5.5 What is the Fourier transform doing?

5.5 What is the Fourier transform doing?#

The definition of the Fourier transform above can feel like it appears out of nowhere. That is okay. Let us unpack it more intuitively.

How can a single integral possibly pick out the amount of one specific frequency \(\omega\) hiding inside an arbitrary signal? Look again at the transform, \(X(\omega) = \int x(t)\, e^{-j\omega t}\, dt\), and read it in three steps:

  1. Synthesize a phasor at \(-\omega\). The term \(e^{-j\omega t}\) is a complex sinusoid rotating at frequency \(\omega\), just in the clockwise direction (the minus sign reverses the direction of rotation).

  2. Multiply to measure similarity. Multiplying the signal \(x(t)\) by this phasor “winds” the signal around the complex plane at rate \(\omega\). Wherever the signal’s own oscillation matches the winding rate, the product reinforces in a consistent direction.

  3. Integrate to sum over time. The integral adds up the wound signal across all time, accumulating that reinforcement (or lack of it) into a single complex number.

Intuitively, we are measuring the correlation between \(x(t)\) and a phasor probing at frequency \(\omega\). The cleanest way to see this is to look at the wound-up signal in the complex plane and track its center of mass (the average of all the wound points). When the probe frequency matches a frequency present in the signal, the winding lines up and the center of mass is pulled far from the origin, yielding a large \(|X(\omega)|\). When the probe frequency does not match, the winding smears symmetrically around the origin, the contributions cancel, and the center of mass sits near zero:

Three complex-plane plots of the same signal wound at different probe frequencies. At a probe of 2 Hz the curve forms balanced lobes and the center of mass sits at the origin. At 3 Hz, matching the signal, the curve bunches to one side and the center of mass is pulled well away from the origin. At 4 Hz the center of mass returns to near the origin.

Fig. 21 Winding a signal (here a 3 Hz oscillation) around the complex plane at three probe frequencies. The red dot is the center of mass. Only at the matching probe of 3 Hz (middle) is the center of mass pulled away from the origin, signaling a large amplitude at that frequency. At non-matching probes (2 and 4 Hz), the contributions cancel and the center of mass stays near zero.#

The center of mass is an average, and the integral in the Fourier transform is a (continuous) sum, so the two are proportional. The Fourier transform sweeps this probe across every frequency \(\omega\) and records, for each one, how far off-origin the center of mass lands.