Sampling Theorem & Aliasing Visualization

Interactive visualization of the Nyquist-Shannon sampling theorem and aliasing phenomenon

Signal Parameters

2.0
8.0
1.0
0

Presets

f_s / 2f_max
2.00
Adequate Sampling
Original Signal
Samples
Reconstructed
Aliased (if any)
Original Spectrum
Sampled (with replicas)
Aliased Components
Sampling Rate: 8.0 Hz
Nyquist Rate: 4.0 Hz
Max Frequency: 2.0 Hz
Aliased Frequency: --
Reconstruction Error (MSE): --

Key Concepts

Nyquist-Shannon Sampling Theorem

A continuous signal can be perfectly reconstructed from its samples if the sampling rate is at least twice the maximum frequency component of the signal.

Condition:

Sampling Process:

  • f_s: Sampling frequency (samples per second)
  • f_max: Maximum frequency component in the signal
  • When f_s ≥ 2f_max: Perfect reconstruction is possible
  • When f_s < 2f_max: Aliasing occurs, information is lost

Aliasing Phenomenon

When a signal is sampled at too low a rate, high-frequency components appear as low-frequency components.

Aliased Frequency:

Examples:

  • 10 Hz signal sampled at 15 Hz appears as 5 Hz
  • 8 Hz signal sampled at 10 Hz appears as 2 Hz

Wagon Wheel Effect:

In movies, wagon wheels can appear to rotate backwards due to temporal aliasing. This is the same phenomenon that causes aliasing in signal processing!

Signal Reconstruction Methods

Zero-Order Hold

Holds each sample value constant until the next sample. Creates a staircase approximation.

Linear Interpolation

Connects adjacent samples with straight lines. Simple but not accurate for high frequencies.

Sinc Interpolation (Ideal)

Uses Whittaker-Shannon interpolation formula. Perfect reconstruction if Nyquist condition is met.

Cubic Spline

Uses piecewise cubic polynomials. Smooth results with good accuracy.

Real-World Applications

Audio Sampling

CD audio: 44.1 kHz sampling rate for frequencies up to 22.05 kHz (human hearing range)

Video Frame Rates

24/30/60 fps to avoid motion aliasing in film and video

Image Processing

Pixel grid samples continuous scene; anti-aliasing filters reduce artifacts

Digital Communications

ADC/DAC conversion must satisfy Nyquist criterion for accurate transmission