Differentiable short-time Fourier transform with respect to the hop length

The short-time Fourier transform (STFT) is a frequently used tool for analyzing non-stationary digital signals in various fields including audio Stafford et al. [1998], medicine Huang et al. [2019], and vibration analysis Leclère et al. [2016]. Spectrograms, which are obtained from the STFT magnitude, are essential for visualizing, understanding, and processing non-stationary signals in time-frequency representation. The STFT parameters, including tapering function, window length, and hop length, are critical and dependent on the application and signal characteristics. The tapering function balances frequency resolution and spectral leakage, with a narrower main lobe providing better frequency resolution at the expense of increased spectral leakage, and a wider main lobe reducing spectral leakage but decreasing frequency resolution. The Hann or Hamming window is a common starting point, but the best choice depends on the application's specific requirements. Actually, most studies on STFT parameters have focused on the choice of the window length, as it determines the time-frequency resolution trade-off. A shorter window length provides better time resolution but poor frequency resolution. Conversely, a longer window length provides better frequency resolution but poor time resolution. To provide more precise control over temporal and frequency resolution based on the local characteristics of the input signal, researchers have proposed using variable-length windows.