Rectangular mask short-time Fourier transform

In mathematics and Fourier analysis, a rectangular mask short-time Fourier transform is a simplified form of the short-time Fourier transform which is used to analyze how a signal's frequency content changes over time. In rec-STFT, a rectangular window is used to isolate short time segments of the signal. Other types of the STFT may require more computation time than the rec-STFT.
The rectangular mask function can be defined for some bound over time as
We can change B'' for different tradeoffs between desired time resolution and frequency resolution.
Rec-STFT
Inverse form

Property

Rec-STFT has similar properties with Fourier transform

Integration
Shifting property
Modulation property
special input

When
When

Linearity property

If,and are their rec-STFTs, then

Power integration property
Energy sum property

Example of tradeoff with different B

From the image, when B is smaller, the time resolution is better. Otherwise, when B is larger, the frequency resolution is better.

Advantage and disadvantage

Compared with the Fourier transform:Advantage: The instantaneous frequency can be observed.Disadvantage: Higher complexity of computation.
Compared with other types of time-frequency analysis:Advantage: Least computation time for digital implementation.Disadvantage: Quality is worse than other types of time-frequency analysis. The jump discontinuity of the edges of the rectangular mask results in Gibbs ringing artifacts in the frequency domain, which can be alleviated with smoother windows.