Fourier + Inverse Fourier Transform
Table of Contents
Fourier Transform #
The Fourier transform is used to transform a signal from the time domain to the frequency domain.
The Fourier transform formula is:
$$ X(\omega) = \int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt $$
Where
- \( \small X(\omega) \) is the Fourier transform of the function \( \small x(t) \),
- \( \small x(t) \) is the input signal in the time domain,
- \( \small j \) is the imaginary unit,
- \( \small e \) is Euler’s number,
- \( \small \omega \) is the angular frequency.
Inverse Fourier Transform #
The inverse Fourier transform is used to transform a signal from the frequency domain to the time domain.
The inverse Fourier transform is:
$$ x(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty}X(\omega)e^{j\omega t}d\omega $$
Note that there are many conventions for these two transforms. \(\small i \) is often used as the imaginary unit, and in spatial contexts, \(\small x \) and \(\small k \) are often used instead of \(\small t \) and \(\small \omega \), respectively.