= Rectangular function
{title2=$\operatorname{rect}(t)=\mathbf1_{|t|<1/2}$}
{wiki}
= Rectangular pulse
{synonym}
The rectangular function has value one on $|t|<1/2$ and zero on $|t|>1/2$. Endpoint values are convention-dependent and do not affect its ordinary <integrals> or <Fourier transform>. Shifts, amplitude changes and time rescaling produce a <rectangular pulse> of arbitrary height, center and duration. With $\widehat f(k)=\int f(t)e^{-ikt}dt$, for $a>0$ the width-$2a$ pulse $f(t)=\mathbf1_{|t|<a}$ has <Fourier transform> $2\sin(ak)/k$, with value $2a$ at zero by a <limit>. Its <convolution> with itself is a rescaled <triangular function>, $(2a-|t|)$ on $|t|<2a$ and zero outside, because the <convolution> measures the overlap length of two intervals.
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