Rectangular mask short-time Fourier transform

Concept

rectangular mask Short-time Fourier transform has the simple form of STFT. 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

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

Property

Rec-STFT has similar properties with Fourier transform

• Integration

(a)

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

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

• 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

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

• 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

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

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

Reference

1. Jian-Jiun Ding (2014) Time-Frequency Analysis and Wavelet Transform