Gaussian Function. In one dimension, the Gaussian function is the probability density function of the normal distribution , sometimes also called the frequency curve. The full width at half maximum (FWHM) for a Gaussian is found by finding the half-maximum points . The constant scaling factor can be ignored, so we must solve.. Fourier Transform of a Scaled and Shifted Gaussian. In Equation [1], we must assume K >0 or the function g (z) won’t be a Gaussian function (rather, it will grow without bound and therefore the Fourier Transform will not exist). Observe that we have defined the constant c =sqrt ( 4* pi * K ). We can relate the function h (z) and n (z) by the.
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The Fourier transform is a generalization of the complex Fourier series in the limit as . Replace the discrete with the continuous while letting . Then change the sum to an integral , and the equations become. Here, is called the forward () Fourier transform, and. is called the inverse () Fourier transform. The notation is introduced in Trott.. Fourier Transform of the Gaussian Konstantinos G. Derpanis October 20, 2005 In this note we consider the Fourier transform1 of the Gaussian. The Gaussian function, g(x), is defined as, g(x) = 1 σ √ 2π e −x2 2σ2, (3) where R ∞ −∞ g(x)dx = 1 (i.e., normalized). The Fourier transform of the Gaussian function is given by: G.
