9716

Recovering the Fourier Coefficients

This Demonstration illustrates recovering the Fourier coefficients from a complex wave that you build. With the sliders you can select the weights of five sine wave signals, 1 to 5 Hz. These are summed into a complex signal in the upper graph. You can then selectively choose to multiply the entire output wave by any of the original unweighted signals. When selected, the resultant product waveforms are displayed in the lower graph. When these resultant waveforms are individually integrated or summed, the value of the integral is the original weight applied to the signal (after potentially dealing with scaling issues).
Recovering the Fourier coefficients is fairly straightforward but can consume a large number of calculations. However, symmetries in these calculations can be exploited to drastically cut down the number of calculations, resulting in the fast Fourier transform (FFT).

SNAPSHOTS

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

DETAILS

In this Demonstration the horizontal center of either graph is always zero, because we are only using sine waves that are all in phase. The addition of cosine waves would create more realistic signals.
Snapshot 1: bottom graph is output signal times 1 Hz (top graph) and integrates to 1
Snapshot 2: bottom graph is output signal times 4 Hz (top graph) and integrates to 0
Snapshot 3: input signals weighed such that the middle of the output is flat
Snapshot 4: a squiggly triangle wave
Reference
[1] A. E. Richmond, Calculus For Electronics, 3rd ed., New York: McGraw–Hill, 1983.
    • 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.









 
RELATED RESOURCES
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 © 2014 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+