ΕΠΕΞΕΡΓΑΣΙΑ ΣΗΜΑΤΩΝ (ECE-Y523)

The Fast Fourier Transform is used everywhere but it has a fascinating origin story that could have ended the nuclear arms race.

Discrete convolutions, from probability, to image processing and FFTs.

Εκτός των άλλων και μια εξαιρετική παρουσίαση της συνέλιξης απ' ευθείας ή μέσω του DFT και του FFT.

In this video, we take a look at one of the most beautiful algorithms ever created: the Fast Fourier Transform (FFT). This is a tricky algorithm to understand so we take a look at it in a context that we are all familiar with: polynomial multiplication. You will see how the core ideas of the FFT can be "discovered" through asking the right questions. The key insights that are presented in this video is that polynomial multiplication can be improved significantly by multiplying polynomials in a special value representation. The challenge that presents itself is the problem of converting a polynomial from a standard coefficient representation to value representation. We see that the FFT is an incredibly efficient recursive algorithm that performs this task, and we also discover that a slightly tweaked FFT (Inverse FFT) can also solve the reverse problem of interpolation. If this video doesn't blow your mind, I don't know what will.

0:00 Introduction

2:19 Polynomial Multiplication

3:36 Polynomial Representation

6:06 Value Representation Advantages

7:07 Polynomial Multiplication Flowchart

8:04 Polynomial Evaluation

13:49 Which Evaluation Points?

16:30 Why Nth Roots of Unity?

18:28 FFT Implementation

22:47 Interpolation and Inverse FFT

26:49 Recap