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Rectangular mask short time Fourier transform

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Rectangular mask short-time Fourier transform

In mathematics, a rectangular mask short-time Fourier transform has the simple form of short-time Fourier transform. Other types of the STFT may require more computation time than the rec-STFT. Define its mask function

Contents

w ( t ) = {   1 ; | t | ≤ B   0 ; | t | > B

We can change B for different signal.

Rec-STFT

X ( t , f ) = ∫ t − B t + B x ( τ ) e − j 2 π f τ d τ

Inverse form

x ( t ) = ∫ − ∞ ∞ X ( t 1 , f ) e j 2 π f t d f  where  t − B < t 1 < t + B

Property

Rec-STFT has similar properties with Fourier transform

  • Integration

    • (a)

    ∫ − ∞ ∞ X ( t , f ) d f = ∫ t − B t + B x ( τ ) ∫ − ∞ ∞ e − j 2 π f τ d f d τ = ∫ t − B t + B x ( τ ) δ ( τ ) d τ = {   x ( 0 ) ; | t | < B   0 ; otherwise

    (b)

    ∫ − ∞ ∞ X ( t , f ) e − j 2 π f v d f = {   x ( v ) ; v − B < t < v + B   0 ; otherwise
  • Shifting property(shift along x-axis)
  • Modulation property (shift along y-axis)
    • ∫ t − B t + B [ x ( τ ) e j 2 π f 0 τ ] d τ = X ( t , f − f 0 )
  • special input
    1. When x ( t ) = δ ( t ) , X ( t , f ) = {   1 ; | t | < B   0 ; otherwise
    2. When x ( t ) = 1 , X ( t , f ) = 2 B sinc ⁡ ( 2 B f ) e j 2 π f t
  • Linearity property

    • If h ( t ) = α x ( t ) + β y ( t ) , H ( t , f ) , X ( t , f ) , and Y ( t , f ) are their rec-STFTs, then

    H ( t , f ) = α X ( t , f ) + β Y ( t , f ) .
  • Power integration property
  • Energy sum property(Parseval's theorem)

    • Rectangular mask B's effect

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

    We can choose specified B to decide time resolution and frequency resolution.

    Advantage and disadvantage

  • Compare with the Fourier transform

    • Advantage The instantaneous frequency can be observed.

    Disadvantage Higher complexity of computation.

  • Compared with other types of time-frequency analysis:

    • The rec-STFT has an advantage of the least computation time for digital implementation, but its performance is worse than other types of time-frequency analysis.

    References

    Rectangular mask short-time Fourier transform Wikipedia
    (Text) CC BY-SA


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