## 3 The Design of Sparse Linear Phase FIR Prototype Filter of NPR Cosine-Modulated Filter Banks
The \(L\times\frac{N}{2}\) discretized sampling matrix \(\mathbf{B}\) can be constructed as  
\[|H_{0}(e^{i\omega})|^{2}+|H_{0}(e^{i(\omega-\frac{\pi}{M})})|^{2} = 1,\omega\in[0,\frac{\pi}{M}], \tag{9}\]0  
\[\mathbf{B}_{p} = \begin{bmatrix}\mathbf{c}(\omega_{1}^{(p)})\\ \vdots\\ \mathbf{c}(\omega_{L_{p}}^{(p)})\end{bmatrix} ,\;\left[\omega_{1}^{(p)},\omega_ {2}^{(p)},\cdots,\omega_{L_{p}}^{(p)}\right]\in[0,\omega_{0})\,,\] \[\mathbf{B}_{t} = \begin{bmatrix}\mathbf{c}(\omega_{1}^{(t)})\\ \vdots\\ \mathbf{c}(\omega_{L_{t}}^{(t)})\end{bmatrix} ,\;\left[\omega_{1}^{(t)},\omega_ {2}^{(t)},\cdots,\omega_{L_{t}}^{(t)}\right]\in\left[\omega_{0},\frac{\pi}{M} -\omega_{0}\right],\] \[\mathbf{B}_{s} = \begin{bmatrix}\mathbf{c}(\omega_{1}^{(s)})\\ \vdots\\ \mathbf{c}(\omega_{L_{s}}^{(s)})\end{bmatrix} ,\;\left[\omega_{1}^{(s)},\omega_ {2}^{(s)},\cdots,\omega_{L_{s}}^{(s)}\right]\in\left(\frac{\pi}{M}-\omega_{0},\pi\right]\]  
with \(L=L_{p}+L_{t}+L_{\text{s}}\). Define \(L\times 1\) vector \(\mathbf{d}\) and \(\mathbf{e}\) as  
\[\mathbf{d} =\left[1\quad\cdots\quad 1\quad f(\omega_{1}^{(t)})\quad\cdots \quad f(\omega_{L_{t}}^{(t)})\quad 0\quad\cdots\quad 0\right]^{\text{T}},\] \[\mathbf{e} =\left[\delta_{p}\\[\min \|\mathbf{h}\|_{0} \tag{19a}\] \[s.t. |\mathbf{B}\mathbf{h}-\mathbf{d}|\leq\mathbf{e}\] (19b) \[\mathbf{c}(\frac{\pi}{2M})\mathbf{h}=\frac{\sqrt{2}}{2} \tag{19c}\]  
Choose the initial \(L\times 1\) weight vector \(\mathbf{w}^{(1)}=[1,...,1]^{\mathrm{T}}\). At the \(k\) iteration with \(k\geq 1\), the procedure of computing the sparse FIR prototype filter proceeds as follows. **Step 1:** Normalize column vectors \(\mathbf{b}_{0},\mathbf{b}_{1},\ldots,\mathbf{b}_{\frac{N}{2}-1}\) of \(\mathbf{B}\) under the weight vector \(\mathbf{w}^{(k)}\) as  
\[\mathbf{B}^{(k)}=[\mathbf{b}_{0}^{(k)},\ \mathbf{b}_{1}^{(k)},\ \ldots,\ \mathbf{b}_{n}^{(k)},\ \ldots,\ \mathbf{b}_{\frac{N}{2}-1}^{(k)}], \tag{20}\]  
\[\mathbf{b}_{n}^{(k)}=\mathbf{b}_{n}/[\sum_{i=0}^{L}\mathbf{w}^{(k)}(l)*| \mathbf{b}_{n}(l)|^{2}]^{\frac{1}{2}},0\leq n\leq\frac{N}{2}-1. \tag{21}\]  
**Step 2:** The OMP algorithm [9] is employed to solve the following problem  
\[\min_{\mathbf{h}^{(k)}} \|\mathbf{B}^{(k)}\mathbf{h}^{(k)}-\mathbf{d}\|_{2}^{2} \tag{22a}\] \[s.t. \|\mathbf{h}^{(k)}\|_{0}\leq k. \tag{22b}\]  
Let \(k\times 1\) vector \(\mathbf{s}^{(k)}\) denote the final solution of Equation (22), and the final \(L\times 1\) residual vector is given as  
\[\mathbf{r}^{(k)}=\Phi^{(k)}\mathbf{s}^{(k)}-\mathbf{d}. \tag{23}\]  
where \(\Phi^{(k)}=[\mathbf{b}_{n_{1}}^{(k)},\mathbf{b}_{n_{2}}^{(k)},\ldots,\mathbf{ b}_{n_{k}}^{(k)}]\) is the matrix of chosen atoms and \(\Lambda^{(k)}=\{n_{1},n_{2},\ldots,n_{k}\}\) is the nonzero index set. **Step 3:** Solve the following linear programming problem:  
\[\min_{\mathbf{h}^{(k)},\mu} \mu \tag{24a}\] \[\mathbf{e} =\left[\delta_{p}\\[\min \|\mathbf{h}\|_{0} \tag{19a}\]0 (24b) \[\mathbf{e} =\left[\delta_{p}\\[\min \|\mathbf{h}\|_{0} \tag{19a}\]1 (24c) \[\mathbf{e} =\left[\delta_{p}\\[\min \|\mathbf{h}\|_{0} \tag{19a}\]2  
If \(\mu\leq 0\), then the computed filter \(\mathbf{\hat{h}}^{(k)}\) is the sparse FIR prototype filter of cosine-modulated filter banks which meeting our design specifications. **Step 4:** Compute the new weight vector by formula from reference [10]  
\[\mathbf{e} =\left[\delta_{p}\\[\min \|\mathbf{h}\|_{0} \tag{19a}\]3  
where \(w_{l}^{(k+1)}\) and \(r_{l}^{(k)}\) are the \(i\)-th entries of the new weight vector \(\mathbf{w}^{(k+1)}\) and the residual vector \(\mathbf{r}^{(k)}\) respectively. Replace \(\mathbf{w}^{(k)}\) with \(\mathbf{w}^{(k+1)}\) and repeat the procedure from Steps 1 to 4.