一维高斯函数的FFT变换

 

Gaussian FT

 The Fourier transform of a Gaussian function, G(x) is also a Gaussian function, H(k). We illustrate this with the position wavevector conjugate pair.

Substituting in the value of G(x) = exp{-x²/a²} we have

We can obtain a Gaussian integrand by completing the square

Here is the trick. We multiply by exp{-A²k²}

We need to find the constant A such that we can factor the term inside exponential function of the integrand. This procedure is known as completing the square.

The cross term ?ikx determines the value of A

Thus,

The result is

The result shows that the Fourier transform of a Gaussian is also a Gaussian (in the conjugate variable space). Note the inverse relationship of the variable a in the two spaces. As a increases the width of Gaussian in the x (position) space increases and the width in the k (wavevector) space decreases. If we assume that the uncertainty of the Gaussian is proportional to the square root of the variance then

Dx = a/Ö2

Dk = Ö2/a

Thus, Dx Dk = 1 which is a statement of the uncertainty principle. This is so because we can multiply k by h to obtain the momentum, p. Therefore, Dx Dp = h. We can see this relationship graphically in the two Figures below.

For example, note the correspondance between the blue curve in x space (broad distribution) and in k space (narrow distribution).