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In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration ...
Taking the Fourier transform (unitary, angular-frequency convention) of a Gaussian function with parameters a = 1, b = 0 and c yields another Gaussian function, with parameters , b = 0 and /. [3] So in particular the Gaussian functions with b = 0 and c = 1 {\displaystyle c=1} are kept fixed by the Fourier transform (they are eigenfunctions of ...
The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced sine and cosine transforms (which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation.
Unlike the sampled Gaussian kernel, the discrete Gaussian kernel is the solution to the discrete diffusion equation. Since the Fourier transform of the Gaussian function yields a Gaussian function, the signal (preferably after being divided into overlapping windowed blocks) can be transformed with a fast Fourier transform , multiplied with a ...
Gaussian window, σ = 0.4. The Fourier transform of a Gaussian is also a Gaussian. Since the support of a Gaussian function extends to infinity, it must either be truncated at the ends of the window, or itself windowed with another zero-ended window. [46]
As in short time Fourier transform, the resolution in time and frequency domain can be adjusted by choosing different window function width. In Gabor transform cases, by adding variance , as following equation: The scaled (normalized) Gaussian window denotes as:
All these extensions are also called normal or Gaussian laws, so a certain ambiguity in names exists. The multivariate normal distribution describes the Gaussian law in the k-dimensional Euclidean space. A vector X ∈ R k is multivariate-normally distributed if any linear combination of its components Σ k j=1 a j X j has a (univariate) normal ...
Special examples are the Gaussian quadrature for polynomials and the Discrete Fourier Transform for plane waves. It should be stressed that the grid points and weights, x i , w i {\displaystyle x_{i},w_{i}} are a function of the basis and the number N {\displaystyle N} .