# Audlet Filter Banks: A Versatile Analysis/Synthesis Framework Using Auditory Frequency Scales
## 3 The Proposed Approach
### Synthesis Stage
```
Initialize\((H_{k},d_{k})_{k},(y_{k})_{k}\) \(x_{0}\in\ell_{2}(\mathbb{Z})\) (arbitrary) \(j=0\) and \(\varepsilon>0\) (error tolerance) \(\mathcal{H}_{0}\leftarrow\sum_{k}d_{k}^{-1}H_{k}\) for\(k=0,\ldots,K+1\)do \(G_{k}\gets H_{k}/\mathcal{H}_{0}\) endfor \(b\leftarrow\widetilde{\mathcal{S}}((y_{k})_{k},(G_{k},d_{k})_{k})\) \(r_{0}\gets b-\widetilde{\mathcal{S}}(\mathcal{A}(x_{0},(H_{k},d_{k})_{k }),(G_{k},d_{k})_{k})\) \(p_{0}\gets r_{0}\) while\(r_{j}>\varepsilon\)do \(q_{j}\leftarrow\widetilde{\mathcal{S}}(\mathcal{A}(p_{j},(H_{k},d_{k})_{k}),(G_{k },d_{k})_{k})\) \(a_{j}\leftarrow|r_{j}|^{2}/\langle p_{j},q_{j}\rangle\) \(x_{j+1}\gets x_{j}+a_{j}p_{j}\) \(r_{j+1}\gets r_{j}-a_{j}q_{j}\) \(b_{j}\leftarrow|r_{j+1}/r_{j}|^{2}\) \(p_{j+1}\gets r_{j+1}+b_{j}p_{j}\) \(j\gets j+1\) endwhile
```  
**Algorithm 1** Synthesis by means of conjugate gradients  
To speed up convergence we use _approximate dual filters_ as an initial choice for \(G_{k}\)'s,  
\[G_{k}(e^{2i\pi\xi}):=\frac{H_{k}(e^{2i\pi\xi})}{\mathcal{H}_{0}(\xi)}. \tag{20}\]  
We interpret \(G_{k}\)'s as approximate dual filters because in the absence of aliasing (i.e., if \(\mathcal{H}_{j}=0\), \(\forall j\
eq 0\)), the application of \(G_{k}\) exactly cancels all ripples in the frequency response \(\mathcal{H}_{0}\)Hence, the analysis-synthesis system \(\widetilde{\mathcal{S}}(\mathcal{A}(x,(H_{k},d_{k})_{k}),(G_{k},d_{k})_{k})\) can be interpreted as a preconditioned variant of \(\widetilde{\mathcal{S}}(\mathcal{A}(x,(H_{k},d_{k})_{k}),(H_{k},d_{k})_{k})\)[48; 57; 58]. Note that in the painless case, evoked in Section 3.2, the operator \(\widetilde{\mathcal{S}}(\mathcal{A}(x,(H_{k},d_{k})_{k}),(G_{k},d_{k})_{k})\) equals the identity and thus, synthesis is performed simply by applying \(\widetilde{\mathcal{S}}((y_{k})_{k},(G_{k},d_{k})_{k})\) once. Although this is not apparent from the iterative inversion scheme described above, the proposed synthesis stage acts in a similar fashion to an FB. More specifically, if \(D=\mathrm{lcm}(\{d_{k}\}_{k})\) and \((\tilde{H}_{j},D)_{j}\) is the equivalent uniform FB associated with \((H_{k},d_{k})_{k}\)[16; 20], then iterating the CG algorithm until convergence is equivalent to computing the FB synthesis with respect to the canonical dual FB of \((\tilde{H}_{j},D)_{j}\), which is of the form \((\tilde{G}_{j},D)_{j}\), for some sequences of filters \((\tilde{G}_{j})_{j}\) (see [21]). Since convergence is achieved within numerical precision in a small number of CG steps we can assume that the proposed synthesis system is characterized by the properties of the filters \((\tilde{G}_{j})_{j}\). We cannot easily compute those filters, but it is well known that the ratio of the optimal frame bounds \(B/A\) (see Section 3.2) is closely related to the similarity of a system and its canonical dual [59]. If \(B/A\approx 1\), then we can expect \(\tilde{G}_{j}\approx\tilde{H}_{j}\), for all \(j\). Since each \(\tilde{H}_{j}\) is just a delayed version of some \(H_{k}\), the time- and frequency-domain localization of the synthesis system matches that of the analysis system. For larger values of \(B/A\), the duality of \((\tilde{H}_{j},D)_{j}\) and \((\tilde{G}_{j},D)_{j}\) implies that \((\tilde{G}_{j},D)_{j}\) has to account for the discrepancies of \((\tilde{H}_{j},D)_{j}\)[59]. These considerations apply to any dual FB pair, Audlet or not. The Audlet FB is constructed in such a way that, given the prototype filter \(w\) and filter density \(V\), the frequency response of \((H_{k},d_{k})_{k}\) is as flat as possible, such that the \(B/A\) depends mostly on the presence of aliasing. The required aliasing compensation often implies a widening of the dual filters' essential support and essential passband, proportional to the amount of aliasing present.