Differentiable adaptive short-time Fourier transform with respect to the window length

Fourier theory is a crucial aspect of signal processing, widely used in science and engineering. The short-time Fourier transform (STFT), also known as the windowed Fourier transform, plays a vital role in analyzing non-stationary signals with time-varying spectral content. Spectrograms, derived from the STFT magnitude, are commonly used for visualizing and processing non-stationary signals. The STFT window length is a critical parameter that determines the trade-off between temporal and frequency resolution, and several post-processing techniques have been developed to improve spectrogram readability, including synchrosqueezing Thakur et al. [2013] and reassignment Auger and Flandrin [1995]. Some researchers have proposed finding the optimal window length based on a given criterion Meignen et al. [2020], Jablonski and Dziedziech [2022], while others have recently proposed a differentiable version of STFT with respect to the window lengthLeiber et al. [2022a,b], Zhao et al. [2021], allowing for the optimization of the criterion using a gradient descent algorithm instead of grid search. Actually, the best window length depends on the signal itself and more particularly on its frequency content. It must therefore adapt to the time-varying spectral structure of the signal. Enhanced versions of STFT are then proposed to set the window length according to the local characteristics of the input signal.