# A Nonlinear Optimization Design Algorithm for Nearly Linear-Phase 2D IIR Digital Filters
## 6 Experimental Results
### 2D Lowpass Filter  
The proposed method is used to design a 2D IIR lowpass filter with the ideal design specifications shown in Figure 6. The filter was presented in [8], and its order was selected as (4,4). The transition band constraint is included in this example, and \(H_{max}^{t}\) is computed for both filters, where the results are shown in Table 2.  
This 2D lowpass filter is designed by the proposed method using the following values: Max. iterations = 140, \(\epsilon=1\times 10^{-7}\), \(\varepsilon_{s}=0.1\), \(\Gamma_{pb}=0.0001\), \(\Gamma_{tb}=1\), \(\Gamma_{sb}=0.01\), \(H_{0}=1\), \(\tau_{1}=\tau_{2}=2\). The algorithm is implemented for different values of \(\Gamma_{g}\), starting from 0.1 to 4. The filter with \(\Gamma_{g}=2.35\) has the best quality in this example.  
Figure 7 shows the impulse and magnitude responses of the filter obtained by using the proposed method and the filter presented in [8]. Figures 8 and 9 show the group delays, \(\tau_{1}\) and \(\tau_{2}\), of the obtained 2D lowpass filter for \(\Gamma_{g}=2.35\). The group delays, \(\tau_{1}\) and \(\tau_{2}\), of the 2D lowpass filter presented in [8] are shown in Figure 10 and Figure 11, respectively. Table 2 compares the results of the filter designed by the proposed method and the filter obtained in [8]. The quality measure of the magnitude response in the passband, defined by Equation (15) of the 2D lowpass filter obtained by the proposed method is \(Q_{h}=14.25\), where \(H_{min}=0.843\) and \(H_{max}=1.120\), and for the filter presented in [8] is \(Q_{h}=25.26\), where \(H_{min}=0.642\) and \(H_{max}=1.069\). The quality of the magnitude response in the stopband defined by Equation (16) is \(Q_{s}=0.916\) for the filter designed by the proposed method, and \(Q_{s}=2.275\) for the filter presented in [8]. The quality of the group delay characteristic, defined by Equation (14), for the 2D lowpass filter obtained by the proposed method is \(Q_{\tau}=47.650\), where \(\tau_{min}=2.217\) and \(\tau_{max}=6.864\). The same quality measure for the 2D lowpass filter presented in [8] is found to be \(Q_{\tau}=93.774\), where \(\tau_{min}=1.738\) and \(\tau_{max}=54.091\). We can see that the filter designed with the proposed method is better than the one designed in [8], especially with respect to the passband phase linearity.  
Figure 6: Ideal specifications for 2D lowpass filter in [8].  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Proposed Method** & **Method in [8]** \\ \hline Orders & (4,4) & (4,4) \\ \(H_{\rm min}\) & 0.843 & 0.642 \\ \(H_{\rm max}\) & 1.120 & 1.069 \\ \(Q_{h}\) & 14.25 & 25.26 \\ \(H_{\rm max}^{t}\) & 1.0079 & 1.020 \\ \(H_{\rm max}^{s}\) & 0.403 & 0.446 \\ \(Q_{s}\) & 0.916 & 2.275 \\ \(\tau_{\rm min}\) & 2.217 & 1.738 \\ \(\tau_{\rm max}\) & 6.864 & 54.091 \\ \(Q_{\tau}\) & 47.650 & 93.774 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison results for lowpass filter.  
Figure 8: Group delays in the passband of the obtained lowpass filter (\(\Gamma_{g}=2.35\)).  
Figure 7: Magnitude response of the 2D lowpass filter [8].