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<H1><A NAME="SECTION001300000000000000000"></A>
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<A NAME="chapter-fourier"></A><A NAME="chapter-fft"></A>
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<BR>
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Fourier analysis and resynthesis
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</H1>
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<P>
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Among the applications of filters discussed in Chapter <A HREF="node127.html#chapter-filter">8</A>, we
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saw how to use heterodyning, combined with a low-pass filter, to find the
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amplitude and phase of a sinusoidal component of a signal (Page
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<A HREF="node158.html#sect8.heterodyning"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="crossref.png"></A>). In this chapter we will refine this technique
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into what is called
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<A NAME="12298"></A><I>Fourier analysis</I>.
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In its simplest form, Fourier analysis takes as input any periodic
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signal (of period <IMG
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SRC="img3.png"
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ALT="$N$">) and outputs the complex-valued
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amplitudes of its <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$"> possible sinusoidal components.
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These <IMG
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ALT="$N$"> complex amplitudes can theoretically be used to
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reconstruct the original signal exactly. This reconstruction is called
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<I>Fourier resynthesis</I>.
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<P>
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In this chapter we will start by developing the theory of Fourier analysis and
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resynthesis of periodic sampled signals. Then we will go on to show how
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to apply the same techniques to arbitrary signals, whether periodic or not.
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Finally, we will develop some standard applications such as the phase vocoder.
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<P>
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<BR><HR>
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<!--Table of Child-Links-->
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<UL>
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<LI><A NAME="tex2html2999"
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HREF="node164.html">Fourier analysis of periodic signals</A>
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<UL>
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<LI><A NAME="tex2html3000"
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HREF="node165.html">Periodicity of the Fourier transform</A>
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<LI><A NAME="tex2html3001"
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HREF="node166.html">Fourier transform as additive synthesis</A>
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</UL>
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<BR>
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<LI><A NAME="tex2html3002"
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HREF="node167.html">Properties of Fourier transforms</A>
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<UL>
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<LI><A NAME="tex2html3003"
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HREF="node168.html">Fourier transform of DC</A>
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<LI><A NAME="tex2html3004"
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HREF="node169.html">Shifts and phase changes</A>
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<LI><A NAME="tex2html3005"
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HREF="node170.html">Fourier transform of a sinusoid</A>
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</UL>
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<BR>
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<LI><A NAME="tex2html3006"
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HREF="node171.html">Fourier analysis of non-periodic signals</A>
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<LI><A NAME="tex2html3007"
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HREF="node172.html">Fourier analysis and reconstruction of audio signals</A>
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<UL>
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<LI><A NAME="tex2html3008"
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HREF="node173.html">Narrow-band companding</A>
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<LI><A NAME="tex2html3009"
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HREF="node174.html">Timbre stamping (classical vocoder)</A>
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</UL>
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<BR>
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<LI><A NAME="tex2html3010"
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HREF="node175.html">Phase</A>
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<UL>
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<LI><A NAME="tex2html3011"
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HREF="node176.html">Phase relationships between channels</A>
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</UL>
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<BR>
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<LI><A NAME="tex2html3012"
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HREF="node177.html">Phase bashing</A>
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<LI><A NAME="tex2html3013"
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HREF="node178.html">Examples</A>
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<UL>
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<LI><A NAME="tex2html3014"
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HREF="node179.html">Fourier analysis and resynthesis in Pd</A>
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<LI><A NAME="tex2html3015"
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HREF="node180.html">Narrow-band companding: noise suppression</A>
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<LI><A NAME="tex2html3016"
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HREF="node181.html">Timbre stamp (``vocoder")</A>
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<LI><A NAME="tex2html3017"
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HREF="node182.html">Phase vocoder time bender</A>
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</UL>
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<BR>
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<LI><A NAME="tex2html3018"
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HREF="node183.html">Exercises</A>
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</UL>
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<!--End of Table of Child-Links-->
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<BR><HR>
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<ADDRESS>
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Miller Puckette
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2006-12-30
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</ADDRESS>
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