i and the Fourier Transform; what do they have to do with each other? The answer is the complex exponential. It's called complex because the "i" turns an exponential function into a spiral containing within it a cosine wave and a sine wave. By using convolution, these two functions allow the Fourier Transform to model almost any signal as a collection of sinusoids.
In this video, we look at an intuitive way to understand what "i" is and what it is doing in the Fourier Transform.
Other videos of interest:
Convolution and the Fourier Transform:
[ Ссылка ]
Convolution playlist:
[ Ссылка ]
How Imaginary Numbers were invented:
[ Ссылка ]
0:00 - Introduction
1:15 - Ident
1:20 - Welcome
1:29 - The history of imaginary numbers
3:48 - The origin of my quest to understand imaginary numbers
4:32 - A geometric way of looking at imaginary numbers
9:37 - Looking at a spiral from different angles
10:39 - Why "i" is used in the Fourier Transform
10:44 - Answer to the last video's challenge
11:39 - How "i" enables us to take a convolution shortcut
13:05 - Reversing the Cosine and Sine Waves
15:01 - Finding the Magnitude
15:12 - Finding the Phase
15:20 - Building the Fourier Transform
15:38 - The small matter of a minus sign
16:34 - This video's challenge
17:10 - End Screen
The imaginary number i and the Fourier Transform
Теги
i and the Fourier Transformi adn the fourier transformi and teh fourier transformi and the fourier tranformi adn teh fourier tranformConvolutionImaginary NumbersImaginary NumberComplex ExponentialCosine WavesCosine WaveSinusoidSineCosineconvolution integralconvolution signals and systemsconvolution theorem fourier transformcosine functionsimaginary numbers explainedcomplex exponential functionThe imaginary number i and the Fourier Transform