Note that a “signal” in this context is just a trigonometric polynomial over a periodic interval. If you think of your periodic interval as representing angle

Note that a “signal” in this context is just a trigonometric polynomial over a periodic interval. If you think of your periodic interval as representing angle measure, and the points in the interval as points on the unit circle in the complex plane, then your trigonometric polynomial can alternately be thought of as a Laurent polynomial in the complex plane. https://en.wikipedia.org/wiki/Laurent_polynomial https://en.wikipedia.org/wiki/Laurent_polynomial What the FFT does is convert between the values of your function at n roots of unity in the complex plane -> the coefficients of the Laurent polynomial interpolating those values.