DSP (digital signal processing) is the field of applied mathematics and engineering dedicated to transforming and manipulating digital signals.
Examples of real digital signals include audio files, image files, video files, and digitized recordings of various physical quantities by computers like the configuration of a robot as it moves in time, measurements of the processes in a factory, the trajectory of a spacecraft — almost anything that can be periodically sampled and take on a finite set of values [1] can be seen as a digital signal.
DSP includes using tools like the Discrete Fourier Transform (DFT), the Z-transform, wavelet analysis, probability, statistics, and linear algebra to do things such as filter a signal (example: audio equalizer), predict future values (example: weather forecasting), data compression (example: JPEGs), system identification (example: fit a model of the earth to predict seismic activity), control (example: make a DC motor to respond to position commands), and stabilization (example: keep plane from “wanting” to smash into the ground). Particularly, it requires a careful consideration of the effect of sampling a signal (example: if done carelessly, you can make the sampled system unstable [read: explode]), as well as an interpolation process of some kind if you plan on using that signal outside your computer (example: you want to hear an audio signal stored on your computer).
I got into DSP because I was an audio engineer and musician [2], and I wanted to design my own audio plugins. IMO I think almost everyone would benefit from some knowledge of DSP, but the math is really intense. Personally, I found out late in life that I have a nearly infinite appetite for math, so it’s a good fit for me.
[1] Actually, a lot of basic DSP books don’t restrict the signal to be in a finite set because it makes the math easier if the signal could be any real number. However, certain structures that would be exactly equivalent in theory are not equivalent on a real computer because ordinary computer arithmetic is approximate.
[2] I still play music, but not as much as before engineering school.
Infinite-dimensional vector spaces also show up in another context: functional analysis.
From an engineering perspective, functional analysis is the main mathematical framework behind (1) and (2) in my previous comment. Although they didn’t teach functional analysis for real in any of my coursework, I kinda picked up that it was going to be an important topic for what I want to do when I kept seeing textbooks for it cited in PDE and “signals and systems” books. I’ve been learning it on my own since I finished Calc III like four years ago.
Such an incredibly interesting and deep topic IMO.
I actually designed a (digital) equalizer using an IIR filter this semester, which actually does theoretically work on sequences of numbers, which constitutes an infinite dimensional vector space. The actual math was just algebra and programming, but it was an implementation of a Z-transform transfer function which is a sequence operator (maps input sequence to output sequence).
IMO infinite-dimensional stuff shows up in two types of problems:
For some reason, you need to solve the partial differential equation you started with, i.e. you can’t use symmetry or approximations to simplify it into an ordinary differential equation.
When you’re dealing with signals that change in time or space, you have to decompose those signals into simpler signals that are easier to analyze.
If your signal looks like f(t) = K•u(t)e^at with u(t) = {1 if t≥0, 0 else}:
If Real(a) > 0, then your signal will eventually blow up.
If Real(a) < 0, then you signal will not blow up. In fact, your signal will have a maximum absolute value of |K|, and it will approach zero as time goes on.
If Real(a) = 0, it is either a complex sinusoid or a constant. In either case, it is bounded with maximum absolute value of |K|. It very much does not blow up.
So e pops up all the time in stable systems and bounded signals because the function e^at solves the common differential equation dx/dt = ax(t) with x(0)=1 regardless of the value of a, particularly regardless of whether or not the real part of a causes the solution to blow up.
I don’t judge anyone by their weight, but it’s sure hard to direct that same acceptance toward myself.
Yeah same here, but I’m at a weight where I’m exceeding weight limits of things like ladders, furniture, etc. And I’m in terrible physical shape on top of all that. It’s really more of a “tactical” thing for me at this point. Just gotta get it done.
Sounds like you’re doing well, though.
Thank you. Could be better, could be a lot worse. I’m still a social disaster.
And I make my own frozen meals
Me too. The other day I made like half my meals for the entire summer in one giant cook.
Do most people generally eat the same things all the time?
Yeah. IMO variety is expensive because it’s usually cheaper to buy a few things in bulk.
consulting a professional before doing so.
I consulted a nutritionist before doing my first weight loss [1] because I wanted to make sure my diet was nutritionally sound. Surprisingly, it was fine, just too much of everything. Very surprising considering that I’m a picky eater with texture issues, but I’ll take it.
In contrast, my sister had to see a nutritionist to go on hormones and apparently her diet was nutritionally whack, so she had to make a bunch of changes.
vegetables
Please God no (at least not raw)
[1] I put it back on when I went to engineering school, but I managed to keep it off for a couple years. Oh well. I’ll get around to losing it again soon.