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<H2><A NAME="SECTION001732000000000000000"> </A><A NAME="s:psac"> </A>
<BR>
The power spectrum and covariance statistics
</H2>
<P>
Let us define power spectrum, covariance and autocovariance statistics
<I>P</I>, <I>Cov</I> and <I>ACF</I>:
<BR>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{eqnarray}
P[x](\nu) & = & |{\cal F}x|^2 \\
Cov[x,y](l) & = & x(t)*y(-t)\\
ACF[x](l) & = & Cov[x,x](l)
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="32" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
SRC="img509.gif"
ALT="$\displaystyle P[x](\nu)$"></TD>
<TD ALIGN="CENTER" NOWRAP>=</TD>
<TD ALIGN="LEFT" NOWRAP><IMG
WIDTH="12" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
SRC="img510.gif"
ALT="$\displaystyle \vert{\cal F}x\vert^2$"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(12.7)</TD></TR>
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><I>Cov</I>[<I>x</I>,<I>y</I>](<I>l</I>)</TD>
<TD ALIGN="CENTER" NOWRAP>=</TD>
<TD ALIGN="LEFT" NOWRAP><I>x</I>(<I>t</I>)*<I>y</I>(-<I>t</I>)</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(12.8)</TD></TR>
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><I>ACF</I>[<I>x</I>](<I>l</I>)</TD>
<TD ALIGN="CENTER" NOWRAP>=</TD>
<TD ALIGN="LEFT" NOWRAP><I>Cov</I>[<I>x</I>,<I>x</I>](<I>l</I>)</TD>
<TD WIDTH=10 ALIGN="RIGHT">
(12.9)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
The power spectrum is special among the periodograms in that it is
the square of a linear operator and reveals the important
correspondence between frequency and time domain analyses:
<BR>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{eqnarray}
P[x](\nu) & = & {\cal F}[ACF[x](l)](\nu),
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="32" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
SRC="img511.gif"
ALT="$\displaystyle P[x](\nu)$"></TD>
<TD ALIGN="CENTER" NOWRAP>=</TD>
<TD ALIGN="LEFT" NOWRAP><IMG
WIDTH="124" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
SRC="img512.gif"
ALT="$\displaystyle {\cal F}[ACF[x](l)](\nu),$"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(12.10)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
by virtue of Eq. (<A HREF="node232.html#e:fcon">12.5</A>).
<P>
Let us consider which linear operators or matrices convert series of
independent random variables into series of independent variables.
For the discrete, evenly sampled observations the ACF is computed as
the scalar product of vectors obtained by circularly permutating the
data of the series. For a series of independent random variables,
e.g. white noise, the vectors are orthogonal. It is known from linear
algebra that only orthogonal matrices preserve orthogonality. So, only
in the special case of evenly spaced discrete observations and
frequencies (Sect. <A HREF="node232.html#s:iftr">12.3.1</A>) are
<!-- MATH: ${\cal F}[x]$ -->
<IMG
WIDTH="72" HEIGHT="74" ALIGN="MIDDLE" BORDER="0"
SRC="img513.gif"
ALT="${\cal F}[x]$">
(and <I>P</I>[<I>x</I>])
independent for each frequency. In the next subsection we discuss the
case of dependent and correlated values of <I>P</I>[<I>x</I>].
<P>
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<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
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