## SincImpulse

The preceding Fourier pair can be used to show that

 (B.35)

Proof: The inverse Fourier transform of sinc is

In particular, in the middle of the rectangular pulse at , we have

 (B.36)

This establishes that the algebraic area under sinc is 1 for every . Every delta function (impulse) must have this property.

We now show that sinc also satisfies the sifting property in the limit as . This property fully establishes the limit as a valid impulse. That is, an impulse is any function having the property that

 (B.37)

for every continuous function . In the present case, we need to show, specifically, that

 (B.38)

Define sinc . Then by the power theoremB.9),

 (B.39)

Then as , the limit converges to the algebraic area under , which is as desired:

 (B.40)

We have thus established that

 (B.41)

where

 sinc (B.42)

For related discussion, see [36, p. 127].

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