Understanding the Fourier Transform
The Fourier Transform is based on a foundational idea introduced in 1822 by mathematician Joseph Fourier in his Theory of Heat. It states that any periodic signal—one that repeats identically after a period T—can be expressed as a sum of sine and cosine waves.
📈 A Simple Example
Let’s consider a function f(x) that takes only two values:
- −1 when x is less than 0
- 1 when x is greater than or equal to 0
This function represents a sudden jump from −1 to 1. Although it looks nothing like a sine wave, it can still be approximated using a Fourier series—a weighted sum of sinusoidal functions.
Fourier Series Representation
The Fourier series of f(x) (restricted to a finite domain) can be written as:
%20en%20serie%20de%20Fourier.png "Formule série de Fourrier")
Visualizing the Approximation
As we add more terms to the series:
- The approximation becomes more accurate
- The sharp jump in f(x) is better captured
- The graph begins to resemble the original function
You can illustrate this with a graph showing:
- The approximation using the first term
- The first two terms
- The first ten terms
