Fourier Transform ~ Engineering Definition And Students

Wednesday, September 4, 2019

Fourier Transform

By Suryakant PrasadSeptember 04, 2019 No comments:

The Fourier Transform

1.1 Fourier transforms as integrals
There are several ways to define the Fourier transform of a function f : R → C. In this section, we define it using an integral representation and state some basic uniqueness and inversion properties, without proof. Thereafter, we will consider the transform as being defined as a suitable limit of Fourier series, and will prove the results stated here.

The Fourier Transform is one of deepest insights ever made. Unfortunately, the meaning is buried within dense equations:

$\displaystyle{X_k = \sum_{n=0}^{N-1} x_n \cdot e^{-i 2 \pi k n / N}}$

$\displaystyle{x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k \cdot e^{i 2 \pi k n / N}}$

Yikes. Rather than jumping into the symbols, let's experience the key idea firsthand. Here's a plain-English metaphor:

What does the Fourier Transform do? _Given a smoothie, it finds the recipe.
How? _Run the smoothie through filters to extract each ingredient.
Why? _Recipes are easier to analyze, compare, and modify than the smoothie itself.
How do we get the smoothie back? _Blend the ingredients.

Here's the "math English" version of the above:The Fourier Transform takes a time-based pattern, measures every possible cycle, and returns the overall "cycle recipe" (the amplitude, offset, & rotation speed for every cycle that was found).

Time for the equations? No! _Let's get our hands dirty and experience how any pattern can be built with cycles, with live simulations.

If all goes well, we'll have an aha!_moment and intuitively realize why the Fourier Transform is possible. We'll save the detailed math analysis for the follow-up.

This isn't a force-march through the equations, it's the casual stroll I wish I had. Onward!

function
Fourier transform--1	1
Fourier transform--cosine
Fourier transform--delta function
Fourier transform--exponential function
Fourier transform--Gaussian
Fourier transform--Heaviside step function
Fourier transform--inverse function
Fourier transform--Lorentzian function
Fourier transform--ramp function
Fourier transform--sine