The discrete Fourier transform

Having learned about Fourier series and Fourier transforms, it seems we have strayed a bit from the path of data science. Where are the data aspects of all of this? Fourier series and Fourier transforms are about decomposing functions, not data. Yes, it is useful to be able to decompose and reconstruct a function, but what happens if we don’t have the exact mathematical equation of our function and only have data points taken from the function? Is there a Fourier-like decomposition that works with data, not mathematical expressions? This is where the DFT comes in.

Often, we only have observations from a function, say the value of the function, , taken at regularly spaced intervals. In this case, we would have the observations taken at values, , with corresponding function values, . Without loss of generality, we can assume the spacing between the values is 1 as we can simply rescale the X-axis if not. This means our values are . We’...