This repository contains two iPython notebooks (based on SageMath 7.5.1), showing two applications of the Discrete Fast Fourier Transformation.
DTMF tones, the distinct sounds associated with a certain number on most telephone keyboards can be identified from a continuous audio stream using the Fourier Transformation. This is how a telephone computer might work.
Don't forget to listen to the audio sample (dtmf.wav) first.
The original JPEG standard uses the Discrete Cosine Transformation (very similar to the DFT). This notebook shows what's at least necessary to lossfully compress an example image. The notebook shows how different compression ratios affect file size and image quality.
Note: The example image (commonly known as Lena, called Lisa.png in the notebook) is not MIT-licensed. See https://en.wikipedia.org/wiki/Lenna