Rectangular mask short-time Fourier transform

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In mathematics and Fourier analysis, a rectangular mask short-time Fourier transform (rec-STFT) 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 (a simple on/off time-limiting function) is used to isolate short time segments of the signal. Other types of the STFT may require more computation time ( refers to the amount of time it takes a computer or algorithm to perform a calculation or complete a task) than the rec-STFT.

Contents

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

B = 50, x-axis (sec) SquareWave.jpg
B = 50, x-axis (sec)

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

(a)

(b)

  1. When
  2. When

If ,and are their rec-STFTs, then

Example of tradeoff with different B

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.) DifferentB.JPG
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

Compared with the Fourier transform:

Compared with other types of time-frequency analysis:

See also

References

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