# Audlet Filter Banks: A Versatile Analysis/Synthesis Framework Using Auditory Frequency Scales
## 5 Evaluation
### Construction of Perfect-Reconstruction Gammatone FBs  
#### 5.1.1 Method  
To construct a gammatone FB we use the prototype filter shape in the frequency domain of a complex gammatone filter of order \(\gamma\) centered at zero [42; 43]  
\[H_{GT,\gamma A}(e^{2i\pi\xi})=\left(1+i\alpha^{-1}\xi\right)^{-\gamma}. \tag{21}\]  
An order \(\gamma=4\) and bandwidth factor \(\alpha=1.019\) are usually chosen for emulating the human auditory filters [38]. Because \(H_{GT,\gamma A}\) has an infinite support in the frequency domain, it can be truncated to become a compactly-supported gammatone filter shape by  
\[w_{csGT,\gamma A}(\xi)=\begin{cases}H_{GT,\gamma A}(e^{2i\pi\xi})&\text{ if }|H_{GT,\gamma A}(e^{2i\pi\xi})|\geq\epsilon,\\ 0&\text{otherwise.}\end{cases} \tag{22}\]  
where \(\epsilon\) is a threshold that allows to trade accuracy for computational efficiency. Once an essential frequency range and a filter density are chosen, the set of gammatone filters is generated according to (7) using \(w(\xi)=H_{GT,\gamma A}(e^{2i\pi\xi})\) (or \(w=w_{csGT,\gamma A}\) if a painless system is desired) and \(\beta=1\). In Figure S2 in supplementary material, the frequency response and impulse response of two gammatone filters computed using (7) and (21) with center frequencies \(f_{k}=258\) and \(4000\) Hz are displayed.  
To examine the stability and reconstruction property of the proposed gammatone construction, we replicated the two simulations described in [44] (Section IV). The first simulation considers uniform FBs and the second simulation considers non-uniform FBs. The uniform FBs were evaluated by two measures: the ratio \(B/A\) and reconstruction error in terms of signal-to-noise ratio (SNR). The non-uniform FBs were evaluated only by the SNR. We compared our results to those from [PERSON] and Mertins (S-M) [44] where available.  
The FB settings were as follows. The sampling rate was \(f_{s}=44.1\) kHz, the essential frequency range was \([f_{\min}=20\text{ Hz},f_{\max}=20000\text{ Hz}]\), and the scale was ERB. The gammatone filters in [44] were implemented as FIR filters, that is, the \(H_{k}\)'s had an infinite frequency response. Thus, in the following simulations we used \(w(\xi)=H_{GT,4,1019}(e^{2i\pi\xi})\). In the uniform case, the downsampling factors \(d_{k}\)'s, \(k\in\{1,\dots,K-1\}\), were set to a constant \(D\); \(d_{0}\) and \(d_{K}\) were chosen according to (17) and (18), respectively. The evaluation was performed for all combinations of \(D\in\{1,2,4,6,8\}\) and \(K\in\{51,76,101,151\}\) (our \(K\) corresponds to \(M+1\) in [44]). For the synthesis stage, Algorithm 1 was used with an error tolerance \(\epsilon=10^{-9}\). The ratio \(B/A\) was calculated for the _full_ frequency range (i.e., from 0 to \(f_{N}\)) by iteratively computing the eigenvalues of the operator \(\mathcal{S}\) associated with the system \((H_{k},d_{k})_{k}\)[65]. The SNR was calculated as \(||x||^{2}/||x-\hat{x}||^{2}\) in dB for \(x\) being a Gaussian white noise with a length of 30,000 samples.  
In the non-uniform case, \(K\) was fixed to 51 and the FBs were evaluated for various values of \(R\). We considered the oversampling factors \(O\in\{1,2,4,6,8\}\) used in [44]. The relationship between \(O\) and \(R\) is \(O=R/2-\frac{1}{2}(d_{0}^{-1}+d_{K}^{-1})\) because in [44], \(O\) was \(\sum_{k=1}^{K-1}d_{k}^{-1}\), which considers only the real part of the coefficients, and \(h_{0}\) and \(h_{K}\) were not included. For simplicity, our FBs were designed for \(R_{t}\in\{2,4,8,12,16\}\). Similar to [44], two sets of \(d_{k}\) were used to achieve the various \(R_{t}\)'s. The first set consisted of \(d_{k}\)'s that were inversely proportional to the filters' bandwidth according to (15). The second set was exactly that mentioned in [44] (Appendix B). For each set, \(d_{0}\) and \(d_{k}\) were chosen according to (17) and (18), respectively. All other FB and signal parameters were as in the uniform case.