Window Functions With Minimum-Sidelobe Derivatives for Computing Instantaneous Frequency

In the field of nonstationary signal analysis and processing, the short-time Fourier transform (STFT) is frequently used to convert signals into the time-frequency domain. The instantaneous frequency (IF) of the STFT is defined as the time derivative of the STFT phase and plays an important role in the reassignment method and the synchrosqueezing transform (SST). In this paper, we propose a framework to design a window function for computing the IF of STFT. Computing the IF requires the STFT with a window and STFT with its derivative, i.e., the IF computation depends on both the window function and its derivative. To design a window suitable for computing the IF, we formulate the window design problem as a sidelobe minimization problem of the corresponding derivative. Two windows are designed considering the sidelobe energy or the highest sidelobe level as cost functions for minimizing the sidelobes of their derivatives. The SST using the proposed window provides a sharper time-frequency representation compared to those produced using ordinary bandwidth-adjustable windows.