15 Perspectives on the Fourier transform
I get so excited when Fourier transforms come up because there are so many truths that are useful! But it’s another tangled web of facts. So here are some things I love about Fourier transforms (and background facts that you need to work with Fourier transforms).
Quadrature pairs
When something is rotating in 2D, it can be projected onto x- and y-axes. By convention, theta is the angle that describes the direction that something is pointing in, it’s measured relative to the positive x-axis, and it rotates counter-clockwise. The part that’s projected onto the horizontal axis is “cosine-phase” — it starts positive, crosses 0, goes negative, comes back. The part that’s projected onto the vertical axis is “sine phase” — it starts at 0, grows, then crosses back through θ to negative territory.
1D Fourier Transforms
Here is an interactive Colab notebook for exploring 1D Fourier transforms of different time series.
Complex numbers
You can get the gist of Fourier transforms without using complex numbers, but to do the math, you need to be acquainted with complex numbers. Folks who thrived in Complex Analysis would find my descriptions here inadequate, I am sure … but here’s how I think of them.
i is the marker for complex numbers (or j if you’re an engineer). It is defined as the square root of -1. That’s impossible, right? When you square a number, you multiply it by itself … a negative number times a negative number is a positive number … there’s no way to get a negative number when you square something, right? But i2 = -1. At some point I knew how this definition was derived, but I’ve long since forgotten. I’m sure there’s much more, but because I don’t do math for a living, to me i is a marker for something that is spinning or cyclical. Let me explain!
It’s standard to use z to represent a complex number, which is a number with 2 parts:
z = x + iy
x is the real part of the complex number, y is the imaginary part. That’s just a definition, but now we’re going to think of x as a distance on one axis and y as a distance on another axis.
If we look at that quadrature pair movie, we can see how complex numbers might be used to describe things that are spinning. The point at the tip of the black line is z — a complex number. I could tell you how to find the point at the tip of that line two ways: either I could tell you to travel a certain distance along the x-axis and then a certain distance up parallel to the y-axis … or I could tell you to travel in a certain direction for a certain distance (that’s a polar coordinate system).
So complex numbers, as it turns out, can be described two ways. They have a magnitude, written as |z|, and a direction, θ (the angle between z and the x-axis). At any given moment, z has a projection onto the x-axis: x = |z|·cos(θ). That’s called the “real” part. The projection to the y-axis is y = |z|·sin(θ). Now, if we define the y-axis as the “imaginary” axis, then i becomes just a marke