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<H2><A NAME="SECTION001733000000000000000">
Sampling patterns</A>
</H2>
<P>
The effect of a certain sampling pattern in the frequency analysis is
particularly transparent for the power spectrum. Let <I>s</I> be the
sampling function taking on the value 1 at the (unevenly spaced) times
of the observations observation and 0 elsewhere. The power spectrum of
the sampling function
<BR>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{eqnarray}
W(\nu) & = & |{\cal F}s|^2
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="12" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
SRC="img514.gif"
ALT="$\displaystyle W(\nu)$"></TD>
<TD ALIGN="CENTER" NOWRAP>=</TD>
<TD ALIGN="LEFT" NOWRAP><IMG
WIDTH="11" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
SRC="img515.gif"
ALT="$\displaystyle \vert{\cal F}s\vert^2$"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(12.11)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
is an ordinary, non-random function called the spectral window
function. The discrete observations are the product of <I>s</I> and the
model function <I>f</I>: <I>x</I> = <I>s f</I> so that their transform is a convolution
of transforms:
<!-- MATH: ${\cal F}x = [{\cal F}s]*[{\cal F}f]\equiv S*F$ -->
<IMG
WIDTH="204" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
SRC="img516.gif"
ALT="${\cal F}x = [{\cal F}s]*[{\cal F}f]\equiv S*F$">,
where
<!-- MATH: $S\equiv{\cal F}s$ -->
<IMG
WIDTH="33" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
SRC="img517.gif"
ALT="$S\equiv{\cal F}s$">
and
<!-- MATH: $F={\cal F}f$ -->
<IMG
WIDTH="38" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
SRC="img518.gif"
ALT="$F={\cal F}f$">.
For
<!-- MATH: $f=A\cos{2\pi\lambda{t}}\equiv
A(e^{+2\pi\lambda{t}}+e^{-2\pi\lambda{t}})/2$ -->
<IMG
WIDTH="330" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
SRC="img519.gif"
ALT="$f=A\cos{2\pi\lambda{t}}\equiv
A(e^{+2\pi\lambda{t}}+e^{-2\pi\lambda{t}})/2$">
and
<!-- MATH: $F={\cal F}f =
A(\delta_{+\nu}+\delta_{-\nu})/2$ -->
<IMG
WIDTH="214" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
SRC="img520.gif"
ALT="$F={\cal F}f =
A(\delta_{+\nu}+\delta_{-\nu})/2$">
we obtain the result
<!-- MATH: ${\cal F}x =
A(S(\nu-\lambda)+S(\nu+\lambda))/2$ -->
<IMG
WIDTH="269" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
SRC="img521.gif"
ALT="${\cal F}x =
A(S(\nu-\lambda)+S(\nu+\lambda))/2$">.
Because of the linearity of
<IMG
WIDTH="48" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img522.gif"
ALT="${\cal F}$">
our result extends to any combination of frequencies.
Taking the square modulus of the result equation, we obtain both
squared and mixed terms. The mixed terms
<!-- MATH: $S(\nu+\lambda_k)S(\nu+\lambda_j)$ -->
<IMG
WIDTH="146" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
SRC="img523.gif"
ALT="$S(\nu+\lambda_k)S(\nu+\lambda_j)$">
correspond to an interference of
frequencies <IMG
WIDTH="52" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
SRC="img524.gif"
ALT="$\lambda_k$">
and <IMG
WIDTH="52" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
SRC="img525.gif"
ALT="$\lambda_j$">
differing by either sign or
absolute value. Therefore, <EM>if interference between frequencies is
small, the power spectrum reduces to the sum of the window functions
shifted in frequency:</EM>
<BR>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{eqnarray}
P(\nu) & \approx &
\sum |[{\cal F}s](\nu+\lambda_k)|^2 \equiv \sum W(\nu+\lambda_k)
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="6" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
SRC="img526.gif"
ALT="$\displaystyle P(\nu)$"></TD>
<TD ALIGN="CENTER" NOWRAP><IMG
WIDTH="46" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img527.gif"
ALT="$\textstyle \approx$"></TD>
<TD ALIGN="LEFT" NOWRAP><IMG
WIDTH="296" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
SRC="img528.gif"
ALT="$\displaystyle \sum \vert[{\cal F}s](\nu+\lambda_k)\vert^2 \equiv \sum W(\nu+\lambda_k)$"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(12.12)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
<P>
In the opposite case of strong interference, ghost patterns may arise
in the power spectrum due to interference of window function patterns
belonging to positive as well as negative frequencies. The ghost
patterns produced at frequencies nearby or far from the true frequency
are called <EM>aliases</EM> and <EM>power leaks</EM>, respectively.
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<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
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