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<H2><A NAME="SECTION001731000000000000000"> </A><A NAME="s:iftr"> </A>
<BR>
Fourier transforms
</H2>
<P>
Transformations which take functions, e.g. <I>x</I>, <I>y</I> as arguments and
return functions as results are called operators. The direct and
inverse Fourier transform,
<!-- MATH: ${\cal F}^{\pm 1}$ -->
<IMG
WIDTH="70" HEIGHT="58" ALIGN="BOTTOM" BORDER="0"
SRC="img485.gif"
ALT="${\cal F}^{\pm 1}$">,
and the convolution,
*, are operators defined in the following way:
<BR>
<DIV ALIGN="CENTER"><A NAME="e:fou"> </A>
<!-- MATH: \begin{eqnarray}
{{\cal F}^{\pm 1}[x]}(\nu) &{\displaystyle = C_{\pm}
\int_{-\infty}^{+\infty} e^{\pm 2\pi{i}t\nu}x(t) dt} &=
{1\over{n_o^{1\pm 1\over{2}}}}
\sum_{k=1}^{n_o} x_ke^{\pm 2\pi{i}t_k\nu}\\{[x*y]}(l)
&{\displaystyle = C \int_{-\infty}^{+\infty} x(t)y(l-t) dt} &=
{1\over{n_o}} \sum_{k=1}^{n_o} x_ky_{l-k},
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="54" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
SRC="img486.gif"
ALT="$\displaystyle {{\cal F}^{\pm 1}[x]}(\nu)$"></TD>
<TD ALIGN="CENTER" NOWRAP><IMG
WIDTH="193" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
SRC="img487.gif"
ALT="$\textstyle {\displaystyle = C_{\pm}
\int_{-\infty}^{+\infty} e^{\pm 2\pi{i}t\nu}x(t) dt}$"></TD>
<TD ALIGN="LEFT" NOWRAP><IMG
WIDTH="168" HEIGHT="80" ALIGN="MIDDLE" BORDER="0"
SRC="img488.gif"
ALT="$\displaystyle =
{1\over{n_o^{1\pm 1\over{2}}}}
\sum_{k=1}^{n_o} x_ke^{\pm 2\pi{i}t_k\nu}$"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(12.2)</TD></TR>
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT">[<I>x</I>*<I>y</I>](<I>l</I>)</TD>
<TD ALIGN="CENTER" NOWRAP><IMG
WIDTH="186" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
SRC="img489.gif"
ALT="$\textstyle {\displaystyle = C \int_{-\infty}^{+\infty} x(t)y(l-t) dt}$"></TD>
<TD ALIGN="LEFT" NOWRAP><IMG
WIDTH="116" HEIGHT="80" ALIGN="MIDDLE" BORDER="0"
SRC="img490.gif"
ALT="$\displaystyle =
{1\over{n_o}} \sum_{k=1}^{n_o} x_ky_{l-k},$"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(12.3)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
where square brackets, [ ], indicate the order of the operators and
round brackets, (), indicate the arguments of the input and output
functions. Without loss of generality we consider here functions
with zero mean value. Note that because of the finite and infinite
correlation length of stochastic and periodic series, respectively,
no unique normalization <I>C</I> applies in the continuous case.
<P>
The discrete operators
<!-- MATH: ${\cal F}^{-1}$ -->
<IMG
WIDTH="70" HEIGHT="58" ALIGN="BOTTOM" BORDER="0"
SRC="img491.gif"
ALT="${\cal F}^{-1}$">
and * are well defined only
for observations and frequencies which are spaced evenly by <IMG
WIDTH="48" HEIGHT="56" ALIGN="BOTTOM" BORDER="0"
SRC="img492.gif"
ALT="$\delta t$">and
<!-- MATH: $\delta\nu = 1/\Delta t$ -->
<IMG
WIDTH="61" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
SRC="img493.gif"
ALT="$\delta\nu = 1/\Delta t$">,
respectively, and span ranges <IMG
WIDTH="56" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img494.gif"
ALT="$\Delta t$">and
<!-- MATH: $\Delta\nu=1/\delta t$ -->
<IMG
WIDTH="61" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
SRC="img495.gif"
ALT="$\Delta\nu=1/\delta t$">.
