Editing Rectangular function (section)

== Dirac delta function ==
The rectangle function can be used to represent the [[Dirac delta function]] <math>\delta (x)</math>.<ref name=":0">{{Cite book |last1=Khare |first1=Kedar |title=Fourier Optics and Computational Imaging |last2=Butola |first2=Mansi |last3=Rajora |first3=Sunaina |publisher=Springer |year=2023 |isbn=978-3-031-18353-9 |edition=2nd |pages=15–16 |chapter=Chapter 2.4 Sampling by Averaging, Distributions and Delta Function |doi=10.1007/978-3-031-18353-9}}</ref> Specifically,<math display="block">\delta (x) = \lim_{a \to 0} \frac{1}{a}\operatorname{rect}\left(\frac{x}{a}\right).</math>For a function <math>g(x)</math>, its average over the width ''<math>a</math>'' around 0 in the function domain is calculated as,

<math display="block">g_{avg}(0) = \frac{1}{a} \int\limits_{- \infty}^{\infty} dx\ g(x) \operatorname{rect}\left(\frac{x}{a}\right).</math>
To obtain <math>g(0)</math>, the following limit is applied,

<math display="block">g(0) = \lim_{a \to 0} \frac{1}{a} \int\limits_{- \infty}^{\infty} dx\ g(x) \operatorname{rect}\left(\frac{x}{a}\right)</math>
and this can be written in terms of the Dirac delta function as,
<math display="block">g(0) = \int\limits_{- \infty}^{\infty} dx\ g(x) \delta (x).</math>The Fourier transform of the Dirac delta function <math>\delta (t)</math> is

<math display="block">\delta (f)
= \int_{-\infty}^\infty \delta (t) \cdot e^{-i 2\pi f t} \, dt
= \lim_{a \to 0} \frac{1}{a} \int_{-\infty}^\infty \operatorname{rect}\left(\frac{t}{a}\right)\cdot e^{-i 2\pi f t} \, dt
= \lim_{a \to 0} \operatorname{sinc}{(a f)}.</math>
where the [[sinc function]] here is the normalized sinc function. Because the first zero of the sinc function is at <math>f = 1 / a</math> and <math>a</math> goes to infinity, the Fourier transform of <math>\delta (t)</math> is

<math display="block">\delta (f) = 1,</math>
means that the frequency spectrum of the Dirac delta function is infinitely broad. As a pulse is shorten in time, it is larger in spectrum.