This page provides an overview of my series on Fourier transformations. It will provide a summary of my separate posts.
Fourier is a transformation that can be used to figure out what individual signals make up a signal based on a series of measurements. An example of this is figuring out how many ducks there is in a pond and how fast they bob up and down based on a measured wave height. When inputting a series of measurements into Fourier, you’ll get out a list of complex numbers. The imaginary and real part of those complex numbers make up a vector, from which the original signal can be calculated. Fourier gives the cleanest result when using either a big set of samples (which also makes it incredibly slow), or if the samples describe one full cycle of the signal (sin(0) through sin(2*PI)).
How fourier works
Fourier works by generating the output signal to be tested, and multiplying this by the measurements at the same position. Given no start offset (e.g. sin(5x)*3) these two signals will multiply in a way that they strengthen each other, in a sense resonating with each other. When measuring with an input signal of sin(5x) this will give us the output vector (0,3).
Given a start offset (e.g. sin(x+0.5*PI)*3, shifting the signal 90 degrees) the aforementioned will still apply, but instead we’ll get out an output vector (3,0). The direction of the vector will indicate the “start offset”, while the length of the vector will indicate the strength of the signal.