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

w(t)={\begin{cases}\ 1;&|t|\leq B\\\ 0;&|t|>B\end{cases}}

B = 50, x-axis (sec)

We can change B for different signal.

Rec-STFT

X(t,f)=\int _{{t-B}}^{{t+B}}x(\tau )e^{{-j2\pi f\tau }}\,d\tau

Inverse form

x(t)=\int _{{-\infty }}^{\infty }X(t_{1},f)e^{{j2\pi ft}}\,df{\text{ where }}t-B<t_{1}<t+B

Property

Rec-STFT has similar properties with Fourier transform

Integration

(a)

\int _{{-\infty }}^{\infty }X(t,f)\,df=\int _{{t-B}}^{{t+B}}x(\tau )\int _{{-\infty }}^{\infty }e^{{-j2\pi f\tau }}\,df\,d\tau =\int _{{t-B}}^{{t+B}}x(\tau )\delta (\tau )\,d\tau ={\begin{cases}\ x(0);&|t|<B\\\ 0;&{\text{otherwise}}\end{cases}}

(b)

\int _{{-\infty }}^{\infty }X(t,f)e^{{-j2\pi fv}}\,df={\begin{cases}\ x(v);&v-B<t<v+B\\\ 0;&{\text{otherwise}}\end{cases}}

Shifting property(shift along x-axis)

\int _{{t-B}}^{{t+B}}x(\tau +\tau _{0})e^{{-j2\pi f\tau }}\,d\tau =X(t+\tau _{0},f)e^{{j2\pi f\tau _{0}}}

Modulation property (shift along y-axis)

\int _{{t-B}}^{{t+B}}[x(\tau )e^{{j2\pi f_{0}\tau }}]d\tau =X(t,f-f_{0})

special input

When $x(t)=\delta (t),X(t,f)={\begin{cases}\ 1;&|t|<B\\\ 0;&{\text{otherwise}}\end{cases}}$
When $x(t)=1,X(t,f)=2B\operatorname {sinc}(2Bf)e^{{j2\pi ft}}$

Linearity property

If $h(t)=\alpha x(t)+\beta y(t)\,$ , $H(t,f),X(t,f),$ and $Y(t,f)\,$ are their rec-STFTs, then

H(t,f)=\alpha X(t,f)+\beta Y(t,f).

Power integration property

\int _{{-\infty }}^{\infty }|X(t,f)|^{2}\,df=\int _{{t-B}}^{{t+B}}|x(\tau )|^{2}\,d\tau

\int _{{-\infty }}^{\infty }\int _{{-\infty }}^{\infty }|X(t,f)|^{2}\,df\,dt=2B\int _{{-\infty }}^{\infty }|x(\tau )|^{2}\,d\tau

Energy sum property(Parseval's theorem)

\int _{{-\infty }}^{\infty }X(t,f)Y^{*}(t,f)\,df=\int _{{t-B}}^{{t+B}}x(\tau )y^{*}(\tau )\,d\tau

\int _{{-\infty }}^{\infty }\int _{{-\infty }}^{{\infty }}X(t,f)Y^{*}(t,f)\,df\,dt=2B\int _{{-\infty }}^{\infty }x(\tau )y^{*}(\tau )\,d\tau

Rectangular mask B's effect

comparison of different B

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

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

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Rectangular mask short-time Fourier transform

Property

Rectangular mask B's effect

Advantage and disadvantage

See also

References