Fourier Transform

Tldr

The Fourier transform takes a function defined in space (for example, a physical quantity like temperature or fluid velocity) and converts it into a frequency domain.

Fourier Transform

The continuous Fourier transform of a time-domain signal $f(t)$ is given by:

$$\hat{f}(\omega )={\int }_{-\infty }^{\infty }f(t)e^{-i\omega t}dt$$

Where:

• $f(t)$ is the original time-domain function.
• $\hat{f}(\omega )$ is the Fourier transform of $f(t)$, which is the frequency-domain function.
• $\omega$ is the angular frequency, and $\omega =2\pi f$, where $f$ is the frequency.
• $e^{-i\omega t}$ is the complex exponential kernel.

Inverse Fourier Transform

The inverse Fourier transform, which converts a frequency-domain function $\hat{f}(\omega )$ back to the time-domain, is given by:

$$f(t)=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }\hat{f}(\omega )e^{i\omega t}d\omega$$