Let us look at spectra in terms of Z-transforms.
Let a spectrum be denoted , where

(15)

Expressing this in terms of a three-point Z-transform, we have

(16)

(17)

(18)

It is interesting to multiply out
the polynomial with B(Z) in order
to examine the coefficients of S(Z):

(19)

The coefficient s_{k} of Z^{k} is given by

(20)

Equation (20) is the
autocorrelation formula.
The autocorrelation
value s_{k} at lag 10 is s_{10}.
It is a measure of the similarity of b_{i}
with itself shifted 10 units in time.
In the most
frequently occurring case, b_{i} is real;
then, by inspection of (20),
we see that the autocorrelation coefficients are real,
and s_{k}=s_{-k}.

Specializing to a real time series gives

(21)

(22)

(23)

(24)

(25)

This proves a classic theorem that for real-valued signals
can be simply stated as follows:

For any real signal, the cosine transform of the autocorrelation
equals the magnitude squared of the Fourier transform.