What is Aliasing?

In digital signal processing (DSP), aliasing is a phenomenon that occurs when a continuous-time signal is sampled at a rate that is too low to capture its frequency content accurately. When this happens, higher-frequency components of the signal “fold back” into lower frequencies, causing distortion and misinterpretation of the signal in the discrete domain.

The Nyquist-Shannon sampling theorem states that a continuous signal must be sampled at a rate at least twice its maximum frequency (the Nyquist rate) to reconstruct it accurately. Mathematically, if ( f_s ) is the sampling frequency and ( f_{max} ) is the maximum frequency of the signal, then:

f_s \geq 2 f_{max}

Failing to meet this criterion results in aliasing.

Example

Consider a continuous sinusoidal signal:

x(t) = \sin(2 \pi f t)

If this signal is sampled at $f_s$ < 2f, the sampled points cannot distinguish between the original frequency $f$ and a lower frequency. This creates an alias frequency $f_a$ given by:

f_a = | f - n f_s | \quad \text{for some integer } n

This explains why high-frequency components can appear as lower frequencies in sampled data.

Applications and Implications

Audio DSP: If a 20 kHz tone is sampled below 40 kHz, it can appear as a completely different tone, causing audible distortion.
Image processing: When resizing or downsampling images, aliasing manifests as moiré patterns.
Radar and communication systems: Accurate sampling is crucial; aliasing can lead to false target detection or misinterpretation of frequency channels.

The standard method to prevent aliasing is pre-filtering the signal with a low-pass filter before sampling. This ensures no frequency above $f_s/2$ is present.

Visualization

Below is a conceptual figure showing aliasing:

A high-frequency sine wave sampled at a low rate
The resulting discrete samples appear to represent a lower frequency

Aliasing Plot Figure 1: Signal sampled at a Nyquist rate vs Signal sampled at less than the Nyquist Rate.