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

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

File:SquareWave.jpg

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

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 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)

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \int_{-\infty}^\infty X(t, f)\, df = \int_{t-B}^{t+B} x(\tau)\int_{-\infty}^\infty e^{-j 2 \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)

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \int_{-\infty}^\infty X(t, f)e^{-j 2 \pi f v} \,df =\begin{cases} \ x(v); & v-B<t< v+B \\ \ 0; & \text{otherwise} \end{cases}

• Shifting property(shift along x-axis)

$\int_{t-B}^{t+B} x(\tau+\tau_0) e^{-j 2 \pi f \tau}\, d\tau = X(t+\tau_0,f)e^{j 2 \pi f \tau_0}$

• Modulation property (shift along y-axis)

$\int_{t-B}^{t+B} [x(\tau) e^{j 2 \pi f_0 \tau}] d\tau = X(t,f-f_0)$

• special input

1. When Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): x(t)=\delta(t), X(t,f)=\begin{cases} \ 1; & |t|< B \\ \ 0; & \text{otherwise} \end{cases}

1. When $x(t)=1,X(t,f)=2B\operatorname{sinc}(2Bf)e^{j 2 \pi f t}$

• 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

File:DifferentB.JPG

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.

See also

• Uncertainty principle

References

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