Unit rectangle

The unit rectangle is a pulse of height 1 and width 1, centered at the origin:

$\mathrm{rect}(t)=\{\begin{array}{ll}1,& |t|<1/2\\ 1/2,& |t|=1/2\\ 0,& |t|>1/2.\end{array}$

Image: Rectangular function, CC BY-SA 3.0

Equivalently, $\mathrm{rect}(t)=u(t+1/2)-u(t-1/2)$, a unit step that turns on at $t=-1/2$ and off at $t=+1/2$. The values at the boundaries $|t|=1/2$ are a convention and rarely matter (they affect only a measure-zero set in any integral).

Multiplying a signal by $\mathrm{rect}(t)$ gates it: the product equals the signal on $(-1/2,+1/2)$ and is zero outside. By scaling and shifting we can place the rect window anywhere.

For the general windowing function with arbitrary endpoints, see Rectangular window function.

Reading rect(t/w)

This is the subtlety that bites students later. The expression $\mathrm{rect}(t/w)$ does not mean “rectangle of width $w$” in the direct sense. It means “rectangle that is $1$ when $|t/w|<1/2$ and $0$ when $|t/w|>1/2$.” Solving $|t/w|<1/2$ gives $|t|<w/2$, so $\mathrm{rect}(t/w)$ is a rectangle of width $w$ centered at the origin, going from $t=-w/2$ to $t=+w/2$.

When you see $\mathrm{rect}(t/w)$, mentally read off the boundaries as $\pm w/2$, not $\pm w$. Same for $\mathrm{rect}((t-t_0)/w)$: centered at $t=t_0$, with edges at $t_0\pm w/2$. This rule matters constantly in Fourier series and transform problems.

A consequence: $\mathrm{rect}(2t)$ has $|2t|<1/2⟺|t|<1/4$, so it’s a rectangle of width $1/2$ centered at zero. Width $=$ denominator (for $\mathrm{rect}(t/w)$) or width $=1/\text{multiplier}$ (for $\mathrm{rect}(at)$).

Fourier transform

The rectangle and the sinc form one of the transform pairs that comes up constantly:

$\mathrm{rect}(t)\stackrel{\mathcal{F}}{\leftrightarrow }\mathrm{sinc}(f),\mathrm{rect}(t/w)\stackrel{\mathcal{F}}{\leftrightarrow }w\mathrm{sinc}(wf).$

A narrow rect (small $w$) has a wide sinc spectrum; a wide rect (large $w$) has a narrow sinc spectrum. This is the time-bandwidth tradeoff.

Convolution with itself

$\mathrm{rect}(t)\ast \mathrm{rect}(t)=\mathrm{tri}(t),$

where $\mathrm{tri}(t)$ is the unit triangle, a triangle of height 1 and base 2 centered at the origin. By the convolution theorem, the Fourier transform of this identity gives ${\mathrm{sinc}}^2(f)$ as the spectrum of the triangle.