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This is how a circle produces sine and cosine waves.

93,943 просмотров • 1 месяц назад •via X (Twitter)

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Most people first see Euler’s Formula as a strange equation in a textbook. Then years later, they realize it quietly powers the modern world. Leonhard Euler discovered that these seemingly unrelated mathematical ideas: → exponential growth → imaginary numbers → sine waves → cosine waves → rotation … are all deeply connected through one identity: e^(iθ) = cos(θ) + i·sin(θ) At first glance it looks impossible. How can an exponential function suddenly produce circles and waves? The key insight is that multiplying by a complex exponential creates rotation. As the angle θ changes: - the cosine term tracks horizontal motion - the sine term tracks vertical motion - together they trace a perfect circle in the complex plane Euler showed that waves and rotation are mathematically the same phenomenon viewed differently. That single idea changed science and engineering forever. Today, Euler’s Formula sits underneath: → Fourier Transforms → signal processing → wireless communication → MRI scanners → quantum mechanics → electrical engineering → audio compression → radar systems → GPS → neural network frequency analysis Even modern AI systems indirectly rely on mathematics built on top of these foundations. The famous special case is Euler’s Identity: e^(iπ) + 1 = 0 Richard Feynman reportedly called it: "our jewel." Because it revealed that mathematics is not a collection of separate topics. It is one connected language describing reality itself.

Tech with Mak

25,383 просмотров • 3 месяцев назад

Discrete Fourier Transform by hand ✍️ ~ 12 steps walkthrough below Here is a little-known secret about the DFT and the inverse DFT: it is just matrix multiplication in both directions, one the transpose of the other, exactly like the forward pass and backpropagation I drew in other examples. Goal: recover which cosine waves a signal is made of, using nothing but multiplication and addition. = 1. Given = Three signals written as sums of cosines, and a fourth, X, that we do not know yet. = 2. Frequency matrix F = Let us write the coefficients as a matrix. Each signal is a row, each frequency a column, so A = cos(w) + 2cos(2w) becomes [1, 2, 0, 0]. = 3. Sample the waves = We read the four cosine waves at ten discrete time points. That word "discrete" is the whole difference between this and the continuous transform. = 4. Cosine matrix W = Let us write those samples as a matrix: each frequency a row, each time point a column. = 5. Frequency to time = We multiply F by W. That combines the four cosine waves in the proportions F specifies, and the result T is the three signals as they would look in time. = 6. Transpose = Let us stand each signal up as a column. = 7. Time to frequency = We multiply W by that transpose. Every cell is the dot product of one signal with one cosine wave, which measures how much of that wave the signal contains. Zero means none of it. = 8. Scale = Let us multiply by 2/n, with n = 10. The projections come out five times too large, and this is the correction. = 9. Transpose back = We turn it back around, and it is F again, exactly. That is the check: the transform recovered the coefficients we started from. = 10. Now solve for X = Let us run the same multiplication on the one signal whose recipe we never knew. = 11. Scale = We divide by 5 again. = 12. Transpose back = And X reads [0, 0, 3, 2], which says X = 3cos(3w) + 2cos(4w). Note: I originally drew this to show that the DFT is a special case of a convolution layer, its filters fixed to sine and cosine waves rather than learned. No wonder, then, that a convolution layer free to learn its own filters can be trained to process signals. 💾 Save this post!

Tom Yeh

25,575 просмотров • 15 дней назад