Nyquist Theorem

What Is the Nyquist Theorem?

The Nyquist theorem, also known as the Nyquist–Shannon sampling theorem, defines the conditions under which a continuous-time signal can be sampled and perfectly reconstructed from its samples, without losing any information. The Nyquist theorem holds that a continuous-time signal can be perfectly reconstructed from its samples if it is sampled at a rate greater than twice its highest frequency components. This is known as the Nyquist rate.

The Nyquist theorem is the cornerstone of digital signal processing. This principle enables reliable reconstruction, manipulation, and analysis of real-world signals using digital systems, forming the basis for technologies such as audio and video recording, communications systems, and medical imaging. Without the Nyquist theorem, the transition from analog to digital processing would be prone to errors, such as aliasing.

Aliasing causes different signals to become indistinguishable from each other after sampling. When a signal is sampled below the Nyquist rate, high-frequency components are “folded” back into lower frequencies, creating inaccurate data in the reconstructed digital signal. This can result in distortion, loss of important details, and the appearance of artifacts that were not present in the original signal. These issues can degrade audio quality, distort images, and corrupt measurements in engineering applications.

Filter Design with MATLAB: Preventing Aliasing with the Nyquist Theorem

Aliasing is a fundamental challenge in digital signal processing—once it occurs, it cannot be reversed. To prevent aliasing, it’s essential to understand the Nyquist theorem.

An anti-aliasing filter is a low-pass filter applied to a signal before it is sampled for digital processing. The filter’s main purpose is to remove frequency components that are higher than half the sampling rate. By attenuating or eliminating these high-frequency components, the anti-aliasing filter ensures the sampled signal does not contain frequencies that would be misrepresented as lower frequencies after sampling.

In practical systems anti-aliasing filters are typically implemented as analog electronic circuits, or as digital filters during resampling. You can design filters, such as anti-aliasing filters, in MATLAB^®.