Take a signal g(t) and wrap it around a cylinder, completing one turn every 1/f seconds. At most winding frequencies the wrapped curve is balanced and its center of mass sits near the axis. But when f matches a frequency actually present in g, the coils stack in phase and the whole shape lurches off-center — that off-centeredness is |X(f)|, the Fourier transform. Sweep f and the peaks of the spectrum light up.
wound signal — g(t) wrapped at freq fcenter of mass — average of the coilFormally X(f) = ∫ g(t) e−2πift dt — multiplying by e−2πift is exactly "wind g around the origin at frequency f," and the integral is "find the center of mass." When f is off-resonance the coil is symmetric and the integral cancels to nearly zero; when f hits a tone in the signal every wrap lands on top of the last, the mass piles up on one side, and |X(f)| spikes. That is why the spectrum below has sharp peaks at 3 and 7 Hz (the second half as tall, matching its ½ amplitude). This transform is the workhorse of signal processing: filtering, audio EQ, MRI reconstruction, and detecting a motor's vibration signature all come down to reading these peaks. Keys: ← → tune f.