Poisson Summation Formula

The Poisson summation formula relates a sum of a function at integer points to a sum of its Fourier transform at integer points.

Statement

For a function $f(x)$ with Fourier transform $F(f_x)$:

$$ \sum_{n=-\infty}^{\infty} f(x - n) = \sum_{m=-\infty}^{\infty} F(m) \, e^{j2\pi m x} $$

Setting $x = 0$:

$$ \sum_{n=-\infty}^{\infty} f(n) = \sum_{m=-\infty}^{\infty} F(m) $$

Comb Form

Equivalently, the Fourier transform of a train of equally spaced delta functions is another train of delta functions:

$$ \mathcal{F}\left\{\sum_{n=-\infty}^{\infty} \delta(x - n\Delta)\right\} = \frac{1}{\Delta}\sum_{m=-\infty}^{\infty} \delta\left(f_x - \frac{m}{\Delta}\right) $$

Spacing $\Delta$ in the spatial domain becomes spacing $1/\Delta$ in the frequency domain.

See Also

• Fourier transform
• Dirac delta function