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:

poisson

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.

 

Advertisements
This entry was posted in Uncategorized and tagged . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s