## 3 The Design of Sparse Linear Phase FIR Prototype Filter of NPR Cosine-Modulated Filter Banks  
Without loss of generality, the linear phase FIR filter \(H(e^{i\omega})\) is assumed to be the Type \(II\) which suggests that the length \(N\) of \(H(e^{i\omega})\) is even and its tap weights satisfying \(h(n)=h(N-1-n)\) for all \(0\leq n\leq N-1\). For the other types of the linear phase filters, an argument similar to the one developed in this paper can be followed. The frequency response corresponding to an \(N\)-th order Type \(II\) linear phase FIR prototype filter can be represented as  
\[H(e^{i\omega}) = e^{-j\frac{N-1}{2}\omega}H_{0}(\omega) \tag{8}\] \[= e^{-j\frac{N-1}{2}\omega}\left[2\sum_{n=0}^{\frac{N-1}{2}}h(n) \cos((\frac{N{-}1}{2}{-}n)\omega)\right].\]  
where \(H_{0}(\omega)\) is the real-valued amplitude response of prototype filter. The perfect-reconstruction conditions Equations (6) and (7) for linear phase cosine-modulated filter banks can be reduced to  
\[|H_{0}(e^{i\omega})|^{2}+|H_{0}(e^{i(\omega-\frac{\pi}{M})})|^{2} = 1,\omega\in[0,\frac{\pi}{M}], \tag{9}\] \[|H_{0}(e^{i\omega})| = 0,\omega\in(\frac{\pi}{M},\pi]. \tag{10}\]  
Thus, the problem of sparse linear phase FIR prototype filter design for cosine-modulated filter banks can be formulated as  
\[\min\quad\|\mathbf{h}\|_{0} \tag{11a}\] \[s.t.\quad|\mathbf{c}(\omega)\mathbf{h}|^{2}+|\mathbf{c}(\omega{- \frac{\pi}{M}})\mathbf{h}|^{2}=1,\omega\in[0,\frac{\pi}{M}],\] (11b) \[|\mathbf{c}(\omega)\mathbf{h}|=0,\omega\in(\frac{\pi}{M},\pi] \tag{11c}\]  
where we have  
\[\mathbf{c}(\omega) = [\cos\!\!\frac{N{-}1}{2}\omega,\cdots,\cos(\frac{N{-}1}{2}{-}n) \omega,\cdots,\cos\!\!\frac{1}{2}\omega], \tag{12}\] \[\mathbf{h} = 2[h(0),h(1),\cdots,h(n),\cdots,h(\frac{N}{2}{-}1)]^{\mathsf{T}}. \tag{13}\]  
Define a approximation function as \(f(\omega)\), which is required to satisfy the following equations:  
\[f(\omega_{0})=1, \tag{14a}\] \[= e^{-j\frac{N-1}{2}\omega}\left[2\sum_{n=0}^{\frac{N-1}{2}}h(n) \cos((\frac{N{-}1}{2}{-}n)\omega)\right].\]0 (14b) \[= e^{-j\frac{N-1}{2}\omega}\left[2\sum_{n=0}^{\frac{N-1}{2}}h(n) \cos((\frac{N{-}1}{2}{-}n)\omega)\right].\]1  
\[= e^{-j\frac{N-1}{2}\omega}\left[2\sum_{n=0}^{\frac{N-1}{2}}h(n) \cos((\frac{N{-}1}{2}{-}n)\omega)\right].\]2 \(M\)-channel maximally decimated filter bank. where \(\omega_{0}=\alpha\pi/2M\) and \(0\leq\alpha\leq 1\). There are lots of methods to obtain the function \(f(\omega)\). In this paper, we establish a mathematical model like  
\[= e^{-j\frac{N-1}{2}\omega}\left[2\sum_{n=0}^{\frac{N-1}{2}}h(n) \cos((\frac{N{-}1}{2}{-}n)\omega)\right].\]3  
Substituting Equation (14a), Equation (14c) and \(f\left(\pi/2M\right)=\sqrt{2}/2\) into Equation (15), we can obtain  
\[= e^{-j\frac{N-1}{2}\omega}\left[2\sum_{n=0}^{\frac{N-1}{2}}h(n) \cos((\frac{N{-}1}{2}{-}n)\omega)\right].\]4  
Utilizing the approximation function \(f(\omega)\), the problem of designing sparse NPR cosine-modulated filter banks can be expressed as a constrained optimization problem for the coefficients of prototype filter in the following form:  
\[= e^{-j\frac{N-1}{2}\omega}\left[2\sum_{n=0}^{\frac{N-1}{2}}h(n) \cos((\frac{N{-}1}{2}{-}n)\omega)\right].\]5 \[= e^{-j\frac{N-1}{2}\omega}\left[2\sum_{n=0}^{\frac{N-1}{2}}h(n) \cos((\frac{N{-}1}{2}{-}n)\omega)\right].\]6 (17b) \[= e^{-j\frac{N-1}{2}\omega}\left[2\sum_{n=0}^{\frac{N-1}{2}}h(n) \cos((\frac{N{-}1}{2}{-}n)\omega)\right].\]7 (17c) \[= e^{-j\frac{N-1}{2}\omega}\left[2\sum_{n=0}^{\frac{N-1}{2}}h(n) \cos((\frac{N{-}1}{2}{-}n)\omega)\right].\]8 (17d) \[= e^{-j\frac{N-1}{2}\omega}\left[2\sum_{n=0}^{\frac{N-1}{2}}h(n) \cos((\frac{N{-}1}{2}{-}n)\omega)\right].\]9  
where \(\delta_{p}\)\(\delta_{t}\) and \(\delta_{s}\) represent the ripples of pass band, transition band and stop band respectively. To compute a solution of problem Equation (17), we follow the standard discretized procedure and replace the continuous parameter \(\omega\) with discretized samples distributed in the frequency set \([0,\pi]\).