## 4 Examples  
In this section, the performance of the proposed scheme is evaluated by several design examples on a Intel Core i7-6700 3.40 GHz PC with MATLAB, and compares to other NPR cosine-modulated filter banks design algorithms. **Example 1**. : _This example is concerned with the design of NPR cosine-modulated filter banks with \(M=4\). The filter order of the FIR linear phase prototype filter is \(N=140\). The numbers of discretized samples for pass band, transition band and stop band are \(L_{p}=6\), \(L_{t}=20\) and \(L_{s}=800\), respectively. Choose \(\delta_{p}=\delta_{t}=\delta_{s}=1\times 10^{-3}\). The amplitude distortion function magnitude \(e_{am}(\omega)=1-\left|A_{0}(e^{i\omega})\right|\), the total aliasing transfer function magnitude \(e_{a}(\omega)=\sqrt{\sum_{l=1}^{M-1}\left|A_{l}\left(e^{i\omega}\right)\right|^ {2}}\), and the amplitude response of the sparse prototype filter derived from the proposed scheme are shown in Figure 2. The amplitude distortion \(e_{am}=\underset{\omega}{\max}\left|e_{am}(\omega)\right|\) and total aliasing error \(e_{a}=\underset{\omega}{\max}\left|e_{a}(\omega)\right|\) of the corresponding cosine-modulated filter bank are \(e_{am}=1.70\times 10^{-3}\) and \(e_{a}=5.06\times 10^{-7}\), respectively. It takes 31 iterations for proposed method to converge, with the CPU time equals to 0.2716 seconds in each iteration._  
We compare the performance of the proposed scheme with the WCLS [6] scheme and QBPSO [7] scheme. At the similar reconstruction performances, the number of nonzero tap weights of the prototype filter yielded from the proposed method is 18 less than that of the WCLS scheme and the QBPSO scheme. In Table 1, we compare the filter order, the number of nonzero tap weights, the amplitude distortion \(e_{am}\) and the total aliasing error \(e_{a}\) corresponding to the proposed scheme, the WCLS scheme and the QBPSO scheme. Figure 2: \(M=4\) channel cosine-modulated filter banks in example 1. (**a**) The amplitude distortion function magnitude \(e_{am}(\omega)\); (**b**) The total aliasing transfer function magnitude \(e_{a}(\omega)\); (**c**) Amplitude response of sparse prototype filter. **Example 2**. : _This example is concerned with the design of NPR cosine-modulated filter banks with \(M=8\). The filter order of the FIR linear phase prototype filter is \(N=160\). The numbers of discretized samples for pass band, transition band and stop band are \(L_{p}=6\), \(L_{t}=93\) and \(L_{s}=91\), respectively. Choose \(\delta_{p}=\delta_{t}=\delta_{s}=1\times 10^{-3}\). The amplitude distortion function magnitude \(e_{am}(\omega)\), the total aliasing transfer function magnitude \(e_{a}(\omega)\), and the amplitude response of the sparse prototype filter derived from the proposed scheme are shown in Figure 3. The amplitude distortion \(e_{am}\) and total aliasing error \(e_{a}\) of the corresponding cosine-modulated filter bank are \(e_{am}=5.2\times 10^{-4}\) and \(e_{a}=9.85\times 10^{-6}\). It takes 49 iterations for proposal method to converge, with the CPU time equals to 0.0591 seconds in each iteration._  
We compare the performance of the proposed scheme with the WCLS [6] scheme and GI [8] scheme. At the similar reconstruction performances, the number of nonzero tap weights of the prototype filter yielded from the proposed method is 46 less than that of the WCLS scheme, the number of nonzero tap weights of the prototype filter yielded from the proposed method is 34 less than that of the GI scheme. In Table 1, we compare the filter order, the number of nonzero tap weights, the amplitude distortion \(e_{am}\) and the total aliasing error \(e_{a}\) corresponding to the proposed scheme, the WCLS scheme and the GI scheme.