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What Is the Fourier Transform?

(www.quantamagazine.org)
474 points rbanffy | 2 comments | 04 Sep 25 22:11 UTC | HN request time: 0s | source
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anyfoo ◴[04 Sep 25 23:48 UTC] No.45133536[source]▶
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.

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zozbot234 ◴[05 Sep 25 09:34 UTC] No.45136685[source]▶
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).
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segfault99 ◴[05 Sep 25 10:48 UTC] No.45137146[source]▶
True dat. But you see there's this thing called 'Engineering Maths'. Apparently it's really bad for real mathematicians' blood pressure.
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1. zozbot234 ◴[05 Sep 25 12:34 UTC] No.45137792[source]▶
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.
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2. segfault99 ◴[07 Sep 25 09:11 UTC] No.45156621[source]▶
I forgot to mention the converse also applies. Mathematicians talking about stuff we engineers learned the paint by numbers way makes our heads hurt!