Hello,(adsbygoogle = window.adsbygoogle || []).push({});

I'd like to prove the orthogonality of two "shifted" Sinc functions, but I can't find the mistake.

Here is my attempt:

[tex]\int_{-\infty}^{+\infty}sinc(x)sinc(x-x_0)dx[/tex]

Observing this quantity can be obtained by evaluating the Fourier transform at zero, we have:

[tex]\mathcal{F}\{ sinc(x)sinc(x-x_0) \}(0)[/tex]

and using the convolution theorem and the shift theorem (for the second sinc), we get:

[tex](rect(\omega)\otimes e^{-i2\pi\omega x_0}rect(\omega))(0) = [/tex]

[tex]= \int_{-1/2}^{+1/2} e^{-i2\pi\omega x_0}d\omega = [/tex]

[tex]= \left[ \frac{sin(2\pi\omega x_0)}{2\pi x_0} \right]_{-1/2}^{1/2} = [/tex]

[tex]= \frac{sin(\pi x_0)}{\pi x_0} = [/tex]

[tex]= sinc(x_0) [/tex]

Now, this quantity is 0iff[itex]x_0[/itex] is a non-zero integer!

Is this the correct result?

Aren't two sinc functions supposed to be orthogonal for any [itex]x_0[/itex] real?

**Physics Forums - The Fusion of Science and Community**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Sinc functions orthogonality

Loading...

Similar Threads - Sinc functions orthogonality | Date |
---|---|

Projection of a Jinc is a Sinc | Apr 5, 2015 |

Proof that sinc function is not elementary? | Dec 7, 2013 |

How to integrate a sinc function? | Sep 8, 2011 |

Convolution with a sinc gives uniform approximation to a function | Dec 30, 2005 |

2D Integral, Gaussian and 2 Sinc Functions | Sep 1, 2005 |

**Physics Forums - The Fusion of Science and Community**