Spectra in terms of Z-transforms

Let us look at spectra in terms of Z-transforms. Let a spectrum be denoted $S(\omega)$, where  

$$S(\omega) \quad =\quad\vert B(\omega)\vert^2 \quad =\quad\overline{B(\omega)}B(\omega)$$ (15)

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

$$\begin{eqnarray} S(\omega) & = & (\bar{b}_0+\bar{b}_1 e^{-i\omega} + \bar{b}_2... ... b_2Z^2 ) \\ S(Z) & = & \overline{B} \left(\frac{1}{Z}\right) B(Z)\end{eqnarray}$$ (16) (17) (18)

It is interesting to multiply out the polynomial $\bar{B}(1/Z)$ with B(Z) in order to examine the coefficients of S(Z):

$$\begin{eqnarray} S(Z) &=& \frac{\bar{b}_2b_0}{Z^2} + \frac{(\bar{b}_1b_0 + \ba... ...) &=& \frac{s_{-2}}{Z^2} + \frac{s_{-1}}{Z} + s_0 + s_1Z + s_2 Z^2\end{eqnarray}$$ (19)

The coefficient s_k of Z^k is given by  

$$s_k \quad =\quad\sum_{i} \bar{b}_i b_{i+k}$$ (20)

Equation (20) is the autocorrelation formula. The autocorrelation value s_k at lag 10 is s₁₀. 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

$$\begin{eqnarray} S(Z) & = & s_0 + s_1\left(Z+\frac{1}{Z} \right) + s_2\left(Z^2... ...os k\omega \\ S(\omega ) & = & \mbox{cosine transform of }\;\; s_k\end{eqnarray}$$ (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.