Rectangular window function

The rectangular window function is a function that equals $1$ on an interval $[a,b]$ and $0$ everywhere else. It’s built from two Heaviside step functions:

${\Pi }_{a,b}(t):=u(t-a)-u(t-b)=\{\begin{array}{ll}0& t<a\\ 1& a<t<b\\ 0& t>b\end{array}$

Sometimes written $\text{rect}((t-c)/w)$ where $c=(a+b)/2$ is the center and $w=b-a$ is the width.

Why “window”

It “windows” — gates — a function. Multiplying $g(t)$ by ${\Pi }_{a,b}(t)$ keeps $g$ inside $[a,b]$ and zeroes it elsewhere:

$g(t)\cdot {\Pi }_{a,b}(t)=\{\begin{array}{ll}g(t)& a<t<b\\ 0& \text{otherwise}\end{array}$

This lets you express a piecewise function as a sum of pieces, each multiplied by its window.

Laplace transform

For $0<a<b$:

$\mathcal{L}\{{\Pi }_{a,b}\}(s)=\mathcal{L}\{u(t-a)-u(t-b)\}=\frac{e^{-as}-e^{-bs}}{s}$

(Subtraction is linear; each piece transforms via the standard Heaviside formula.)

Composing piecewise functions

A piecewise function with pieces $f_0,f_1,f_2,\dots$ on intervals $(-\infty ,a_1),(a_1,a_2),(a_2,a_3),\dots$ can be written:

$f(t)=f_0(t){\Pi }_{-\infty ,a_1}(t)+f_1(t){\Pi }_{a_1,a_2}(t)+f_2(t){\Pi }_{a_2,a_3}(t)+\cdots$

Each window picks out one piece. The $-\infty$ window can be replaced by $u(a_1-t)$ (left-going step) if needed for the formal definition.

For functions defined only on $t\ge 0$ (the natural domain for Laplace problems), you can simplify:

$f(t)=f_0(t){\Pi }_{0,a_1}(t)+f_1(t){\Pi }_{a_1,a_2}(t)+\cdots +f_n(t)u(t-a_n)$

The last piece is $f_n(t)u(t-a_n)$ since it’s “on” forever after $a_n$.

Example

The function

$f(t)=\{\begin{array}{ll}\sin t& 0\le t\le \pi /4\\ \sin t+\cos (t-\pi /4)& t>\pi /4\end{array}$

can be written as

$f(t)=\sin t\cdot {\Pi }_{0,\pi /4}(t)+(\sin t+\cos (t-\pi /4))\cdot u(t-\pi /4)$

Or simplified by recognizing that $\sin t$ is “on” both before and after $\pi /4$:

$f(t)=\sin t+\cos (t-\pi /4)\cdot u(t-\pi /4)$

The window only adds $\cos (t-\pi /4)$ starting at $t=\pi /4$.

Taking Laplace:

$F(s)=\mathcal{L}\{\sin t\}+\mathcal{L}\{\cos (t-\pi /4)u(t-\pi /4)\}$

The first term is $1/(s^2+1)$. The second uses the second shifting theorem:

$\mathcal{L}\{\cos (t-\pi /4)u(t-\pi /4)\}=e^{-\pi s/4}\cdot \mathcal{L}\{\cos t\}=\frac{s{e}^{-\pi s/4}}{s^2+1}$

So $F(s)=\frac{1}{s^2+1}+\frac{s{e}^{-\pi s/4}}{s^2+1}=\frac{1+s{e}^{-\pi s/4}}{s^2+1}$.

Use cases

The rectangular window appears in:

• Piecewise-defined ODE forcing functions — discontinuous inputs can be written as a sum of windowed pieces, then Laplace-transformed.
• Signal processing — windowing a signal to look at it over a finite interval.
• Pulse description — a finite-duration rectangular pulse is exactly ${\Pi }_{a,b}(t)$ scaled by amplitude.

For the building-block step function, see Heaviside step function. For the limit case (an infinitely narrow window with infinite amplitude), see Dirac delta function.