Non linear time frequency transforms with adaptive window

Time-frequency transforms are powerful signal processing tools. They enable us to analyse signals locally and extract key information, such as frequency or scaling components. The most popular transforms are the short-time Fourier transform (STFT) and the continuous wavelet transform (CWT). However, one issue with these transforms is that they have a fixed resolution; the time-frequency resolution is fully determined by the window function or the mother wavelet, regardless of the properties of the analysed signal. This can lead to a lack of precision when signals move from highly contained transient parts to highly contained harmonic parts, depending on the size of the window. This presentation introduces a solution to this problem: adaptive time-frequency transforms based on modifying the window function according to the analysed signal. We refer to this behaviour as the 'focus phenomenon' and it is linked to objects called 'focus functions'. We will define the adaptive transforms and introduce fundamental results such as the frame theorem, stability estimates, and focus function construction. We will also provide a few examples of applications of these transforms for audio processing.