Whittaker-Shannon Sampling Theorem

The Whittaker-Shannon sampling theorem (also known as the Nyquist-Shannon sampling theorem) establishes the conditions under which a continuous band-limited signal can be perfectly reconstructed from discrete samples.

One-Dimensional Statement

A signal $g(t)$ that is band-limited to frequencies below $B$ Hz can be perfectly reconstructed from samples taken at rate $f_s$ if and only if:

$$ f_s \geq 2B $$

The minimum sampling rate $f_s = 2B$ is called the Nyquist rate.

Two-Dimensional Statement

For a two-dimensional function $g(x, y)$ band-limited to $|f_x| \leq B_x$ and $|f_y| \leq B_y$, sampling on a rectangular grid with spacings $X$ and $Y$ allows perfect reconstruction if:

$$ X \leq \frac{1}{2B_x}, \qquad Y \leq \frac{1}{2B_y} $$

Reconstruction Formula

The continuous signal can be reconstructed from samples $g(nX, mY)$ using:

$$ g(x, y) = \sum_{n=-\infty}^{\infty} \sum_{m=-\infty}^{\infty} g(nX, mY) \, \mathrm{sinc}\left(\frac{x - nX}{X}\right) \mathrm{sinc}\left(\frac{y - mY}{Y}\right) $$

where $\mathrm{sinc}(u) = \frac{\sin(\pi u)}{\pi u}$.

Aliasing

When the sampling conditions are violated (undersampling), spectral replicas overlap in the frequency domain. This phenomenon is called aliasing and causes irreversible distortion that prevents perfect reconstruction.

Historical Note

The theorem is attributed to E.T. Whittaker (1915), V.A. Kotelnikov (1933), and C.E. Shannon (1949), who independently developed versions of this result.