# Audlet Filter Banks: A Versatile Analysis/Synthesis Framework Using Auditory Frequency Scales
## 6 Conclusions  
A framework for the construction of oversampled perfect-reconstruction FBs with filters distributed on auditory frequency scales has been presented. This framework was motivated by auditory perception and targeted at audio signal processing; it has thus been named \"Audlet\". The proposed approach has its foundation in the mathematical theory of frames. The analysis FB design is directly performed in the frequency domain and allows for various filter shapes, and uniform or non-uniform settings with low redundancies. The synthesis is achieved using a (heuristic) preconditioned conjugate-gradient iterative algorithm. The convergence of the algorithm has been observed for Audlet FBs that constitute a frame. This is possible even for redundancies close \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**System**} & \multirow{2}{*}{\(R_{\text{t}}\)} & \multicolumn{2}{c}{**SDR**} & \multicolumn{2}{c}{**SAR**} & \multicolumn{2}{c}{**OPS**} & \multicolumn{2}{c}{**TPS**} \\ \cline{2-9}  & & \(\beta=1\) & **1/6** & **1** & **1/6** & **1** & **1/6** & **1** & **1/6** \\ \hline trev\_gfb & & 0.1 & 5.8 & 3.2 & 9.2 & 0.26 & 0.26 & 0.06 & 0.12 \\ Audlet\_gfb & 1.1 & 4.7 & 10.7 & 8.5 & 19.0 & 0.25 & 0.31 & 0.11 & 0.20 \\ Audlet\_hann & & 4.7 & 11.8 & 7.6 & 18.3 & 0.26 & 0.34 & 0.05 & 0.26 \\ STFT\_hann & & \(-1.7\) & & 0.5 & & 0.46 & & 0.02 \\ \hline trev\_gfb & & 2.4 & 8.5 & 5.7 & 13.5 & 0.24 & 0.30 & 0.11 & 0.17 \\ Audlet\_gfb & 1.5 & 6.9 & 11.1 & 12.5 & 20.5 & 0.24 & 0.35 & 0.13 & 0.29 \\ Audlet\_hann & & 7.0 & 12.8 & 11.1 & 20.1 & 0.22 & 0.36 & 0.07 & 0.35 \\ STFT\_hann & & 2.4 & & 9.2 & & 0.22 & & 0.04 \\ \hline trev\_gfb & & 7.0 & 10.7 & 12.0 & 18.9 & 0.24 & 0.37 & 0.24 & 0.34 \\ Audlet\_gfb & 4 & 9.0 & 11.4 & 18.3 & 21.6 & 0.27 & 0.38 & 0.32 & 0.39 \\ Audlet\_hann & & 11.1 & 13.1 & 19.4 & 21.7 & 0.25 & 0.37 & 0.21 & 0.32 \\ STFT\_hann & & 11.4 & & 20.5 & & 0.38 & & 0.34 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Objective quality measures for the separated voice signal. The signal-to-distortion ratio (SDR) and signal-to-artifact ratio (SAR) are in dB; the larger the ratio, the better the separation result. Overall perceptual score (OPS) and target perceptual score (TPS) are without unit; they indicate scores between 0 (bad quality) and 1 (excellent quality). The corresponding audio files are available on the companion webpage. STFT: short-time Fourier transform. to 1. For higher redundancies and filters with a compact support in the frequency domain, a so-called \"painless\" system can be achieved. In this case the exact dual FB can be calculated, which in turn results in a computationally more efficient synthesis. We showed how to construct a gammatone FB with perfect reconstruction. The proposed gammatone FB was compared to widely used state-of-the-art implementations of gammatone FB with approximate reconstruction. The results showed the better performance of the proposed approach in terms of reconstruction error and stability, especially at low redundancies. An example application of the framework to the task of audio source separation demonstrated its utility for audio processing. Overall, the Audlet framework provides a versatile and efficient FB design that is highly suitable for audio applications requiring stability, perfect reconstruction, and a flexible choice of redundancy. The framework is implemented in the free Matlab/Octave toolbox LTFAT [61; 62]. Supplementary material available online at www.mdpi.com/2076-3417/8/1/96/s1 is provided by the authors. Archive S1: Matlab functions and test audio files to perform Audlet analyses/syntheses in various FB settings and reproduce all results presented in the manuscript. The archive, about 2.6 MB in size, also includes a brief documentation. Figure S2: Frequency response and impulse response of two gammatone filters computed using the proposed framework. **Acknowledgments:** The authors would like to thank [PERSON] for insightful discussions and help on the theoretical development on non-uniform FBs.