# Audlet Filter Banks: A Versatile Analysis/Synthesis Framework Using Auditory Frequency Scales
## 3 The Proposed Approach
### Analysis Filter Bank
#### 3.1.3 Construction of the Set \(\{d_{k}\}\)  
Downsampling the filters' outputs, i.e., using \(d_{k}>1\) for some or all \(k\in\{0,\ldots,K\}\), has the advantage of reducing \(R\) but introduces aliasing. The amount of aliasing can be determined from the frequency domain representation of the output signal \(\widehat{X}(z)=\sum_{k}\mathcal{Z}\left(g_{k}*\uparrow_{d_{k}}\{y_{k}\}\right)[n]\), also called the _alias domain_ representation [16; 17]. For \(\xi\in(-1/2;1/2]\), \(\widehat{X}(z)\) reduces to the following ([20] Section 4)  
\[\widehat{X}(e^{2i\pi\xi})=\frac{1}{D}[X(e^{2i\pi(\xi+0/D)})\cdots X(e^{2i\pi( \xi+(D-1)/D)})]\mathcal{H}_{j}(\xi) \tag{13}\]  
where \(D=\text{lcm}(\{d_{k}\}_{k})\) and  
\[\mathcal{H}_{j}(\xi):=\widehat{\mathcal{H}}_{j}(\xi)+\widehat{ \mathcal{H}_{j}(-\xi)}\quad\text{ with} \tag{14}\] \[\widehat{\mathcal{H}}_{j}(\xi)=\sum_{\begin{subarray}{c}k\in\{0,\ldots,K\},\\ \text{s.t.}\ j\in\frac{D}{k}\mathcal{Z}\\ \end{subarray}}d_{k}^{-1}H_{k}(e^{2i\pi\xi})\overline{H_{k}(e^{2i(\xi+j/D)})},\]  
for all \(j\in\{0,\ldots,D-1\}\). The term \(\mathcal{H}_{0}\) in (14) represents the frequency response of the FB, while the terms \(\mathcal{H}_{j},\,j\
eq 0\), represent the alias components. Thus, an alias-free system is obtained when \(\mathcal{H}_{0}=C>0\) and \(\mathcal{H}_{j}=0,\,\forall j\
eq 0\). While this is not always achievable, choosing \(d_{k}\)'s to be _inversely proportional to the filters' bandwidth_ yields a close-to-optimal solution [19], i.e.,  
\[d_{k}=\left\lfloor\frac{c_{bw}f_{s}}{\Gamma_{k}}\right\rfloor\text{ for }k=1,\ldots,K-1. \tag{15}\]  
For a targeted redundancy \(R_{\text{t}}\), combining (15) and (12) while disregarding the floor operator \(\lfloor\cdot\rfloor\) leads to  
\[c_{bw}=\frac{2}{R_{\text{t}}f_{s}}\sum_{k=1}^{K-1}\beta B_{\text{scale}}(f_{k }). \tag{16}\]  
Since the \(H_{k}\) values are strictly decaying away from \(f_{k}\) with a bandwidth of \(\Gamma_{k}\), choosing \(d_{k}\)'s according to (15) ensures an even distribution of the overall aliasing across channels.  
Figure 4: Illustration of the frequency allocations of the filters \(H_{0}\) (red line) and \(H_{K}\) (green line) given the restricted frequency response \(\mathcal{H}_{0}^{(\prime)}(\xi)\) (dashed line) of an FB.  
Using (15) to derive \(d_{0}\) and \(d_{K}\) may result in a large amount of aliasing because \(H_{0}\) and \(H_{K}\) may feature large plateaus depending on \(f_{\min}\) and \(f_{\max}\). We propose instead choosing \(d_{0}\) and \(d_{K}\) according to  
\[d_{0} = \left|\frac{f_{s}}{2f_{p,s}^{-}+\frac{\beta B_{\mathrm{scale}}(f_{p,s}^{-})}{c_{\mathrm{low}}}}\right| \tag{17}\] \[d_{K} = \left|\frac{f_{s}}{2(f_{N}-f_{p,s}^{+})+\frac{\beta B_{\mathrm{ scale}}(f_{p,s}^{-})}{c_{\mathrm{low}}}}\right|\,. \tag{18}\]  
Note that \(R_{\mathrm{t}}\) controls the \(d_{k}\) only for \(k=1,\ldots,K-1\), while the actual redundancy \(R\) depends on all \(d_{k}\), i.e., including \(d_{0}\) and \(d_{K}\). As a result, the value of \(R\) is slightly larger than \(R_{\mathrm{t}}\).