The Fourier Transform is a mathematical operation that transforms a function of time (or space) into a function of frequency. It decomposes a complex signal into its constituent sinusoidal components, each with a specific frequency, amplitude, and phase. This is particularly useful in many fields, such as signal processing, physics, and engineering, because it allows for analysing the frequency characteristics of signals. The Fourier Transform provides a bridge between the time and frequency domains, enabling the analysis and manipulation of signals in more intuitive and computationally efficient ways. The result of applying a Fourier Transform is often represented as a spectrum, showing how much of each frequency is present in the original signal.
\[\Large\boxed{\boxed{\widehat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\ e^{-i 2\pi \xi x}\,\mathrm dx, \quad \forall\xi \in \mathbb{R}.}}\]
Inverse Fourier Transform:
\[\Large\boxed{\boxed{ f(x) = \int_{-\infty}^{\infty} \widehat f(\xi)\ e^{i 2 \pi \xi x}\,\mathrm d\xi,\quad \forall x \in \mathbb R.}}\]
The equation allows us to listen to mp3s today. Digital Music Couldn’t Exist Without the Fourier Transform: http://bit.ly/22kbNfi
@kentpitman @rwxrwxrwx
could I get you two's (et al.'s) opinions on my conference paper idea for #ELS2025:
https://codeberg.org/tfw/cl-series-fft#headline-4
I eventually scrapped together a 3 element Winograd #FFT that was compatible with Waters' Series 1400 line expansion. This year's theme is "going beyond macros" and Series is kinda beyond other macros.
I did it by hand to figure things out, but later I could cover/follow Sidney Burrus' PFA program-generation book, but for #lisp, with Series (instead of Fortran).
@hairylarry @nosrednayduj @me @rat @pkw @prahou @northernlights @SDF @kentpitman @mdhughes @Contrapunctus
https://web.archive.org/web/20081220142747/http://cnx.org/content/m17626/latest/
<- the #lisp code of https://codeberg.org/tfw/lispmoo2/commit/8ff729418304b78bb2e6bda1f1ac627704b25596
#fft #podcast #live https://anonradio.net:8443/anonradio
sharpsign
first hour of the new year UTC
@northernlights @SDF @kentpitman @mdhughes @Contrapunctus
Join as always:
telnet lambda.moo.mud.org 8888
co guest
@join screwtape
Currently me'n Slak hanging out in paradise sushi.
I ran out of characters, but I also want to talk a lot about my experiences implementing a pipelined Winograd #fft using rooms and $things in my own #lispmoo2 https://codeberg.org/tfw/lispmoo2 and my hopes, dreams, secret plans and resolutions for the new year. Dynamic spatial inheritance plus currying and script-descriptio
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