# Nearly Linear-Phase 2-D Recursive Digital Filters Design Using Balanced Realization Model Reduction
## 5 Illustrative Examples and Numerical Evaluation
### 2-D Lowpass Filters  
For the sake of comparison, we carried out a study of an example of separable-denominator 2-D IIR lowpass filter with nearly linear phase in the passband. This example was presented by [PERSON] and [PERSON] in [6], where it is required to design a 2-D lowpass filter to satisfy the following specifications:  
\[\mid H(\omega_{1},\omega_{2})\mid=\left\{\begin{array}{ll}1&{\omega_{1}}^{2 }+{\omega_{2}}^{2}\leq{\omega_{p}^{2}}\\ 0&{\omega_{1}}^{2}+{\omega_{2}}^{2}\geq{\omega_{s}^{2}}\end{array}\right. \tag{17}\]  
where the passband edge \(\omega_{p}=0.4\pi\) and the stopband edge \(\omega_{s}=0.6\pi\). First, the linear-phase 2-D FIR filter prototype of order (24,24) was designed using the window method. After that, the proposed algorithm is guaranteed to obtain a stable 2-D IIR filter with nearly linear phase in the passband. The solution was computable in a reasonable amount of time on a PC with an Intel (R) Core (TM) i5-2450M CPU @ 2.50 GHz and RAM of 6.00 GB. The computation time for the controllability Gramian \(P^{s}\) and the observability Gramian \(Q^{s}\) was 7.46 s. The magnitude response of the full-order, (24,24), 2-D FIR lowpass filter is shown in Figure 2a. Figure 2b shows the magnitude response of the reduced-order (13,13) 2-D IIR filter presented in [6]. Figure 2c shows the magnitude response of the reduced-order (13,13) 2-D IIR filter using the proposed method. The magnitude contour of the obtained filter is shown in Figure 2d. Figure 2e,f show the passband group delays, \(\tau_{1}=\tau_{2}\approx 12\) samples, of the reduced-order 2-D IIR filter. Table 1 summarizes the results of the filter designed by the proposed method, the filter presented in [6], and the results of the filter reported in Table 1 of [52]. These results are presented in terms of the reduced order, the obtained group delays \(\tau_{1}\) and \(\tau_{2}\) and their ripple \(\Delta_{\tau}\), the maximum ripple \(\Delta_{p}\) in the passband region, and the maximum ripple in the stopband region \(\Delta_{s}\). The proposed method gives a filter with comparable performance to that of the same order filter obtained in [6, 52], with improvements in the maximum passband ripple \(\Delta_{p}\), the stopband \(\Delta_{s}\), and the group delay ripple \(\Delta_{\tau}\), as shown in Table 1 below. The table also shows that a smaller number of coefficients is required by the obtained 2-D IIR filter than the original 2-D FIR.  
We also analyze the complexity of the implementation based on the number of non-zero coefficients in the transfer function of the filter. When implementing a general \(N_{1}\times N_{2}\) 2-D FIR filter, the number of filter coefficients C is \((N_{1}+1)(N_{2}+1)\), so the cost of implementation increases rapidly as \(N_{1}\) or \(N_{2}\) increases. In this example, for the 2-D FIR filter of order (24,24), C is 625, whereas for the reduced-order (13,13) 2-D IIR filter with a separable  
\begin{table}
\begin{tabular}{c c c c} \hline  & **Proposed** & **Method in** & **Method in** \\  & **Method** & **[6]** & **[52]** \\ \hline Orders & (13,13) & (13,13) & (18,18) \\ Group Delay & (12,12) & (12,12) & (12,12) \\ \(\Delta_{p}\) & 0.0102 & 0.0124 & 0.0128 \\ \(\Delta_{s}\) & 0.0060 & 0.0111 & 0.0350 \\ \(\Delta_{\tau}\) & 0.0425 & 0.0725 & - \\ \(C_{FIR}\) & 625 & 625 & 676 \\ \(C_{IIR}\) & 224 & 224 & 399 \\ \hline \end{tabular}
\end{table}
Table 1: Comparison results of the 2-D IIR filters.  
denominator, C is \(14\times 14\) for the numerator plus \(2\times 14\) for the denominator (i.e., \(C=224\)). We can see that implementing the 2-D IIR filter designed by the proposed method requires less than half of the number of coefficients required by the 2-D FIR filter.