5.6 Where we are going next

We now have a complete mathematical picture of the frequency domain. But several practical problems stand between this picture and a tool we can run on a computer:

• The Fourier transform is defined over continuous signals, whereas digital audio is sampled.

• It integrates over infinite time, from \(-\infty\) to \(\infty\), which we can never do in practice.

• It is defined over a continuum of frequencies, infinitely many of them.

In the coming chapters, we will resolve each of these. We will derive the discrete Fourier transform in Chapter 8, which replaces the integral with a finite sum over samples and turns the Fourier transform from a mathematical object into a tractable computation. We will also see how to preserve a notion of time, so that instead of one spectrum for an entire signal, we can watch how its frequencies evolve from moment to moment, which is how tools like spectrograms are built. But for now, we understand enough about the frequency domain to study some other important concepts first (modulation synthesis and sampling theory).