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Rectangular mask short-time Fourier transform

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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

Property

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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 B

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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 disadvantage

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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 also

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References

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