1. correlation Correlation of g(t) with f(t):  -  g(t)h(t+  ) d  which is the Fourier transform of  G(f * )H(f)  Closely related to the convolution theorem, relevant for finding features in data. Properties of Fourier transforms (4):

2. The Central Limit Theorem The Central Limit Theorem says: The convolution of the convolution of the convolution etc. approaches a Gaussian. Mathematically, f(x) * f(x) * f(x) * f(x) *... * f(x)  exp[(-x/a) 2 ] or: f(x)* n  exp[(-x/a) 2 ] The Central Limit Theorem is why nearly everything has a Gaussian distribution.

3. 5. Encontrar la transformada de Fourier de la función:

4. =

5. The Central Limit Theorem for a square function Note that  (x)* 4 already looks like a Gaussian!

6. The autoconvolution of a function f(x) is given by: Suppose, however, that prior to multiplication and integration we do not reverse one of the two component factors; then we have the integral: which may be denoted by f f. A single value of f f is represented by: The Autocorrelation The shaded area is the value of the autocorrelation for the displacement x. In optics, we often define the autocorrelation with a complex conjugate:

7. The Autocorrelation Theorem The Fourier Transform of the autocorrelation is the spectrum! Proof:

8. The Autocorrelation Theorem in action t t  

9. The Autocorrelation Theorem for a light wave field The Autocorrelation Theorem can be applied to a light wave field, yielding an important result: Remarkably, the Fourier transform of a light-wave field’s autocorrelation is its spectrum! This relation yields an alternative technique for measuring a light wave’s spectrum. This version of the Autocorrelation Theorem is known as the “Wiener- Khintchine Theorem.” = the spectrum!

10. The Autocorrelation Theorem for a light wave intensity The Autocorrelation Theorem can be applied to a light wave intensity, yielding a less important, but interesting, result: Many techniques yield the intensity autocorrelation of an ultrashort laser pulse in an attempt to measure its intensity vs. time (which is difficult). The above result shows that the intensity autocorrelation is not sufficient to determine the intensity—it yields the magnitude, but not the phase, of