Gaussian Function

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook

GaussianFunction1D

GaussianReIm

GaussianContours

In one dimension, the Gaussian function is the probability density function of the normal distribution,

[画像: f(x)=1/(sigmasqrt(2pi))e^(-(x-mu)^2/(2sigma^2)), ]

(1)

sometimes also called the frequency curve. The full width at half maximum (FWHM) for a Gaussian is found by finding the half-maximum points x_0. The constant scaling factor can be ignored, so we must solve

e^(-(x_0-mu)^2/(2sigma^2))=1/2f(x_(max))

(2)

But f(x_(max)) occurs at x_(max)=mu, so

e^(-(x_0-mu)^2/(2sigma^2))=1/2f(mu)=1/2.

(3)

Solving,

e^(-(x_0-mu)^2/(2sigma^2))=2^(-1)

(4)

[画像: -((x_0-mu)^2)/(2sigma^2)=-ln2 ]

(5)

(x_0-mu)^2=2sigma^2ln2

(6)

x_0=+/-sigmasqrt(2ln2)+mu.

(7)

The full width at half maximum is therefore given by

FWHM=x_+-x_-=2sqrt(2ln2)sigma approx 2.3548sigma.

(8)

GaussianFunction2D

In two dimensions, the circular Gaussian function is the distribution function for uncorrelated variates X and Y having a bivariate normal distribution and equal standard deviation sigma=sigma_x=sigma_y,

[画像: f(x,y)=1/(2pisigma^2)e^(-[(x-mu_x)^2+(y-mu_y)^2]/(2sigma^2)). ]

(9)

The corresponding elliptical Gaussian function corresponding to sigma_x!=sigma_y is given by

[画像: f(x,y)=1/(2pisigma_xsigma_y)e^(-[(x-mu_x)^2/(2sigma_x^2)+(y-mu_y)^2/(2sigma_y^2)]). ]

(10)

GaussianApodization

The Gaussian function can also be used as an apodization function

A(x)=e^(-x^2/(2sigma^2)),

(11)

shown above with the corresponding instrument function. The instrument function is

I(k)=e^(-2pi^2k^2sigma^2)sigmasqrt(pi/2)[erf((a-2piiksigma^2)/(sigmasqrt(2)))+erf((a+2piiksigma^2)/(sigmasqrt(2)))],

(12)