<!DOCTYPE html>

<!--Converted with LaTeX2HTML 2002-2-1 (1.71)
original version by:  Nikos Drakos, CBLU, University of Leeds
* revised and updated by:  Marcus Hennecke, Ross Moore, Herb Swan
* with significant contributions from:
  Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
<HTML>
<HEAD>

<meta name="viewport" content="width=device-width, initial-scale=1.0">


<TITLE>Fourier transform as additive synthesis</TITLE>
<META NAME="description" CONTENT="Fourier transform as additive synthesis">
<META NAME="keywords" CONTENT="book">
<META NAME="resource-type" CONTENT="document">
<META NAME="distribution" CONTENT="global">

<META NAME="Generator" CONTENT="LaTeX2HTML v2002-2-1">
<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">

<LINK REL="STYLESHEET" HREF="book.css">

<LINK REL="previous" HREF="node165.html">
<LINK REL="up" HREF="node164.html">
<LINK REL="next" HREF="node167.html">
</HEAD>

<BODY >
<!--Navigation Panel-->
<A NAME="tex2html3059"
  HREF="node167.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
 SRC="next.png"></A> 
<A NAME="tex2html3053"
  HREF="node164.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
 SRC="up.png"></A> 
<A NAME="tex2html3049"
  HREF="node165.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
 SRC="prev.png"></A> 
<A NAME="tex2html3055"
  HREF="node4.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
 SRC="contents.png"></A> 
<A NAME="tex2html3057"
  HREF="node201.html">
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
 SRC="index.png"></A> 
<BR>
<B> Next:</B> <A NAME="tex2html3060"
  HREF="node167.html">Properties of Fourier transforms</A>
<B> Up:</B> <A NAME="tex2html3054"
  HREF="node164.html">Fourier analysis of periodic</A>
<B> Previous:</B> <A NAME="tex2html3050"
  HREF="node165.html">Periodicity of the Fourier</A>
 &nbsp; <B>  <A NAME="tex2html3056"
  HREF="node4.html">Contents</A></B> 
 &nbsp; <B>  <A NAME="tex2html3058"
  HREF="node201.html">Index</A></B> 
<BR>
<BR>
<!--End of Navigation Panel-->

<H2><A NAME="SECTION001312000000000000000"></A>
<A NAME="sect9-IFT"></A>
<BR>
Fourier transform as additive synthesis
</H2>

<P>
Now consider an arbitrary signal <IMG
 WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img669.png"
 ALT="$X[n]$"> that repeats every <IMG
 WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img3.png"
 ALT="$N$">
samples.  (Previously we
had assumed that <IMG
 WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img669.png"
 ALT="$X[n]$"> could be obtained as a sum of sinusoids, and we haven't
yet found out whether every periodic <IMG
 WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img669.png"
 ALT="$X[n]$"> can be obtained that way.)  Let
<IMG
 WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img1062.png"
 ALT="$Y[k]$"> denote the Fourier transform of <IMG
 WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img670.png"
 ALT="$X$"> for <!-- MATH
 $k = 0, ..., N-1$
 -->
<IMG
 WIDTH="110" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img1063.png"
 ALT="$k = 0, ..., N-1$">:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
Y[k] = {\cal FT}\left \{ X[n] \right \} (k)
\end{displaymath}
 -->

<IMG
 WIDTH="151" HEIGHT="28" BORDER="0"
 SRC="img1064.png"
 ALT="\begin{displaymath}
Y[k] = {\cal FT}\left \{ X[n] \right \} (k)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
= {{\left [ {U^{-k}} \right ]} ^ {0}} X[0] +
    {{\left [ {U^{-k}} \right ]} ^ {1}} X[1] +
    \cdots +
    {{\left [ {U^{-k}} \right ]} ^ {N-1}} X[N-1]
\end{displaymath}
 -->

<IMG
 WIDTH="393" HEIGHT="30" BORDER="0"
 SRC="img1065.png"
 ALT="\begin{displaymath}
= {{\left [ {U^{-k}} \right ]} ^ {0}} X[0] +
{{\left [ {U^...
...X[1] +
\cdots +
{{\left [ {U^{-k}} \right ]} ^ {N-1}} X[N-1]
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
= {{\left [ {U^{0}} \right ]} ^ {k}} X[0] +
    {{\left [ {U^{-1}} \right ]} ^ {k}} X[1] +
    \cdots +
    {{\left [ {U^{-(N-1)}} \right ]} ^ {k}} X[N-1]
\end{displaymath}
 -->

