Waves and Fourier sums

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## Spectra in terms of Z-transforms

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.

 Waves and Fourier sums

Next: Two ways to compute Up: CORRELATION AND SPECTRA Previous: CORRELATION AND SPECTRA

2013-01-06