# Audlet Filter Banks: A Versatile Analysis/Synthesis Framework Using Auditory Frequency Scales
## 5 Evaluation
### Construction of Perfect-Reconstruction Gammatone FBs
#### 5.1.2 Results and Discussion  
The ratios \(B/A\) computed for the uniform gammatone FBs for various combinations of \(D\) and \(K\) and those reported in [44] (Figure 5) are listed in Table 1. For \(K\) = 51-101, our ratios \(B/A\) decreased with increasing \(K\). This is a consequence of the increasing overlap between filters with increasing \(K\), which in turn yields a flatter FB response. Increasing \(K\) to 151 did not result in smaller ratios. This can be attributed to the steep flank of \(H_{K}\) in that setting. This can be counteracted by increasing the values of \(\kappa_{1}\) and \(\kappa_{2}\) when very small filter spacing \(V\) (equivalently, large \(K\)) is used. Our framework generally achieved comparable or smaller ratios than those from [44]. Note that in [44], \(B/A\) was calculated for the frequency range from 0.06 to 17 kHz. These ratios, when calculated for the full frequency range, might have been larger than those listed in Table 1. Consequently, the actual difference between Audlet and S-M ratios might be larger than that reflected in Table 1.  
The SNRs achieved with our framework were 180 dB (or larger) for all tested combinations of \(D\) and \(K\). The limit of 180 dB is the consequence of the error tolerance of \(10^{-9}\) in the PCG algorithm. In comparison, SNRs reported in [44] for \(D\) = 1 ranged between 30 and 72 dB and increased with increasing \(K\) (SNRs for other \(D\)'s were not reported). The SNRs computed for the non-uniform gammatone FBs for various \(R\) are listed in Table 2 together with those reported in [44]. In all conditions, our framework achieved SNRs of at least 170 dB. In contrast, the system from [44] offered decent reconstruction (SNR \(\geq\) 15 dB) only in configurations involving small downsampling factors (i.e., at large \(R\)). Overall, we conclude that the reconstruction quality of currently available gammatone FB implementations deteriorates at low redundancies. This may hinder the quality of sub-channel \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(K\) & **Framework** & \(D\) = 1 & \(D\) = 2 & \(D\) = 4 & \(D\) = 6 & \(D\) = 8 \\ \hline \multirow{2}{*}{51} & Audlet & 1.124 & 1.124 & 1.125 & 1.134 & 1.157 \\  & S–M & 1.100 & \(>10\) & \(>10\) & \(>10\) & \(>10\) \\ \hline \multirow{2}{*}{76} & Audlet & 1.007 & 1.007 & 1.009 & 1.021 & 1.073 \\  & S–M & 1.100 & 2 & 2 & 3 & 6 \\ \hline \multirow{2}{*}{101} & Audlet & 1.003 & 1.003 & 1.005 & 1.017 & 1.068 \\  & S–M & 1.003 & 1.003 & 1.003 & 2 & 4 \\ \hline \multirow{2}{*}{151} & [PERSON] & 1.015 & 1.015 & 1.016 & 1.025 & 1.066 \\  & S–M & 1.003 & 1.003 & 1.100 & 2 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Ratios \(B/A\) for various combinations of \(D\) and \(K\) obtained for the proposed Audlet framework and reported in [44] (S–M). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{\(d_{k}\)**Based on (15)–(18)} & \multicolumn{2}{c}{\(d_{k}\)**from [44]**} \\ \hline \(R_{t}\) & \(R\) & [PERSON] & S–M & \(R\) & [PERSON] & S–M \\ \hline
2 & 2.40 & \(>180\) & 5 & 2.38 & \(>170\) & 10 \\
4 & 4.46 & \(>180\) & 7 & 4.38 & \(>190\) & 13 \\
8 & 8.60 & \(>180\) & 10 & 8.38 & \(>200\) & 17 \\
12 & 12.73 & \(>220\) & 9 & 12.38 & \(>210\) & 18 \\
16 processing in audio applications but, as it seems, the reconstruction quality can be improved by using the Audlet framework. It might appear intriguing that we obtained larger SNRs than in [44] even in conditions with similar ratios \(B/A\) (compare the condition with \(K=151\) and \(D=1\) in Table 1). The good performance achieved by our framework can mostly be explained by the design of our synthesis stage. In contrast, most analysis-synthesis systems based on gammatone filters, such as [44], use synthesis filters that are time-reversed versions of the analysis filters, i.e., \(G_{k}(e^{2i\pi\xi})=\overline{H_{k}}(e^{2i\pi\xi})\) that translates to \(g_{k}[n]=\overline{h_{k}}[-n]\) in the discrete-time domain (e.g., [35; 39; 40]).