## Sinc Impulse

The preceding Fourier pair can be used to show that(B.35) |

*Proof:*The inverse Fourier transform of

**sinc**is

(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].

**Next Section:**

Impulse Trains

**Previous Section:**

Rectangular Pulse