Beyond Fourier

Fourier Analysis · Axiom Academy

Discovering the limitations of Fourier analysis and when wavelets shine. The Fourier transform tells us what frequencies are in a signal, but not when they occur. Adjust the time slider to see this limitation. A chirp is a signal whose frequency changes continuously over time. Try adjusting the chirp rate to see how frequency evolves. A spectrogram divides the signal into time windows and computes the Fourier transform for each window. This gives us frequency vs. time information. Wavelets Provide Adaptive Resolution Unlike the fixed-window spectrogram, wavelets use different window sizes for different frequencies. Compare the two approaches below. Same resolution at all frequencies Better time resolution at high frequencies Use Fourier when you only care about frequency content, not timing. Perfect for steady-state signals, finding dominant frequencies, or filtering applications. Use wavelets when timing matters: transient events, chirps, edge detection in images, compression (JPEG2000), or any signal where frequency changes over time. You can't have perfect time AND frequency resolution simultaneously. This is a fundamental limit in signal processing, not just a limitation of our tools. Wavelets give us the flexibility to trade off resolution where we need it most.

This is the written version of the interactive lesson above. See the full Fourier Analysis course.