Window function

A function used to restrict consideration of an arbitrary function or signal in some way. The terms time-frequency localization, time localization or frequency localization are often used in this context. For instance, the windowed Fourier transform is given by

\begin{equation*} ( F _ { \text{win} } f ) ( \omega , t ) = \int f ( s ) g ( s - t ) e ^ { - i \omega s } d s, \end{equation*}

where $g ( t )$ is a suitable window function. Quite often, scaled and translated versions of $g ( t )$ are considered at the same time, [a1], [a3]. An example is the Gabor transform. (See also Balian–Low theorem; Calderón-type reproducing formula.) Such window functions are also used in numerical analysis.

More specifically, the phrase window function refers to the function $r ( t )$ that equals $1$ on the interval $( - 1,1 )$ and zero elsewhere (at $- 1$ and $+ 1$ it is arbitrarily defined, usually $1/2$ or $0$). This function, as well as its scaled and translated versions, is also called the rectangle function or pulse function [a2], pp. 30, 35, 60, 61. However, the phrase "pulse function" is also sometimes used for the delta-function, see also Transfer function.

The Fourier transform of the specific rectangle function $r ( t )$ (with $r ( \pm 1 ) = 1 / 2$) is the function

\begin{equation*} g ( y ) = \left\{ \begin{array} { l l } { \frac { 1 } { \pi y } \operatorname { sin } 2 \pi y , } & { y \neq 0, } \\ { 2 , } & { y = 0, } \end{array} \right. \end{equation*}

a version of the sinc function ($\operatorname{sinc}( 0 ) = 1$, $\operatorname { sinc } ( x ) = x ^ { - 1 } \operatorname { sin } x$ for $x \neq 0$), see [a2], pp. 61, 104. In terms of the Heaviside function $H ( x )$ ($H ( x ) = 0$ for $x < 0$, $H ( 0 ) = 1 / 2$, $H ( x ) = 1$ for $x > 0$), $r ( x )$ is given by

\begin{equation*} r ( x ) = H ( x + 1 ) - H ( x - 1 ). \end{equation*}

There is also a relation with the Dirac delta-function $\delta ( x )$:

\begin{equation*} \operatorname { lim } _ { n \rightarrow \infty } \frac { n } { 2 } r ( n x ) = \delta ( x ). \end{equation*}

References

[a1]	I. Daubechies, "Ten lectures on wavelets" , SIAM (1992) pp. Chap. 1
[a2]	D.C. Champeney, "A handbook of Fourier transforms" , Cambridge Univ. Press (1989)
[a3]	A.I. Saichev, W.A. Woyczyński, "Distributions in the physical and engineering sciences" , 1: Distribution and fractal calculus, integral transforms and wavelets , Birkhäuser (1997) pp. 195ff