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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