Fourier Transform

Colorized Definition

fourier_transform

\newcommand{\energy}{\color{c1}}
\newcommand{\freq}{\color{c2}}
\newcommand{\spin}{\color{c3}}
\newcommand{\signal}{\color{c4}}
\newcommand{\Circle}{\color{c5}}
\newcommand{\average}{\color{c6}}

\displaystyle{\energy X_{\freq k} \plain =
\average \frac{1}{N} \sum_{n=0}^{N-1}
\signal x_n
\spin e^{\mathrm{i} \Circle 2\pi \freq k
\average \frac{n}{N}}}

\plain     To find
\energy    the energy
\freq      at a particular frequency,
\spin      spin
\
\signal    your signal
\Circle    around a circle
\freq      at that frequency,
\plain     and
\
\average   average a bunch of points along that path.

Plain English

  • 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.

## Read More - [Interactive Guide to the Fourier Transform](https://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/)

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