Chapter 19 Interactive Lab

Fourier: Hidden Waves in Signals

A signal is a vector. Fourier analysis rewrites that vector in a wave language. Use the controls below to build signals, reveal frequencies, filter noise, compress information, and see the linear algebra behind the FFT.

1. Build a signal from waves

Adjust amplitudes and frequencies. The signal below is a linear combination of sine waves.

Linear algebra view: each wave is a vector. The signal is a weighted sum of wave vectors.

2. Frequency spectrum

The spectrum shows how much each frequency is present. Peaks should appear near the frequencies you selected above.

3. Low-pass filtering

A low-pass filter keeps slow waves and removes fast waves. Try changing the cutoff.

Think: What happens when the cutoff is below one of the true signal frequencies?

4. Compression by keeping coefficients

Fourier compression keeps the largest wave coordinates and discards the rest.

5. Sharp edges need many frequencies

Smooth signals compress well. Signals with jumps need many frequencies. This is why edges and discontinuities are expensive in Fourier language.

6. Tiny image and 2D frequency intuition

This synthetic image contains horizontal and vertical wave patterns plus a bright object. The right panel shows a simplified Fourier magnitude picture.

Low spatial frequencies describe smooth background; high spatial frequencies describe edges and texture.

7. Chapter summary

Time language

Coordinates are sample values: what happens at each time.

Frequency language

Coordinates are wave strengths: which frequencies are hidden inside.

Linear algebra

The DFT is a matrix transformation. The FFT computes it efficiently.

Applications

Filtering, compression, denoising, audio, images, PDEs, and AI features.