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

 (B.59)

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.

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