Introduction to the Discrete Fourier Transform with Python

Understand the Discrete Fourier Transform – Free Course

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What you’ll learn

  • understand the math behind the Discrete Fourier Transform(DFT), one of the most useful formulas in applied math and computer science.
  • gain a deeper appreciation for the DFT by applying it to simple applications using Python

  • be able to mathematically and programmatically determine note/chord of a sound file using the DFT in Python

  • understand the basics of digital audio
  • be able to filter out noise from a sound file using Python
  • be able to understand the basics of sound compression and filtering
  • understand the difference between time domains vs. frequency domains of digital signals
  • implement your own version of the Discrete Fourier Transform in Python and compare it to the efficient Fast Fourier Transform
  • understand why the DFT works
  • understand the DFT from the geometric viewpoint via the inner product
  • be able to programmatically play sounds from .txt and .wav files
  • understand the basics of sounds, music theory, physics of sounds


  • Some programming in Python is helpful but not necessary. Most of the programming is done with very little/easy Python code.
  • Some precalculus is helpful, including basic complex numbers and basic sine/cosine graphs, all of which is reviewed.


The Discrete Fourier Transform (DFT) is one of the most useful algorithms in computer science and digital signal processing. If you’ve ever opened a JPEG, listened to an MP3, watched a MPEG video, or used voice recognition of Alexa or the Shazam app, you’ve used some variant of the DFT. Its efficient implementation, the Fast Fourier Transform, is considered one of the top 10 algorithms of the 20th century by the American Institute of Physics and the IEEE Computer Society. Fortunately, the math is very accessible and only involves basic complex numbers and basic trigonometry.

This course is a very basic introduction to the Discrete Fourier Transform. We will focus on understanding the math behind the formula and use Python to do some simple applications of the DFT and fully appreciate its utility. . The Python code we are writing is, however, very minimal. Students with little or no programming experience will still be able to learn Python through the examples in the lectures.

Students should have Anaconda installed with the Jupyter Notebook. However, all of the programming assignments can also be run on the cloud on MyBinder.

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