Sinc Impulse
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 theorem (§B.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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Impulse Trains
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Rectangular Pulse