Debug Info:
{
"title": "Fourier Transform",
"symbol": "ℱ",
"definition": "The Fourier transform decomposes a function of time into its constituent frequencies.",
"usage": "Used in signal processing, image analysis, and solving partial differential equations.",
"examples": [
{
"description": "Continuous Fourier Transform",
"latex": "\\hat{f}(\\xi) = \\int_{-\\infty}^{\\infty} f(x)e^{-2\\pi i x \\xi} dx"
},
{
"description": "Discrete Fourier Transform",
"latex": "X_k = \\sum_{n=0}^{N-1} x_n e^{-\\frac{2\\pi i}{N}kn}"
}
],
"tags": [
"analysis",
"signal processing",
"differential equations"
],
"relatedConcepts": [
"Laplace transform",
"wavelets",
"frequency domain"
]
}
Fourier Transform
Definition
The Fourier transform decomposes a function of time into its constituent frequencies.
Usage
Used in signal processing, image analysis, and solving partial differential equations.
Examples
-
Description: Continuous Fourier Transform
LaTeX: \hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)e^{-2\pi i x \xi} dx
-
Description: Discrete Fourier Transform
LaTeX: X_k = \sum_{n=0}^{N-1} x_n e^{-\frac{2\pi i}{N}kn}