Learning about the DFT

I’ve been learning a bit about discrete signals from Brian McFee’s Digital Signals Theory. I’m currently trying to understand the Discrete Fourier Transform. The Fourier Transform is one of those concepts that I feel like I understand until I probe just slightly past the surface. For example: why is the result a complex-valued function? I don’t know (yet)!

I’m currently learning about comparing a discrete signal to various discretizations of sinusoids. A simple way to compare two discrete signals is to sum the products of their respective samples. This is just the familiar dot product, where we treat each signal as a vector in $n$-dimensional space, where $n$ is the number of samples.

By sweeping through a range of sinusoids at varying frequencies, we can build up a representation of our original signal in terms of these sinusoids. Well, kind of.

We need to compare our original signal to both $sin$ and $cos$ functions, because one family of these alone isn’t a basis in the space we’re interested in. Once we have these values we form some kind of spectrum that also incorporates complex numbers at some point. I’m not sure yet.

One thing at a time. In order to get a better intuitive grip on what “similarity” means in this context, I hacked together a little Observable notebook:

The top element is an interactive widget that lets me “paint” a discrete signal using the cursor. We can then compare the input signal against a variety of sinusoids whose frequency and phase are controlled by some sliders. By aggregating these similarity scores (at a particular phase), we can plot a kind of “similarity spectrum”. And finally, we can use that spectrum to (kind of) reproduce the original signal.

Clearly I’m still missing quite a bit: the reconstruction as I’m implementing it here is inherently incomplete, since it only considers samples at a single phase. We need another dimension here (the phase), which I’m assuming is where the complex numbers come into play.