Fourier Transform (FT)

A Fourier transform is a mathematical transform that decomposes functions depending on space or time into functions depending on spatial frequency or temporal frequency.

Decompose it into pure frequencies.

It’s like unmixing different paint colors. wow this is cool.

Take a pure signal. Converts time to frequency.

Learning it formally in CS370.

Concepts

• Fourier Series
• Root of Unity
• Discrete Fourier Transform
• Inverse Discrete Fourier Transform
• Fast Fourier Transform
• Power Spectrum

Resources

• Fourier Analysis (Data-Driven Science and Engineering) by Steve Brunton

Definition 1: Fourier Transform

If $f\in L^1$ then the Fourier transform of $f$ is $F(\omega )=\mathcal{F}\{f(t)\}={\int }_{-\infty }^{\infty }f(t)e^{-\omega tj}dt$ and the inverse Fourier transform of $F(\omega )$ is $f(t)=\mathcal{F}\{F(\omega )\}=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }F(\omega )e^{\omega tj}d\omega$

Connection between Laplace Transform and Fourier Transform

The Fourier transform/inverse transform is just the two-sided Laplace transform in the case that $s=\omega j$.

• Ahh, that’s why the fourier transform is nice, we are only dealing with imaginary numbers, no real dimension

Related

• Laplace Transform