# Audlet Filter Banks: A Versatile Analysis/Synthesis Framework Using Auditory Frequency Scales
## 1 Introduction  
Time-frequency (TF) transforms like the short-time Fourier or wavelet transforms play a major role in audio signal processing. They allow any signal to be decomposed into a set of elementary functions with good TF localization and perfect reconstruction is achieved if the transform parameters are chosen appropriately (e.g., [1, 2]). The result of a signal analysis is a set of TF coefficients, sometimes called sub-band components, that quantifies the degree of similarity between the input signal and the elementary functions. In applications, TF transforms are used to perform sub-band processing, that is, to modify the sub-band components and synthesize an output signal. De-noising techniques [3, 4], for instance, analyze the noisy signal, estimate the TF coefficients associated with noise, deletethem from the set of TF coefficients, and synthesize a clean signal from the set of remaining TF coefficients. Lossy audio codecs like MPEG-2 Layer III, known as MP3 [5], or advanced audio coding (AAC) [6; 7] quantize the sub-bands with a variable precision in order to reduce the digital size of audio files. In audio transformations like time-stretching or pitch-shifting [8; 9], the phases of sub-band components are processed to ensure a proper phase coherence. As a last example, applications of audio source separation [10; 11; 12] or polyphonic transcriptions of music [13] rely on the non-negative matrix factorization scheme: the set of TF coefficients is factorized into several matrices that correspond to various sources present in the original signal. Each source can then be synthesized from its matrix representation. In these applications, the short-time Fourier transform (STFT) is mostly used, although modified discrete cosine transforms (MDCTs) are usually preferred in audio codecs. Because sub-band processing may introduce audible distortions in the reconstructed signal, important properties of the analysis-synthesis system include stability (i.e., the coefficients are bounded if and only if the input signal is bounded), perfect reconstruction (i.e., the reconstruction error is only limited by numerical precision when no sub-channel processing is performed), resistance to noise, and aliasing suppression in each sub-band (e.g., [14; 15] Chap. 10). Furthermore, in all applications, a low redundancy (i.e., a redundancy between 1 and 2) lowers the computational costs. TF transforms are usually implemented as filter banks (FBs) where the set of analysis filters defines the elementary functions and the set of synthesis filters allows signal reconstruction. The TF concentration of the filters together with the downsampling factors in the sub-bands define the TF resolution and redundancy of the transform. FBs come in various flavors and have been extensively treated in the literature (e.g., [16; 17; 18; 19]). The mathematical theory of frames constitutes an interesting alternative background for the interpretation and implementation of FBs (e.g., [20; 21; 22]). Gabor frames (sampled STFT [2; 23]), for instance, are widespread in audio signal processing. For certain applications, such as audio coding [5; 6; 7], audio equalizers [24], speech processing [25], perceptual sparsity [26; 27], or source separation [11; 12; 28; 29], exploiting some aspects of human auditory perception in the signal chain constitutes an advantage. One of the most exploited aspects of the auditory system is the auditory frequency scale, which is a simple means to approximate the frequency analysis performed in the auditory system [30]. Generally, the auditory system is a complex and in many aspects nonlinear system (for a review see, e.g., [31]). Its description ranges from simple collections of linear symmetric bandpass filters [32] through collections of asymmetric and compressive filters [33] to sophisticated models of nonlinear wave propagation in the cochlea [34].