Rectangular mask short-time Fourier transform

Summary

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In mathematics and Fourier analysis, a rectangular mask short-time Fourier transform (rec-STFT) has the simple form of short-time Fourier transform. Other types of the STFT may require more computation time than the rec-STFT.

The rectangular mask function can be defined for some bound (B) over time (t) as

B = 50, x-axis (sec)

We can change B for different tradeoffs between desired time resolution and frequency resolution.

Rec-STFT

Inverse form

PropertyEdit

Rec-STFT has similar properties with Fourier transform

  • Integration

(a)

 

(b)

 
  • Shifting property (shift along x-axis)
 
  • Modulation property (shift along y-axis)
 
  • special input
  1. When  
  2. When  
  • Linearity property

If  , and  are their rec-STFTs, then

 
  • Power integration property
 
 
 
 

Example of tradeoff with different BEdit

 
Spectrograms produced from applying a rec-STFT on a function consisting of 3 consecutive cosine waves. (top spectrogram uses smaller B of 0.5, middle uses B of 1, and bottom uses larger B of 2.)

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

Advantage and disadvantageEdit

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.

See alsoEdit

ReferencesEdit

  1. Jian-Jiun Ding (2014) Time-frequency analysis and wavelet transform