Any signal — sound, image, stock prices, neural spikes — can be decomposed into sine waves of different frequencies. The Fourier transform tells you which frequencies are present and how strongly. The transform is invertible: signal ↔ spectrum is a unitary change of basis. Draw a signal; see its spectrum. Pick a preset; see what makes the signal sharp or smooth.
why "every signal is sine waves" is true
For a periodic signal of period T, Fourier's claim (1807) was: any such signal can be written as a sum of sines and cosines with frequencies 1/T, 2/T, 3/T, …:
f(t) = a₀/2 + Σ_n [aₙ cos(2π n t / T) + bₙ sin(2π n t / T)]
At first this seemed false (Lagrange dismissed it). It turned out to be one of the most consequential ideas in math: the Fourier transform generalizes to any (suitable) signal on the real line, and inversion lets you reconstruct it exactly.
What this picture shows:
• A pure sine has one frequency spike — narrow in spectrum.
• A square wave has odd harmonics at 1/3, 1/5, 1/7, ... — Gibbs phenomenon at edges.
• A pulse train has uniformly-spaced harmonics (related to comb functions).
• A chirp has frequencies that change with time, smearing across the spectrum.
• White noise has a flat spectrum.
Time-frequency uncertainty: a signal sharply localized in time has a wide spectrum, and vice versa. This is the Fourier uncertainty principle — and it's the math behind Heisenberg's quantum uncertainty principle, which is just Fourier uncertainty for wavefunctions.
Applications: MP3 (Fourier on overlapping windows discards inaudible frequencies). JPEG (Fourier-like discrete cosine transform). MRI (image reconstructed from k-space, which IS the 2D Fourier transform of the spin density). NMR, X-ray crystallography, radar, mobile-phone codecs, all of digital signal processing. The Fourier transform is the single most-used mathematical operation in technology.