Rectangular function: Difference between revisions

Revision as of 09:15, 5 August 2023

Revision as of 19:38, 13 November 2023

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== Dirac delta function ==

The rectangle function can be used to represent the [[Dirac delta function]] <math>\delta (x)</math>.<ref name=":0">{{Cite book |last=Khare |first=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 \infty0} \frac{1}{a}\mathrm{rect}(\frac{x}{a}).</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) \mathrm{rect}(\frac{x}{a}).</math>

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<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.

==See also==

*[[Fourier transform]]