But what is the Fourier Transform? A visual introduction.
By 3Blue1Brown
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Introduction to Fourier Transform
🎧 The Fourier Transform is a powerful mathematical tool designed to decompose complex signals (like sound waves) into their constituent pure frequencies.
🔊 It helps to unmix jumbled signals, similar to separating mixed paint colors, by identifying the individual frequencies that combine to form the complex waveform.
🌐 This concept extends far beyond sound, finding ubiquitous application in diverse areas of mathematics and physics.
The Winding Machine Mechanism
🌀 Visualize a signal by wrapping its time-intensity graph around a circle at various "winding frequencies."
⚖️ The center of mass of this wound-up graph provides insight; it spikes significantly when the winding frequency precisely matches a frequency present in the original signal.
💡 This "almost Fourier transform" effectively identifies individual frequencies even within a complex signal composed of multiple pure frequencies due to its linear property.
Mathematical Formalization
📈 The full Fourier Transform considers the 2D center of mass of the wound-up graph, elegantly represented as a complex number in the complex plane.
⚛️ The winding action is precisely described by Euler's formula (`e^(-2πift)`) multiplied by the signal function, encapsulating the rotation and scaling.
📊 The transform is computed as an integral of this complex-valued function, essentially measuring the overall displacement of the center of mass, with its magnitude scaling by the duration of the frequency's presence.
Key Points & Insights
➡️ The Fourier Transform of a signal outputs a new function (`g-hat`) that takes a frequency as input and provides a complex number indicating the strength and phase of that frequency in the original signal.
🔊 A practical application includes sound editing, where the transform allows filtering out specific annoying frequencies by identifying and dampening their corresponding spikes.
🧠 Understanding exponentials as rotation and integrals as center-of-mass calculations provides rich intuitive meaning to the seemingly daunting Fourier Transform formula.
📸 Video summarized with SummaryTube.com on Aug 08, 2025, 04:04 UTC
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📜Transcript
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📄Video Description
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An animated introduction to the Fourier Transform.
Help fund future projects: https://www.patreon.com/3blue1brown
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: http://3b1b.co/fourier-thanks
Learn more about Janestreet: https://janestreet.com/3b1b
Follow-on video about the uncertainty principle: https://youtu.be/MBnnXbOM5S4
Interactive made by a viewer inspired by this video:
https://prajwalsouza.github.io/Experiments/Fourier-Transform-Visualization.html
Also, take a look at this Jupyter notebook implementing this idea in a way you can play with:
https://github.com/thatSaneKid/fourier/blob/master/Fourier%20Transform%20-%20A%20Visual%20Introduction.ipynb
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Russian: xX-Masik-Xx
Vietnamese: @ngvutuan2811
------------------
Animations largely made using manim, a scrappy open-source python library. https://github.com/3b1b/manim
If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and has many other quirks you might expect in a library someone wrote with only their own use in mind.
Music by Vincent Rubinetti.
Download the music on Bandcamp:
https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown
Stream the music on Spotify:
https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u
If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe, and click the bell to receive notifications (if you're into that).
If you are new to this channel and want to see more, a good place to start is this playlist: http://3b1b.co/recommended
Various social media stuffs:
Website: https://www.3blue1brown.com
Twitter: https://twitter.com/3Blue1Brown
Patreon: https://patreon.com/3blue1brown
Facebook: https://www.facebook.com/3blue1brown
Reddit: https://www.reddit.com/r/3Blue1Brown
Full video URL: youtube.com/watch?v=spUNpyF58BY
Duration: 18:40
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AI Summary of "But what is the Fourier Transform? A visual introduction."
Get instant insights and key takeaways from this YouTube video by 3Blue1Brown.
Introduction to Fourier Transform
🎧 The Fourier Transform is a powerful mathematical tool designed to decompose complex signals (like sound waves) into their constituent pure frequencies.
🔊 It helps to unmix jumbled signals, similar to separating mixed paint colors, by identifying the individual frequencies that combine to form the complex waveform.
🌐 This concept extends far beyond sound, finding ubiquitous application in diverse areas of mathematics and physics.
The Winding Machine Mechanism
🌀 Visualize a signal by wrapping its time-intensity graph around a circle at various "winding frequencies."
⚖️ The center of mass of this wound-up graph provides insight; it spikes significantly when the winding frequency precisely matches a frequency present in the original signal.
💡 This "almost Fourier transform" effectively identifies individual frequencies even within a complex signal composed of multiple pure frequencies due to its linear property.
Mathematical Formalization
📈 The full Fourier Transform considers the 2D center of mass of the wound-up graph, elegantly represented as a complex number in the complex plane.
⚛️ The winding action is precisely described by Euler's formula (`e^(-2πift)`) multiplied by the signal function, encapsulating the rotation and scaling.
📊 The transform is computed as an integral of this complex-valued function, essentially measuring the overall displacement of the center of mass, with its magnitude scaling by the duration of the frequency's presence.
Key Points & Insights
➡️ The Fourier Transform of a signal outputs a new function (`g-hat`) that takes a frequency as input and provides a complex number indicating the strength and phase of that frequency in the original signal.
🔊 A practical application includes sound editing, where the transform allows filtering out specific annoying frequencies by identifying and dampening their corresponding spikes.
🧠 Understanding exponentials as rotation and integrals as center-of-mass calculations provides rich intuitive meaning to the seemingly daunting Fourier Transform formula.
📸 Video summarized with SummaryTube.com on Aug 08, 2025, 04:04 UTC
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As an Amazon Associate, we earn from qualifying purchases
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