What Is the Fourier Transform?

If you like Fourier, you're going to love Laplace (or its discrete counterpart, the z transform).

This took me down a very fascinating and intricate rabbit hole years ago, and is still one of my favorite hobbies. Application of Fourier, Laplace, and z transforms is (famously) useful in an incredibly wide variety of fields. I mostly use it for signal processing and analog electronics.

Essentially that's what electrical/electronics engineering is about.

When I did EE, didn't have access to any kind of computer algebra system. Have 'fond' memories of taking Laplace transform transfer functions and converting to z-transform form. Expand and then re-group and factor. Used a lot of pencil, eraser and line printer fanfold paper for doing the very basic but very tedious algebra. Youngsters today don't know how lucky.. (ties onion to belt, etc., etc.)

Years ago, I often struggled to choose between Amazon products with high ratings from a few reviews and those with slightly lower ratings but a large volume of reviews. I used the Laplace Rule of Succession to code a browser extension to calculate Laplacian scores for products, helping to make better decisions by balancing high ratings with low review counts. https://greasyfork.org/en/scripts/443773-amazon-ranking-lapl...

While this is a good idea, I think it's unrelated to the Laplace transform except that they're named after the same dude?

When I think of Laplace Transforms I always think of control theory - poles, zeros etc.

Probably that's why we are learning about it in the "Control Theory" classes at university. :-)

Jokes aside, I graduated as "Computer Engineer" (BSc) and then also did a "Master in Computer Science"; I was (young and) angry at the universe why soooo many classical engineering classes and then theory I had to sit through (Control theory, Electrical engineering, Physics), and we never learned about the cool design patterns etc etc.

Today I see that those formative years helped me a lot with how I develop intuition when looking at large (software) systems, and I also understand that those ever changing best design patterns I can (could have) just look up, learn, and practice in my free time.

I wish a today-me would have told my yesterday-me all this.

I learned about it after I graduated with a CS degree - I mean in true university degree fashion we'd been taught about Laplace and Z transforms (and related things) but with no practical applications.

After graduating I joined an academic research team based mainly in a EE department who were mainly Control Engineers - we were mainly doing stuff around qualitative reasoning and using it for fault diagnosis, training etc.

Then there's the whole mindfuck of fractional order Fourier (and other) transforms.

Did you make sproingies from the tear-off side strips of the printer paper, though? That was the best bit. :P

No worries, as a self proclaimed youngster I didn't manage to understand Fourier in 2 days and never bothered again. Also had no other prior knowledge to algebra so maybe that's why I struggled. Never perceived algebra as useful in anything programming related, will continue to do so as most problems are solvable without it. I'll let the degree havers do all that stuff.

The so-called "Z transform" for discrete sequences is really just a misnomer for the actual method of generating functions (and formal power-series/Laurent-series). You just write a discrete sequence as a power series in z^(-1).

Of course!

True dat. But you see there's this thing called 'Engineering Maths'. Apparently it's really bad for real mathematicians' blood pressure.

You might find LLMs to be a useful crutch for this to an extent, although it's very easy to take the wrong turn and go off into the deep end. But as long as you keep forcefully connecting it back to practical reality, you can get progress out of it. And of course, never actually make it calculate.

I always assume that all the ratings are fake when there is a low count of ratings since it is easy for the seller to place a bunch of game orders when they are starting out.

Just for reference, in case you find yourself in an optimization under uncertainty situation again: The decision-theoretic right way to do this is generate a bayesian posterior over true ranking given ranking count and a prior on true rankings, add a loss function (it can just be the difference between the true rating of the selected item and the true rating of the non-selected item for simplicity) then choose your option to minimize the expected loss. This produces exactly the correct answer.

Analytic combinatorics (the rubric where mathematicians would want to place all the region-of-convergence, zeros-poles, etc. analysis of generating functions–formal power/Laurent series–Z transforms that engineering often focuses on) is not exactly easy-going either. Other common methods (relating convolution to multiplication, inverting transforms etc.) would traditionally be comprised under the Operational Calculus of Mikusiński.

> Never perceived algebra as useful in anything programming related

Image, video, and audio processing and synthesis, compression algorithms, 2D and 3D graphics and geometry, physics simulation, not to mention the entire current AI boom that's nothing but linear algebra… yeah, definitely algebra isn't useful in anything programming related.

Have I got a video for you.

gingerBill – Tools of the Trade – BSC 2025 https://youtu.be/YNtoDGS4uak

My control theory professor (who was also my physics advisor -- it was a small college) explained it like this: Physicists like Fourier transforms because they go from minus to plus infinity, like the universe. Control engineers like Laplace transforms because they start at zero, and a control system also has a starting point.

To be fair (and because I've just remembered - it was ~40 years ago) we did get some practical stuff covered in the maths part of my CS degree in the application of group theory (groups, rings & fields) to coding theory.

Yea that's not what I work with my guy

You said "anything programming related", not "anything related to my work", "my guy".

So you're a programmer but you've never assigned a number to a variable or written any math operations? Do you just do string translations or something?

I just recently got my Computer Engineering degree which is the modern Electronics Engineering and we had a whole class on transforms. We had to do it on paper, but that professor at Cal State LA knew what the heck she was doing. We learned it good.

I'm talking algebra you need a degree for. Well algebra u learn while getting one that is.

"Book learnin' didn't do me no good no how!"

> really just a misnomer

No. Things acquire different names if they are independently discovered by different communities.

Native Americans called Indians. Lol! what was that.

A bigger problem I find is many Amazon listings having a large number of genuine positive reviews, but for a completely different product than the one currently for sale.

Recently I was buying a chromecast dongle thing and one of the listings had some kind of "Amazon recommends" badge on it, from the platform. It had hundreds of 5 start reviews, but if you read them they were all for a jar of guava jam from Mexico.

I'm baffled why Amazon permits and even seemingly endorses this kind of rating farming

I referenced 3B1B for the name: youtube.com/watch?v=8idr1WZ1A7Q

Was this professionally or in school? I still did this in an EE program 15 years ago and I can't imagine things have changed since then. I think kids still have to do lots of ugly math in EE classes.

When I first learned Laplace transform in university, it was my goto for differential equations of any kind. I was even naive enough to believe well this is a solved problem now. Eventually found out this wasnt the case after studying PDEs. Its still my favourite transform. Immensely useful not to mention the whole method of moments in random variables is basically laplace transform.

I don't like Fourier transform but for petty reasons. In the engineering exams, I messed up a Fourier Transform calculation and ended up just a few points short of a perfect score. Hate it ever since :)

Undergrad. Mid-late 1980s.

I wasn't making point about mathematics qua mathematics. Was thinking that if I were doing EE undergrad today, I'd use SageMath or Mathematica to crunch the mechanical algebraic manipulations involved in doing a z-transform.

Plot twist: He's a Haskell guru juggling hylomorphisms blindfolded.

You know that if you have the Laplace transform, you can just insert s = iω and then you have the Fourier transform, right? :-P

(Or jω, if you prefer that notation)

The two-sided Laplace transform would probably have made his head explode.

Can you please provide an example or link to read more? Seems very interesting.

This continued with kids into the 90's. I miss that bit.

https://www.reddit.com/r/nostalgia/comments/b6dptv/folding_t...

I forgot to mention the converse also applies. Mathematicians talking about stuff we engineers learned the paint by numbers way makes our heads hurt!

Tell me about it. But I like my numbers real not imaginary !