Editing Dirac delta function (section)

===Translation===
The integral of any function multiplied by the time-delayed Dirac delta <math> \delta_T(t) {=} \delta(t{-}T)</math> is

<math display="block">\int_{-\infty}^\infty f(t) \,\delta(t-T)\,dt = f(T).</math>

This is sometimes referred to as the ''sifting property''<ref>{{MathWorld|urlname=SiftingProperty|title=Sifting Property}}</ref> or the ''sampling property''.<ref>{{Cite book|last=Karris|first=Steven T.|url={{google books |plainurl=y |id=f0RdM1zv_dkC}}| title=Signals and Systems with MATLAB Applications|date=2003|publisher=Orchard Publications|isbn=978-0-9709511-6-8|language=en| page=[{{google books |plainurl=y |id=f0RdM1zv_dkC&pg=SA1-PA15 }} 15]}}</ref> The delta function is said to "sift out" the value of ''f(t)'' at ''t'' = ''T''.<ref>{{Cite book|last=Roden|first=Martin S.|url={{google books |plainurl=y |id=YEKeBQAAQBAJ}}|title=Introduction to Communication Theory|date=2014-05-17|publisher=Elsevier|isbn=978-1-4831-4556-3|language=en|page=[{{google books |plainurl=y |id=YEKeBQAAQBAJ|page=40}}]}}</ref>

It follows that the effect of [[Convolution|convolving]] a function {{math|''f''(''t'')}} with the time-delayed Dirac delta  is to time-delay {{math|''f''(''t'')}} by the same amount:<ref>{{Cite book|last1=Rottwitt|first1=Karsten|url={{google books |plainurl=y |id=G1jSBQAAQBAJ}}|title=Nonlinear Optics: Principles and Applications|last2=Tidemand-Lichtenberg|first2=Peter| date=2014-12-11| publisher=CRC Press|isbn=978-1-4665-6583-8|language=en|page=[{{google books |plainurl=y |id=G1jSBQAAQBAJ|page=276}}] 276}}</ref>

<math display="block">\begin{align}
(f * \delta_T)(t) \ &\stackrel{\mathrm{def}}{=}\  \int_{-\infty}^\infty f(\tau)\, \delta(t-T-\tau) \, d\tau \\
&= \int_{-\infty}^\infty f(\tau)  \,\delta(\tau-(t-T)) \,d\tau \qquad \text{since}~ \delta(-x) = \delta(x) ~~ \text{by (4)}\\
&= f(t-T).
\end{align}</math>

The sifting property holds under the precise condition that {{mvar|f}} be a [[Distribution (mathematics)#Tempered distributions and Fourier transform|tempered distribution]] (see the discussion of the Fourier transform [[#Fourier transform|below]]). As a special case, for instance, we have the identity (understood in the distribution sense)

<math display="block">\int_{-\infty}^\infty \delta (\xi-x) \delta(x-\eta) \,dx = \delta(\eta-\xi).</math>