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Spectra in terms of Z-transforms

Let us look at spectra in terms of $Z$-transforms. Let a spectrum be denoted $S(\omega)$, where
\begin{displaymath}
S(\omega) \quad =\quad \vert B(\omega)\vert^2 \quad =\quad \overline{B(\omega)}B(\omega)
\end{displaymath} (15)

Expressing this in terms of a three-point $Z$-transform, we have
$\displaystyle S(\omega)$ $\textstyle =$ $\displaystyle (\bar{b}_0+\bar{b}_1 e^{-i\omega} +
\bar{b}_2 e^{-i2\omega})
(b_0 + b_1e^{i\omega} +b_2 e^{i2\omega})$ (16)
$\displaystyle S(Z)$ $\textstyle =$ $\displaystyle \left(\bar{b}_0 +\frac{\bar{b}_1}{Z} +
\frac{\bar{b}_2}{Z^2} \right)
(b_0 + b_1Z + b_2Z^2 )$ (17)
$\displaystyle S(Z)$ $\textstyle =$ $\displaystyle \overline{B} \left(\frac{1}{Z}\right) B(Z)$ (18)

It is interesting to multiply out the polynomial $\bar{B}(1/Z)$ with $B(Z)$ in order to examine the coefficients of $S(Z)$:
$\displaystyle S(Z)$ $\textstyle =$ $\displaystyle \frac{\bar{b}_2b_0}{Z^2} +
\frac{(\bar{b}_1b_0 + \bar{b}_2b_1)}...
...b}_1b_1 + \bar{b}_2 b_2)
+ (\bar{b}_0 b_1 + \bar{b}_1b_2)Z + \bar{b}_0 b_2 Z^2$  
$\displaystyle S(Z)$ $\textstyle =$ $\displaystyle \frac{s_{-2}}{Z^2} + \frac{s_{-1}}{Z} + s_0 + s_1Z + s_2 Z^2$ (19)

The coefficient $s_k$ of $Z^k$ is given by
\begin{displaymath}
s_k \quad =\quad \sum_{i} \bar{b}_i b_{i+k}
\end{displaymath} (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

$\displaystyle S(Z)$ $\textstyle =$ $\displaystyle s_0 + s_1\left(Z+\frac{1}{Z} \right) +
s_2\left(Z^2 +\frac{1}{Z^2}\right)$ (21)
$\displaystyle S(Z(\omega ))$ $\textstyle =$ $\displaystyle s_0 + s_1(e^{i\omega} + e^{-i\omega}) +
s_2(e^{i2\omega} + e^{-i2\omega})$ (22)
$\displaystyle S(\omega)$ $\textstyle =$ $\displaystyle s_0 + 2s_1\cos \omega + 2s_2 \cos 2\omega$ (23)
$\displaystyle S(\omega)$ $\textstyle =$ $\displaystyle \sum_{k} s_k \cos k\omega$ (24)
$\displaystyle S(\omega)$ $\textstyle =$ $\displaystyle \mbox{cosine transform of }\;\; s_k$ (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.


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

2013-01-06