Let us look at spectra in terms of -transforms.
Let a spectrum be denoted , where
(15)
Expressing this in terms of a three-point -transform, we have
(16)
(17)
(18)
It is interesting to multiply out
the polynomial with in order
to examine the coefficients of :
(19)
The coefficient of is given by
(20)
Equation (20) is the
autocorrelation formula.
The autocorrelation
value at lag is .
It is a measure of the similarity of
with itself shifted units in time.
In the most
frequently occurring case, is real;
then, by inspection of (20),
we see that the autocorrelation coefficients are real,
and .
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.