## Poisson Summation Formula

As shown in §B.14 above, the Fourier transform of an impulse train is an impulse train with inversely proportional spacing:

(B.56) |

where

(B.57) |

Using this Fourier theorem, we can derive the continuous-time PSF using the

*convolution theorem*for Fourier transforms:

^{B.1}

(B.58) |

Using linearity and the shift theorem for

*inverse*Fourier transforms, the above relation yields

We have therefore shown

Compare this result to Eq. (8.30). The left-hand side of (B.60) can be interpreted ,

*i.e.*, the time-alias of on a block of length . The function is periodic with period seconds. The right-hand side of (B.60) can be interpreted as the inverse Fourier

*series*of

*sampled*at intervals of Hz. This sampling of in the frequency domain corresponds to the aliasing of in the time domain.

**Next Section:**

Sampling Theory

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Impulse Trains