009 \\ \(\tau_{\text{max}}\) & 4.397 & 11.614 \\ \(Q_{\tau}\) & 64.09 & 99.84 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison results for highpass filter.  
Figure 2: Ideal specifications for 2D highpass filter in [8].  
filter in [8], respectively. Table 1 compares the results of the highpass filter designed by the proposed algorithm with the highpass filter presented in [8].  
The quality measure of the magnitude response in the passband, defined by Equation (15), of the obtained 2D IIR highpass filter is \(Q_{h}=50.76\), where \(H_{min}=0.437\) and \(H_{max}=1.338\). In the same manner, the quality measure of the magnitude response in the passband of the filter presented in [8] is \(Q_{h}=60.54\), where \(H_{min}=0.262\) and \(H_{max}=1.066\). The quality  
Figure 4: Group delays in the passband of the obtained highpass filter ( \(\Gamma_{g}=1.65\)).  
Figure 5: Group delays in the passband of the highpass filter in [8].  
Figure 3: Magnitude response of the 2D highpass filters [8].  
of the magnitude response in the stopband defined by Equation (16) is \(Q_{s}=8.104\) for the filter obtained by the proposed method and for the filter presented in [8], \(Q_{s}\) is negative, indicating too low stopband attenuation compared with the minimum gain in the passband. The quality of the group delay characteristic, defined by Equation (14), of the highpass filter designed by the proposed method is better than the one designed in [8]. It can be seen that \(Q_{\tau}=64.09\), where \(\tau_{min}=0.962\) and \(\tau_{max}=4.397\) for the filter obtained, and \(Q_{\tau}=99.84\), whereas \(\tau_{min}=0.009\) and \(\tau_{max}=11.614\) for the filter designed in [8].