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

: $w(t) =\begin \ 1; & , t, \leq B \\ \ 0; & , t, >B \end$

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

Rec-STFT

: $X(t,f)=\int_^ x(\tau) e^ \, d\tau$

Inverse form

: x(t)=\int_^\infty X(t_1,f)e^ \, df\text t-B

Property

Rec-STFT has similar properties with Fourier transform
* Integration
(a)
: $\int_^\infty X(t, f)\, df = \int_^ x(\tau)\int_^\infty e^\, df \, d\tau = \int_^ x(\tau)\delta(\tau) \, d\tau=\begin \ x(0); & , t, < B \\ \ 0; & \text \end$

(b)
: \int_^\infty X(t, f)e^ \,df =\begin
\ x(v); & v-B
*Shifting property (shift along x-axis)

:: $\int_^ x(\tau+\tau_0) e^\, d\tau = X(t+\tau_0,f)e^$

*Modulation property (shift along ''y''-axis)

:\int_^ (\tau) e^d\tau = X(t,f-f_0)

*special input
#When $x(t)=\delta(t), X(t,f)=\begin \ 1; & , t, < B \\ \ 0; & \text \end$
#When $x(t)=1,X(t,f)=2B\operatorname(2Bf)e^$

*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

:: $\int_^\infty , X(t, f), ^2\, df = \int_^ , x(\tau), ^2\,d\tau$
:: $\int_^\infty \int_^\infty , X(t, f), ^2\,df\,dt = 2B \int_^\infty , x(\tau), ^2\,d\tau$

* Energy sum property (Parseval's theorem)

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

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.

See also

* Uncertainty principle

References

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

Fourier analysis
Time–frequency analysis
Transforms