Nyquist Criterion

The Nyquist criterion (also called Nyquist-Shannon sampling theorem) states the minimum sampling rate required to perfectly reconstruct a band-limited signal.

Statement

If a signal is band-limited with maximum frequency $B$, it must be sampled at a rate:

$$ f_s \geq 2B $$

or equivalently, the sample spacing must satisfy:

$$ a \leq \frac{1}{2B} $$

Key Concepts

• Nyquist frequency: The sampling rate $f_s = 1/a$
• Folding frequency: The bandwidth limit $B$ where aliasing begins
• Aliasing: Spectral overlap that occurs when the Nyquist criterion is violated

In Two Dimensions

For 2D signals with bandlimits $B_x$ and $B_y$:

$$ \frac{1}{a} \geq 2B_x, \quad \frac{1}{b} \geq 2B_y $$

Recovery

If the Nyquist criterion is satisfied, the original signal can be recovered by low-pass filtering:

$$ g(x,y) = g_s(x,y) \ast [4B_x B_y \, \text{sinc}(2B_x x) \, \text{sinc}(2B_y y)] $$

Related Topics

• Whittaker-Shannon sampling theorem
• Aliasing
• Band-limited functions