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

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

(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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