MRI is using fourier transform to obtain an image from the device measurments. Note that there are two possible ways of storing image: the first one – colour based, that we are used to and the second one – using spectrum. You can switch between them using Fourier transform. 

With MP3, the sound samples are transformed using methods that involve Fourier Series Transformations. You take a sound, expand its Fourier Series. It'll most likely be an infinite series BUT it converges so fast that taking the first few terms is enough to reproduce the original sound. The rest of the terms can be ignored because they add so little that a human ear can likely tell no difference. So I just save the first few terms and then use them to reproduce the sound whenever I want to listen to it and it takes much less memory.

Biological Sequence analysis based on Ramanujan-Fourier transform (RFT), in which complete information of DNA sequences is retained. We represent DNA sequences as four binary indicator sequences and apply RFT on the indicator sequences to convert them into frequency domain. The Euclidean distance of the complete RFT coefficients of DNA sequences are used as similarity measure. To address the different lengths in Euclidean space of RFT coefficients, we pad zeros to short DNA binary sequences so that the binary sequences equal the longest length in the comparison sequence data. Thus, the DNA sequences are compared in the same dimensional frequency space without information loss.

 

A signal is any waveform (function of time). This could be anything in the real world - an electromagnetic wave, the voltage across a resistor versus time or sound wave.

Without Fourier, we could only encode data like old FM radios, and we’d have to wait 1 minute to load this page, or leave the laptop all night downloading a single Tube video.