As importance of the phase of complex spectrogram has been recognized widely, many techniques have been proposed for handling it. However, several definitions and terminologies for the same concept can be found in the literature, which has confused beginners. In this paper, two major definitions of the short-time Fourier transform and their phase conventions are summarized to alleviate such complication. A phase-aware signal-processing scheme based on phase conversion is also introduced with a set of executable MATLAB functions (https://doi.org/10/c3qb).
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.INDEX TERMS Short-time Fourier transform (STFT), instantaneous frequency (IF), window derivative, adjustable window, synchrosqueezing transform (SST).
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