Take a signal composed of multiple frequencies, and have its intensity be the length of a vector (rotating at some other frequency) about the origin (i.e. a polar curve). If we take the x-coordinate of the centre of mass of that polar curve, we see a spike at where the frequency of rotation of the vector is equal to the frequency of oscillation of the signal. This is the “almost-Fourier Transform”, and it is linear: the almost-Fourier Transform of the sum of two signals equals the sum of the almost-Fourier transforms of the two signals.
gives a rotating vector with an angular frequency of and a magnitude of the height of . Integrating that (and dividing by the time interval) gives you the centre of mass - the “almost-Fourier Transform”. The actual Fourier transform is this integral, but not divided by the length of the time interval (so scaled up). The Fourier transform is notated as .