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Fourier & the frequency domain
Every signal is a chord. Learn to read the notes.
Fourier's idea — that any signal decomposes into pure sinusoids — may be the most useful single insight in applied mathematics. This course builds it from linear algebra (signals as vectors, spectra as coordinates), makes it computable (DFT and FFT), and confronts its real-world fine print: sampling, aliasing, leakage, and windows.
Your progress
8 lessons
The big ideas
Transforms are changes of basis
Sinusoids form an orthogonal basis for signals; the spectrum is just coordinates in that basis, found by projection. Fourier analysis is linear algebra with headphones on.
Two views, one signal
Time and frequency descriptions are equivalent and invertible. Tangled questions in one view are often trivial in the other — and convolution becomes multiplication.
Sampling has a precise treaty
More than two samples per cycle and the dots hold everything; fewer, and frequencies lie (aliasing). The digital age rests on this theorem.
Finite windows have consequences
Real spectra come from snippets, and cutting causes leakage. Windows, the STFT, and the spectrogram are the honest craft of practical analysis.
The course — start at lesson one
- 01Why sinusoids? Signals as vectorsInner products, orthogonality, projection — the big idea behind everything.Hard
- 02Fourier seriesEvery repeating signal is a sum of harmonics; timbre is the recipe.Hard
- 03The Fourier transformSpectra for everything; convolution becomes multiplication.Hard
- 04DFT & FFTFourier made computable — and the N log N trick that changed the world.Hard
- 05Sampling & aliasingThe Nyquist treaty, and the impostors that punish violations.Hard
- 06Reconstruction & sincFrom dots back to waves — why sinc is the one legal brushstroke.Hard
- 07Windowing, leakage & the STFTHonest spectra from finite data; spectrograms as machine hearing.Hard
- 08The z-transform & pole-zero designThe discrete designer's map: place poles, sculpt filters, read stability.Hard
Out in the world
Compression: MP3 & JPEG
Transform, keep the perceptually large coordinates, discard the rest — projection as a product.
Wi-Fi & 5G
OFDM transmits data inside inverse FFTs, hundreds of orthogonal subcarriers at once.
Medical & scientific imaging
MRI reconstructs bodies by Fourier-transforming spin echoes; spectroscopy reads molecules by their frequencies.
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