November 2026

Nov 1, 2026 Monthly Update

There’s not much to report since last time. I’m still playing around in Cardinal, but not doing a whole lot of programming, sound-related or otherwise.

My copy of Discrete-Time Signal Processing has arrived. Both that and Physical Audio Signal Processing have taken a back seat for the moment while I focus on Mathematics of the Discrete Fourier Transform, which is going pretty well. I feel like I’ve gotten a better handle on $\mathbb{C}^n$ and why certain things are defined the way they are. For example, the minus sign in the exponent in the definition of the DFT,

X(\omega_k) = \sum_{n=0}^{N-1} x[n] e^{-j 2 \pi k n/N},

is there because it’s expressing an inner product, and inner products in $\mathbb{C}^n$ conjugate one of their arguments. That conjugation makes $\langle x, x \rangle$ real for any $x \in \mathbb{C}^n$ , and the inner product induces a norm

\|x\| = \sqrt{\langle x, x \rangle}

just like in $\mathbb{R}^n$ .

After reading this page, I thought it was interesting that fft(eye(N)) gives you the length- $N$ DFT matrix in MATLAB, and had to think for a bit about why this works. Of course, since the DFT is just a linear transformation, this effectively amounts to multiplying its matrix representation by the identity matrix, so the result shouldn’t have been a surprise. Unfortunately for me, I had already written out some summations to convince myself before having this realization.

I tend to be drawn to rabbit holes. That isn’t necessarily a bad thing, but it can make actually achieving a goal difficult. I’m trying to be more deliberate about letting some things remain a mystery for now unless they’re directly related to the task at hand.