# Audlet Filter Banks: A Versatile Analysis/Synthesis Framework Using Auditory Frequency Scales
## 3 The Proposed Approach
### Analysis Filter Bank
#### 3.1.2 Construction of \(H_{0}\) and \(H_{k}\)  
The purpose of the filters \(H_{0}\) and \(H_{K}\) is to stabilize the FB response \(\mathcal{H}_{0}\) by compensating for the potentially low amplitude of \(\mathcal{H}_{0}^{(r)}(\xi)\) in the range \([0,f_{1}[\ \cup\ ]f_{K-1},f_{N}]\). While the content in the frequency bands \(0\) and \(K\) might carry some perceptually relevant information, most applications will not modify the corresponding coefficients. Consequently, it is crucial that \(H_{0}\) and \(H_{K}\) are mostly concentrated outside \([f_{1},f_{K-1}]\), but their time domain behavior is only of secondary importance. Nonetheless, we propose a construction that retains some smoothness in frequency and thus, by Fourier duality, \(h_{0}\) and \(h_{K}\) have appropriate decay. There is no canonical method that provides optimal compensation and time localization for any valid set of Audlet parameters. In [46], for instance, plateau functions with raised cosine flanks were proposed. This method might result in additional ripples in \(\mathcal{H}_{0}\) if \(w\) is not a raised-cosine window. Alternatively, in [47], \(H_{0}\) and \(H_{K}\) were constructed from a set of virtual filters extending the FB beyond \([f_{1},f_{K-1}]\). An adaptation of this method to the Audlet framework is unnecessarily complex and unintuitive. Instead, we propose the following. We define  
\[M=\underset{\xi\in[0,1/2]}{\max}\mathcal{H}_{0}^{(r)}(\xi)\quad\text{ and }\quad\mathcal{H}_{\text{inv}}^{(r)}=\sqrt{(M-\mathcal{H}_{0}^{(r)})_{+}}.\]  
and  
\[P_{K}(\xi)= \begin{cases}1/\sqrt{2}&\text{if }\xi f_{s}\in(-f_{N},-f_{p,s}^{+} ]\cup(f_{p,s}^{+},f_{N}]\\ 1/\sqrt{2}-\cos\left(\pi\frac{|\xi|f_{s}-f_{p,s}^{+}}{f_{p,s}^{+}-f_{p,s}^{-}} \right)/\sqrt{2}&\text{if }|\xi|f_{s}\in[f_{p,e}^{+},f_{p,s}^{+}]\\ 0&\text{elsewhere}.\end{cases} \]  
The frequency \(f_{p,s}^{-}\) (resp. \(f_{p,s}^{+}\)) defines the width of the plateau in \(P_{0}\) (resp. \(P_{K}\)). The region \([f_{p,s}^{-},f_{p,e}^{-}]\) (\([f_{p,e}^{+},f_{p,s}^{+}]\)) defines the transition area of \(P_{0}\) (\(P_{K}\)) (see Figure 4). The filters \(H_{0}\) and \(H_{K}\) are finally defined by their DTFTs as  
\[H_{0}(e^{2\pi i(\cdot)})=P_{0}\cdot\mathcal{H}_{\text{inv}}^{(r)},\quad\text{ and }\quad H_{K}(e^{2\pi i(\cdot)})=P_{K}\cdot\mathcal{H}_{\text{inv}}^{(r)}. \tag{10}\]  
We propose selecting \(0<\kappa_{1}<\kappa_{2}\), such that \(F_{\text{Scale}}(f_{K-1})-F_{\text{Scale}}(f_{1})\geq\kappa_{1}+\kappa_{2}\) and fix  
\[f_{p,s}^{-}=F_{\text{Scale}}^{-1}(F_{\text{Scale}}(f_{1})+\kappa_{1}),\quad f_ {p,e}^{-}=F_{\text{Scale}}^{-1}(F_{\text{Scale}}(f_{1})+\kappa_{2}),\]  
\[f_{p,s}^{+}=F_{\text{Scale}}^{-1}(F_{\text{Scale}}(f_{K-1})-\kappa_{1}),\quad f _{p,e}^{+}=F_{\text{Scale}}^{-1}(F_{\text{Scale}}(f_{K-1})-\kappa_{2}).\]  
This choice ensures that \(P_{0}^{2}+P_{K}^{2}\leq 1\), preventing overcompensation, and is properly adapted to the scale used. By default, we set \(\kappa_{1}=3/V,\kappa_{2}=4/V\), such that \(f_{p,s}^{-}=f_{4},f_{p,e}^{-}=f_{5},f_{p,s}^{+}=f_{K-4},f_{p,e}^{+}=f_{K-5}\). The intuition here is that from \(f_{4}\) (resp. \(f_{K-4}\)) onward, the restricted FB response \(\mathcal{H}_{0}^{(r)}\) is expected to be stable already, and that the size of the transition area ensures a sufficiently smooth roll-off. It should be noted that, although the filters proposed above are chosen to be strictly band-limited, a similar construction with time-limited, but only approximately band-limited, filters is also conceivable, by smoothly truncating \(h_{0},h_{K}\) instead of \(H_{0},H_{K}\). **Remark 1**: _The choice of raised cosine transition areas provides continuously differentiable \(P_{0},P_{K}\). If additional decay of \(h_{0},h_{K}\) is desired, the construction of a compactly supported plateau function of arbitrary differentiability is standard, e.g., through convolution of a characteristic function with a smooth function.