Rectangular mask short-time Fourier transform

In mathematics and Fourier analysis, a rectangular mask short-time Fourier transform (rec-STFT) has the simple form of short-time Fourier transform. Other types of the STFT may require more computation time than the rec-STFT.

The rectangular mask function can be defined for some bound (B) over time (t) as


 * $$ w(t) =\begin{cases}

\ 1; & |t|\leq B \\ \ 0; & |t|>B \end{cases}$$

We can change B for different tradeoffs between desired time resolution and frequency resolution.

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)
 * Integration
 * $$\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}$$
 * 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)


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

\ 1; & |t|< B \\ \ 0; & \text{otherwise} \end{cases}$$
 * special input
 * 1) When $$x(t)=\delta(t), X(t,f)=\begin{cases}
 * 1) 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
 * Linearity property


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


 * 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)


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

Example of tradeoff with different B
From the image, when B is smaller, the time resolution is better. Otherwise, when B is larger, the frequency resolution is better.

Advantage and disadvantage
Compared with the Fourier transform:


 * Advantage: The instantaneous frequency can be observed.
 * Disadvantage: Higher complexity of computation.

Compared with other types of time-frequency analysis:


 * Advantage: Least computation time for digital implementation.
 * Disadvantage: Quality is worse than other types of time-frequency analysis. The jump discontinuity of the edges of the rectangular mask results in Gibbs ringing artifacts in the frequency domain, which can be alleviated with smoother windows.