Then and only then
<!-- MATH: ${\cal F}^{\pm}1$ -->
<IMG
WIDTH="2" HEIGHT="25" ALIGN="BOTTOM" BORDER="0"
SRC="img496.gif"
ALT="${\cal F}^{\pm}1$">reduces to orthogonal matrices. It follows directly from Eq.
(<A HREF="node232.html#e:fou">12.2</A>) that we implicitly assume that the observations and
their transforms are periodic with the periods <IMG
WIDTH="56" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img497.gif"
ALT="$\Delta t$">
and
<IMG
WIDTH="60" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img498.gif"
ALT="$\Delta\nu$">,
respectively. The assumption is of consequence only for
data strings which are short compared to the investigated periods or
coherence lengths or for a sampling which is coarse compared to these
two quantities. Such situations should be avoided also in the general
case of unevenly sampled observations.
<P>
The following properties of <IMG
WIDTH="48" HEIGHT="54" ALIGN="BOTTOM" BORDER="0"
SRC="img499.gif"
ALT="${\cal F}$">
and * are
noteworthy:
<BR>
<DIV ALIGN="CENTER"><A NAME="e:fcon"> </A>
<!-- MATH: \begin{eqnarray}
{\cal F}[x+y] & = & {\cal F}[x]+{\cal F}[y]\\
{\cal F}[x*y] & = & {\cal F}[x]{\cal F}[y]\\
{\cal F}e^{-2\pi{i}\nu_ot} & = & \delta_{\nu_o}(\nu)
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="40" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
SRC="img500.gif"
ALT="$\displaystyle {\cal F}[x+y]$"></TD>
<TD ALIGN="CENTER" NOWRAP>=</TD>
<TD ALIGN="LEFT" NOWRAP><IMG
WIDTH="72" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
SRC="img501.gif"
ALT="$\displaystyle {\cal F}[x]+{\cal F}[y]$"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(12.4)</TD></TR>
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="34" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
SRC="img502.gif"
ALT="$\displaystyle {\cal F}[x*y]$"></TD>
<TD ALIGN="CENTER" NOWRAP>=</TD>
<TD ALIGN="LEFT" NOWRAP><IMG
WIDTH="45" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
SRC="img503.gif"
ALT="$\displaystyle {\cal F}[x]{\cal F}[y]$"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(12.5)</TD></TR>
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="50" HEIGHT="26" ALIGN="BOTTOM" BORDER="0"
SRC="img504.gif"
ALT="$\displaystyle {\cal F}e^{-2\pi{i}\nu_ot}$"></TD>
<TD ALIGN="CENTER" NOWRAP>=</TD>
<TD ALIGN="LEFT" NOWRAP><IMG
WIDTH="18" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
SRC="img505.gif"
ALT="$\displaystyle \delta_{\nu_o}(\nu)$"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(12.6)</TD></TR>
</TABLE></DIV>
<BR CLEAR="ALL"><P></P>
where <IMG
WIDTH="50" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
SRC="img506.gif"
ALT="$\delta_x$">
denotes the Dirac symbol:
<!-- MATH: $\int\delta_xf(y)dy=f(x)$ -->
<IMG
WIDTH="134" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
SRC="img507.gif"
ALT="$\int\delta_xf(y)dy=f(x)$">.
In the discrete case, <IMG
WIDTH="50" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
SRC="img508.gif"
ALT="$\delta_x$">
assumes the value <I>n</I><SUB><I>o</I></SUB> for <I>x</I> and 0elsewhere.
<P>
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<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
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