Here is the complete set of slides that are used during lectures. These slides are compiled at the end of the year and left here for reference. Updated slides are posted below as the class progresses.
This introductory block consists of a single lecture. It previews the class’s contents and introduces the concepts of signals and information. The slides from this lecture can be found here.
This block is an introduction to discrete signals. We start by defining discrete signals and introducing some common types of discrete signals that we will be using in this course. We then discuss some useful properties of discrete signals. Next, we introduce the concept of inner products and what they mean in relation to discrete signals. We finish by introducing discrete complex exponentials, their properties, and why they are useful for us in this class.
Discrete Fourier Transform (DFT)
This set of lectures introduces the DFT. We begin by defining the DFT and seeing how it relates to complex exponentials and inner products. We will look at the periodicity of the DFT and we will learn how to interpret the DFT as a rate of change of a signal. We will study the DFTs of some useful discrete signals.
We will go on to introduce the inverse discrete Fourier transform (iDFT). We will prove that the the iDFT is, in fact, the inverse of the DFT. We will then look at the iDFT both as an inner product (and how this compares to the DFT) and as a series of successive approximations to a signal. We will see how these approximations can be used to reconstruct a square pulse, and we will take this logic to the next step to see how we can use the DFT and iDFT to “clean up” a noisy signal.
Finally, we will investigate and prove several critical properties of the DFT and iDFT, including symmetry, energy conservation (Parseval’s Theorem), and linearity.