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Visualization of the Fourier transform. Showing how a function is resolved into its frequency components through complex phase accumulation, spectral peaks and inverse reconstruction.

Philosophy Of Physics

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15,142 次观看 • 2 个月前 •via X (Twitter)

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A lens can perform the Fourier transform! 🔍 This is the 4F system: the first lens converts an image into its frequency components, and the second lens reconstructs it. By adding an aperture at the Fourier plane, we can filter out high frequencies. #Optics #Physics #b3d

A lens can perform the Fourier transform! 🔍 This is the 4F system: the first lens converts an image into its frequency components, and the second lens reconstructs it. By adding an aperture at the Fourier plane, we can filter out high frequencies. #Optics #Physics #b3d

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The Fourier transform runs inside every MRI machine, every audio compressor, every signal processor on earth. JPMorgan pays $350K to engineers who can derive it from scratch. There is one professor alive who explains it the way nobody else can. His name is Gilbert Strang. MIT. The most brilliant mathematical mind of his generation. His textbook sits in over 10 million homes. No other person on earth holds this combination of depth and clarity in one head. He opens with one confession. The Fourier transform is unreasonably effective. It solves problems it was never designed for. The goal is not to compute it. The goal is to understand why it works at all. Then the core idea. A periodic signal breaks into pure sine waves. Each with a frequency, amplitude, and phase. The transform finds all three simultaneously. A complicated signal in time becomes a simple picture in frequency. Then 1807. Fourier invented the transform not for sound but for heat. How does warmth spread through a metal rod? In time the equation is a partial differential equation - hard. In frequency space it becomes ordinary - easy. Transform in, solve it, transform back. Then convolution. Convolution in time equals multiplication in frequency. Filtering a signal, removing noise, compressing audio - all multiplication in frequency space. Without the transform: thousands of computations. With it: a few multiplications. Then the delta function. Zero everywhere except one point where it is infinite. Integral equals one. Every mathematician in 1900 said it was not a function. Dirac used it anyway. It took 40 years to justify what engineers had been doing the whole time. Watch the moment Strang shows that the Fourier transform of a delta function is a constant - every frequency in equal measure. The more concentrated in time, the more spread in frequency. This is the uncertainty principle. Not quantum physics. A theorem about any signal at all. A signal processing engineer I know rewatched this before their first project at Apple. Said it was the first time the Fourier transform felt like a change of coordinates rather than a formula to memorize. Free on YouTube, MIT OpenCourseWare. bookmark this and watch later - after this lecture every sound, every image, and every signal will feel like a sum of sine waves waiting to be separated

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