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

(a)

\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)

\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}
\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}
\int_{t-B}^{t+B} [x(\tau) e^{j 2 \pi f_0 \tau}] d\tau = X(t,f-f_0)
  1. When x(t)=\delta(t), X(t,f)=\begin{cases} \ 1; & |t|< B \\ \ 0; & \text{otherwise} \end{cases}
  2. When x(t)=1,X(t,f)=2B\operatorname{sinc}(2Bf)e^{j 2 \pi f t}

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) .
\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
\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

Advantage The instantaneous frequency can be observed.

Disadvantage Higher complexity of computation.

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

References

  1. Jian-Jiun Ding (2014) Time-frequency analysis and wavelet transform
This article is issued from Wikipedia - version of the Thursday, January 07, 2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.