This work focuses in the Fast Fourier Transform (FFT). It explains its mathematical deduction, illustrates its mechanics through some Python codes and develops one real-life application. It also gives some historical background to its origins, emphasizing in Gauss' algorithm as the first known formula of the FFT.
There are six sections to this work. Chapter zero gives a superficial background on the Fourier Series and the Fourier Transform. Chapter one explains the Discrete Fourier Transform (DFT) (the FFT is one version of the DFT), gives an example in Python and mentions its limitations regarding speed. Chapter two develops the mathematical deduction of the FFT and presents three different ways of developing it in Python: through recursion, using Numpy and using SciPy. Chapter three presents the link between the modern version of the FFT and Gauss' algorithm. Chapter four shows one real-life application of the FFT in audio correction. A general conclusion is presented in chapter five. Chapter six contains the references for further reading on the topic.