What is the difference between these two?

Truncation is a primitive way to approximate something. In any case, truncating a process means to “cut off” the process after some point.

For example, take the number π ≈ 3.141592653589793… ad infinitum. If you need to do a calculation with π, you typically round it to some number of significant figures. For example, 3.1416 is π rounded to 5 significant figures. Truncation is even more primitive; for example, 3.1415 is π truncated to 5 significant figures by simply “cutting off” the rest.

So Fourier analysis in its most general form requires you to compute an uncountably infinite (one element per real number) number of points to represent a frequency response. If your signal has a start time and an end time, you can get that down to a countable infinity (one element per integer) of points. This is equivalent to truncating the signal in time. Audio signals typically have a beginning and end, so theoretically no work needs to be done. However, for real-time processing, we typically break up the signal into tiny windows and analyze each of these windows piece by piece.

Now you can have a computer start your task, but it will never finish it. It won’t even finish processing one single window. To do that, you have to “ban” any frequencies above a fixed maximum from your analysis. This is equivalent to truncating the signal in its frequency response. For audio signals, this means that if there is anything above 22050Hz (or half your sample frequency), you throw it out before you process it any further. After this, you get an algorithm that will terminate in a finite number of steps.

Truncating the frequency response results in Gibb’s oscillations (audible ringing) if done “irresponsibly”. There are other methods (e.g. Cesaro summation) that can be used to reduce these oscillations, but it’s kind of a technical topic. Practically, we use filters that “slowly” begin to throw out frequencies lower than the desired cutoff. These “gradual” filters don’t cause ringing.

I can only imagine the infinite becoming the limit of the computer or maybe just a very very high number

The infinite would be how many measured values are needed to recover the signal and its frequency response without losing any information. You need to iterate over each of these when doing Fourier analysis on a computer.

(cuz I’m hoping this is in-depth? this is normally when u sqy “actually, that’s not in-depth at all” and fill my DMs with an ocean of math LOL)

So I went to recording school several years before I went to engineering school. We were taught about frequency response and how to manipulate audio signals with canned software, but without any of the mathematics behind it. This is about as in-depth as that. Engineering school went into more depth, but considering that I want to actually write audio software, eventually with my own hand-rolled math, I really feel a need to understand Fourier analysis at a depth that my even engineering education alone has been unable to provide.

So there is a much more in-depth explanation (several books worth at least), but I’ll spare you the details ☺️. But if you ever want those details, feel free to ask for textbook recommendations.

thank god it’s not this complicated cuz that would be too much for my silly head >u<

I used to think math was too complicated for me to learn too. That’s why I waited so long to go to engineering school after high school. And it is complicated. But IMO when I approached math like I approached my creative pursuits, i.e. something I planned on retaining for the rest my life, that is when it clicked for me. I have a terrible memory for random stuff that makes no sense and has no pattern. However, if I understand why something is the way it is, and what motivated mathematicians to come up with some new abstraction, it all of a sudden becomes a lot easier to internalize.