6. Properties of the DFT#

the real and imaginary parts of a discrete Fourier spectrum

In this chapter, we’ll go more in depth with the DFT to understand its fundamental properties. Specifically, we’ll cover the following four topics:

  • Linearity: what happens when we mix signals?

  • The DFT shifting theorem: what happens when we delay signals?

  • Conjugate symmetry: what is going on with negative frequencies?

  • Spectral leakage: what happens to non-analysis frequencies?

A fair word of warning: this chapter will be technical and possibly a bit dry. However, a good understanding of these properties will help us reason about the DFT applied to more complicated signals, and eventually, what the DFT can tell us about filtering.