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.

$$\boxed{{\displaystyle \boxed{{\displaystyle \hat{f}(\xi )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)e^{-i2\pi \xi x}\mathrm{d}x,\mathrm{\forall }\xi \in \mathbb{R}.}}}}$$

Inverse Fourier Transform:
$$\boxed{{\displaystyle \boxed{{\displaystyle f(x)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\hat{f}(\xi )e^{i2\pi \xi x}\mathrm{d}\xi ,\mathrm{\forall }x\in \mathbb{R}.}}}}$$

The equation allows us to listen to mp3s today. Digital Music Couldn’t Exist Without the Fourier Transform: http://bit.ly/22kbNfi

Is The Frequency Domain a Real Place? - When analyzing data, one can use a variety of transformations on the data to massa... - https://hackaday.com/2024/05/16/is-the-frequency-domain-a-real-place/ #fouriertransform #frequencydomain #science

https://www.dynamicmath.xyz/

Your Text Needs More JPEG - We’ve all been victims of bad memes on the Internet, but they’re not all just bad ... - https://hackaday.com/2024/03/19/your-text-needs-more-jpeg/ #discretecosinetransform #fouriertransform #softwarehacks #compression #javascript #lossifizer #lossy #jpeg #text #dct #fft

Since "jct." is short for "junction," every time I read "DCT" (discrete cosine transform) I hear "dunction."

An Interactive Guide to Fourier Series.

https://injuly.in/blog/fourier-series/index.html

#genuary31 - "Generative Audio"

The final day of Genuary is "Generative Audio". Instead of trying to generate an audio clip, I used an audio clip to generate an image. Many different clips within a recording of Beethoven's Pathétique Sonata were passed through an FFT with smoothing and coloring applied.

I've archived all of my Genuary posts on my website: https://codeismycanvas.art/posts/genuary24/

How Shazam recognizes songs so quickly.

https://www.cameronmacleod.com/blog/how-does-shazam-work

Mee-owch!
Groundbreaking Quantum Leap: Physicists Turn Schrödinger’s Cat on Its Head
Researchers have developed a groundbreaking method to perform the fractional Fourier Transform of optical pulses using quantum memory. This unique achievement involved implementing the transformation on a “Schrödinger’s cat” state, having potential applications in telecommunications and spectroscopy.
https://scitechdaily.com/groundbreaking-quantum-leap-physicists-turn-schrodingers-cat-on-its-head/ #fractional #FourierTransform #optical #pulses #quantum #memory #Schrödinger’sCat

The FFT nicely confirms the aperiodicity and an interesting 12-fold symmetry #Fouriertransform #crystallography

Its #Fouriertransform nicely shows the single handedness of the 4-fold axis; what I honestly did not expect are all those sharp peaks!

A Crash Course On How MRI Machines work - Of all the high-tech medical gadgets we read about often, the Magnetic Resonance I... - https://hackaday.com/2023/05/11/a-crash-course-on-how-mri-machines-work/ #fouriertransform #medicalhacks #medical #magnet #coils #mri

The Fourier transform (FT), explained in one sentence: https://blog.revolutionanalytics.com/2014/01/the-fourier-transform-explained-in-one-sentence.html
$$\boxed{{\displaystyle \hat{f}(\xi )={\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-i2\pi \xi t}f(t)\mathrm{d}t}}}$$

Discrete Fourier transform (DFT):
$${\displaystyle x_n=\sum _{k=0}^{N-1}{X}_k\cdot e^{-i2\pi \frac{n}{N}k}}$$

Inverse transform:
$${\displaystyle X_k=\frac{1}{N}\sum _{n=0}^{N-1}{x}_n\cdot e^{i2\pi \frac{n}{N}k}}$$

One of the most powerful #math / #history videos I've ever seen. Veritasium on Fast Fourier Transforms

https://youtu.be/nmgFG7PUHfo

"Improving Spectral, Spatial, and Mechanistic Resolution Using Fourier Transform Nonlinear Optics: A Tutorial Review"
ACS Phys. Chem Au 2022
https://doi.org/10.1021/acsphyschemau.2c00051
#openaccess #Fouriertransform #Kineticparameters #Lasers #Nonlinearoptics #Plasmons

My favourite textbook of all time – had been in storage for a while as we did a long, convoluted, house move…

It’s back, safe and sound, nicely weathered and thumbed from all those years of use…

PS: just realised it’s my wife’s copy, not mine. I Want My One Back! (It’s still in a box, somewhere…)