The Fourier transform extracts a frequency component by untwisting it into DC.

Evidence

One way of thinking about this is to remember that the complex exponential representation of a sine wave, ej t ω , is a spiral as a function of time, as shown in Figure A1.1, and integration is merely adding up all the numbers within the integration range. Equation A1.5a shows that to get the DC content, all one has to do is add up all the signal values. So what e j t −ω is doing, as a spiral in the other direction, is to untwist the waveform at the frequency specified. This converts it to DC, where it can be simply extracted by adding up all the DC values in the integration range. The inverse transform retwists the coefficient to the original frequency, amplitude and phase, and then all the retwisted sine waves are added up to form the waveform. — David M. Howard, Jamie Angus, Acoustics and Psychoacoustics, 2768

Categorization

• XREF: The spiral-as-complex-exponential picture connects to the book’s earlier treatment of sine waves as rotating phasors.
• SEED: Angle: ‘Untwisting sound’ — how a geometric intuition (opposite-wound spirals) replaces the algebra of the transform for non-mathematicians.