Short-time fourier transform: two fundamental properties and an optimal implementation

TLDR

The study investigates shift and rotation invariance properties of linear time‑frequency representations. The authors extend the time‑bandwidth product to fractional Fourier domains, defining a generalized TBP, and then decompose the resulting GTBP‑optimal STFT via a linear canonical transform to relate it to the rotationally invariant STFT. They show that only the STFT family with Hermite‑Gaussian kernels satisfies both shift and rotation invariance, that the GTBP provides a rotation‑independent compactness metric for mono‑component signals, and that the GTBP‑optimal STFT minimizes the increase in GTBP, mirroring the TBP‑optimal STFT.

Abstract

Shift and rotation invariance properties of linear time-frequency representations are investigated. It is shown that among all linear time-frequency representations, only the short-time Fourier transform (STFT) family with the Hermite-Gaussian kernels satisfies both the shift invariance and rotation invariance properties that are satisfied by the Wigner distribution (WD). By extending the time-bandwidth product (TBP) concept to fractional Fourier domains, a generalized time-bandwidth product (GTBP) is defined. For mono-component signals, it is shown that GTBP provides a rotation independent measure of compactness. Similar to the TBP optimal STFT, the GTBP optimal STFT that causes the least amount of increase in the GTBP of the signal is obtained. Finally, a linear canonical decomposition of the obtained GTBP optimal STFT analysis is presented to identify its relation to the rotationally invariant STFT.