(10)

To see how the periodogram is related to the autocorrelation function defined in equation (6) and the spectrum in equation (8), substitute the definition of F(? ) into equation (9):

(note that the limits of the x-integral are ±T , not ±8 ). Note also that I have used the shift theorem here to represent the Fourier transform of f(x+t ). Using the representation of the d-function

from equation (8) of Chapter 8, this reduces to

in the limit as T? 8 . So the periodogram is

Finally, the spectrum is7

Using either equation (5) or equation (8) as its definition, the spectrum is the Fourier transform of the autocorrelation function. This is known as the Wiener-Khintchine theorem.