The Poisson Summation Formula

There are various conventions for normalizing the Fourier transform. For the purposes of this post, let’s write

[\mathscr{F}(f)](s) := \displaystyle\int_{-\infty}^{\infty} e^{i s t} f(t) \, dt

The Poisson Summation Formula concerns the sum

\displaystyle\sum_{n=-\infty}^{\infty} [\mathscr{F}(f)](n)

Where on Earth did that come from? Well, without regard to convergence issues, we have

\displaystyle\sum_{n=-\infty}^{\infty} [\mathscr{F}(f)](n) = \sum_{n=-\infty}^{\infty} \int_{-\infty}^{\infty} e^{i n t} f(t) \, dt = \int_{-\infty}^{\infty} \left[\sum_{n=-\infty}^{\infty} e^{i n t}\right] f(t) \, dt

So we see that we’re just transforming our function with respect to the kernel

\displaystyle\sum_{n=-\infty}^{\infty} e^{i n t}

Problematically, this doesn’t converge to a function. However, it does converge to a distribution. Let’s take a look at how it looks for a few values of n:


Figure: \displaystyle\sum_{n=-N}^N e^{i n t} for N from 0 to 5.

As illustrated by the figure, the sequence is converging to a Dirac Comb, a periodic sequence of Dirac deltas.  From here the Poisson summation formula should be clear.  Alternatively, we could have started by (formally) working out the Fourier series for the Dirac comb, and we’d quickly arrive at the Poisson summation formula.


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