<IMG
 WIDTH="395" HEIGHT="39" BORDER="0"
 SRC="img1066.png"
 ALT="\begin{displaymath}
= {{\left [ {U^{0}} \right ]} ^ {k}} X[0] +
{{\left [ {U^{...
...1] +
\cdots +
{{\left [ {U^{-(N-1)}} \right ]} ^ {k}} X[N-1]
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
In the second version we rearranged the exponents to show that <IMG
 WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img1062.png"
 ALT="$Y[k]$"> is a sum
of complex sinusoids, with complex amplitudes <IMG
 WIDTH="40" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img1067.png"
 ALT="$X[m]$"> and frequencies <IMG
 WIDTH="40" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="img1068.png"
 ALT="$-m\omega$">
for <!-- MATH
 $m = 0, \ldots, N-1$
 -->
<IMG
 WIDTH="123" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img1069.png"
 ALT="$m = 0, \ldots, N-1$">.  In other words, <IMG
 WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img1062.png"
 ALT="$Y[k]$"> can be considered as a
Fourier series in its own right, whose <IMG
 WIDTH="17" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="img111.png"
 ALT="$m$">th component has strength <IMG
 WIDTH="53" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img1070.png"
 ALT="$X[-m]$">.
(The expression <IMG
 WIDTH="53" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img1070.png"
 ALT="$X[-m]$"> makes sense because <IMG
 WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img670.png"
 ALT="$X$"> is a periodic signal). 
We can also express the amplitude of the partials of <IMG
 WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img1062.png"
 ALT="$Y[k]$"> in terms of its own
Fourier transform.  Equating the two gives: 
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
{1 \over N} {\cal FT} \left \{ Y[k] \right \} (m) = X[-m]
\end{displaymath}
 -->

<IMG
 WIDTH="190" HEIGHT="38" BORDER="0"
 SRC="img1071.png"
 ALT="\begin{displaymath}
{1 \over N} {\cal FT} \left \{ Y[k] \right \} (m) = X[-m]
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
This means in turn that <IMG
 WIDTH="53" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img1070.png"
 ALT="$X[-m]$"> can be obtained by summing sinusoids with
amplitudes <IMG
 WIDTH="55" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img1072.png"
 ALT="$Y[k]/N$">.  Setting <IMG
 WIDTH="60" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="img1073.png"
 ALT="$n = -m$"> gives:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
X[n] = {1 \over N} {\cal FT} \left \{ Y[k] \right \} (-n)
\end{displaymath}
 -->

<IMG
 WIDTH="183" HEIGHT="38" BORDER="0"
 SRC="img1074.png"
 ALT="\begin{displaymath}
X[n] = {1 \over N} {\cal FT} \left \{ Y[k] \right \} (-n)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
= {{\left [ {U^{0}} \right ]} ^ {n}} Y[0] +
    {{\left [ {U^{1}} \right ]} ^ {n}} Y[1] +
    \cdots +
    {{\left [ {U^{N-1}} \right ]} ^ {n}} Y[N-1]
\end{displaymath}
 -->

<IMG
 WIDTH="360" HEIGHT="30" BORDER="0"
 SRC="img1075.png"
 ALT="\begin{displaymath}
= {{\left [ {U^{0}} \right ]} ^ {n}} Y[0] +
{{\left [ {U^{...
...Y[1] +
\cdots +
{{\left [ {U^{N-1}} \right ]} ^ {n}} Y[N-1]
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
This shows that any periodic <IMG
 WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img669.png"
 ALT="$X[n]$"> can indeed be obtained as a sum of
sinusoids.  Further, the formula explicitly shows how to reconstruct <IMG
 WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img669.png"
 ALT="$X[n]$">
from its Fourier transform <IMG
 WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img1062.png"
 ALT="$Y[k]$">, if we know its value for the integers <!-- MATH
 $k=0,
\ldots, N-1$
 -->
<IMG
 WIDTH="118" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img1076.png"
 ALT="$k=0,
\ldots, N-1$">.

<P>
<HR>
<!--Navigation Panel-->
<A NAME="tex2html3059"
  HREF="node167.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
 SRC="next.png"></A> 
<A NAME="tex2html3053"
  HREF="node164.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
 SRC="up.png"></A> 
<A NAME="tex2html3049"
  HREF="node165.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
 SRC="prev.png"></A> 
<A NAME="tex2html3055"
  HREF="node4.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
 SRC="contents.png"></A> 
<A NAME="tex2html3057"
  HREF="node201.html">
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
 SRC="index.png"></A> 
<BR>
<B> Next:</B> <A NAME="tex2html3060"
  HREF="node167.html">Properties of Fourier transforms</A>
<B> Up:</B> <A NAME="tex2html3054"
  HREF="node164.html">Fourier analysis of periodic</A>
<B> Previous:</B> <A NAME="tex2html3050"
  HREF="node165.html">Periodicity of the Fourier</A>
 &nbsp; <B>  <A NAME="tex2html3056"
  HREF="node4.html">Contents</A></B> 
 &nbsp; <B>  <A NAME="tex2html3058"
  HREF="node201.html">Index</A></B> 
<!--End of Navigation Panel-->
<ADDRESS>
Miller Puckette
2006-12-30
</ADDRESS>
</BODY>
</HTML>
