Gibbs Phenomenon in the Truncated Discrete-Time Fourier Transform of the Sinc Sequence

Using a finite number of terms of the Fourier series approximating a function gives an overshoot at a discontinuity in the function. This is called the Gibbs phenomenon. This Demonstration shows the same phenomenon with the discrete-time Fourier transform (DTFT) of a sinc sequence. The oscillations around the discontinuity persist with an amplitude of roughly 9% of the original height.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


The Gibbs phenomenon was first observed in the 19th century, when trying to synthesize a square wave with a finite number of Fourier series coefficients. The oscillations around the discontinuities, while they became narrower, remained of constant amplitude even when more terms were added. This behavior was first attributed to flaws in the computation that was synthesizing the square wave, but J. Willard Gibbs in 1899 demonstrated that it was an actual mathematical phenomenon.
[1] M. Vetterli, J. Kovačević, and V. K. Goyal, Foundations of Signal Processing, Cambridge: Cambridge University Press, 2014. www.fourierandwavelets.org.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+