191 lines
5.9 KiB
HTML
191 lines
5.9 KiB
HTML
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
|
|
|
|
<!--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>
|
|
<TITLE>Exercises</TITLE>
|
|
<META NAME="description" CONTENT="Exercises">
|
|
<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="node178.html">
|
|
<LINK REL="up" HREF="node163.html">
|
|
<LINK REL="next" HREF="node184.html">
|
|
</HEAD>
|
|
|
|
<BODY >
|
|
<!--Navigation Panel-->
|
|
<A NAME="tex2html3297"
|
|
HREF="node184.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/next.png"></A>
|
|
<A NAME="tex2html3291"
|
|
HREF="node163.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/up.png"></A>
|
|
<A NAME="tex2html3287"
|
|
HREF="node182.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/prev.png"></A>
|
|
<A NAME="tex2html3293"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/contents.png"></A>
|
|
<A NAME="tex2html3295"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A NAME="tex2html3298"
|
|
HREF="node184.html">Classical waveforms</A>
|
|
<B> Up:</B> <A NAME="tex2html3292"
|
|
HREF="node163.html">Fourier analysis and resynthesis</A>
|
|
<B> Previous:</B> <A NAME="tex2html3288"
|
|
HREF="node182.html">Phase vocoder time bender</A>
|
|
<B> <A NAME="tex2html3294"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A NAME="tex2html3296"
|
|
HREF="node201.html">Index</A></B>
|
|
<BR>
|
|
<BR>
|
|
<!--End of Navigation Panel-->
|
|
|
|
<H1><A NAME="SECTION001380000000000000000">
|
|
Exercises</A>
|
|
</H1>
|
|
|
|
<P>
|
|
|
|
<OL>
|
|
<LI>A signal <IMG
|
|
WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img80.png"
|
|
ALT="$x[n]$"> is 1 for <IMG
|
|
WIDTH="42" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img79.png"
|
|
ALT="$n=0$"> and <IMG
|
|
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img179.png"
|
|
ALT="$0$"> otherwise (an impulse). What is
|
|
its (<IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$">-point) Fourier transform as a function of <IMG
|
|
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img58.png"
|
|
ALT="$k$">?
|
|
|
|
<P>
|
|
</LI>
|
|
<LI>Assuming further that <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$"> is an even number, what does
|
|
the Fourier transform become if <IMG
|
|
WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img80.png"
|
|
ALT="$x[n]$"> is 1 at <IMG
|
|
WIDTH="63" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1246.png"
|
|
ALT="$n=N/2$"> instead of at <IMG
|
|
WIDTH="42" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img79.png"
|
|
ALT="$n=0$">?
|
|
|
|
<P>
|
|
</LI>
|
|
<LI>For what integer values of <IMG
|
|
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img58.png"
|
|
ALT="$k$"> is the Fourier transform of the <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$">-point
|
|
Hann window function nonzero?
|
|
|
|
<P>
|
|
</LI>
|
|
<LI>In order to Fourier analyze a 100-Hertz periodic tone (at a sample rate
|
|
of 44100 Hertz), using a Hann window, what value of <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$"> would be needed to
|
|
completely resolve all the partials of the tone (in the sense of having
|
|
non-overlapping peaks in the spectrum)?
|
|
|
|
<P>
|
|
</LI>
|
|
<LI>Suppose an N-point Fourier transform is done on a complex sinusoid of frequency
|
|
<IMG
|
|
WIDTH="34" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img1247.png"
|
|
ALT="$2.5\omega$"> where <IMG
|
|
WIDTH="74" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1047.png"
|
|
ALT="$\omega=2\pi/N$"> is the fundamental frequency. What percentage
|
|
of the signal energy lands in the main lobe, channels <IMG
|
|
WIDTH="41" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img1248.png"
|
|
ALT="$k=2$"> and <IMG
|
|
WIDTH="41" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img1178.png"
|
|
ALT="$k=3$">? If the signal
|
|
is Hann windowed, what percentage of the energy is now in the main lobe (which
|
|
is then channels 1 through 4)?
|
|
|
|
<P>
|
|
</LI>
|
|
</OL>
|
|
|
|
<P>
|
|
<HR>
|
|
<!--Navigation Panel-->
|
|
<A NAME="tex2html3297"
|
|
HREF="node184.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/next.png"></A>
|
|
<A NAME="tex2html3291"
|
|
HREF="node163.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/up.png"></A>
|
|
<A NAME="tex2html3287"
|
|
HREF="node182.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/prev.png"></A>
|
|
<A NAME="tex2html3293"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/contents.png"></A>
|
|
<A NAME="tex2html3295"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A NAME="tex2html3298"
|
|
HREF="node184.html">Classical waveforms</A>
|
|
<B> Up:</B> <A NAME="tex2html3292"
|
|
HREF="node163.html">Fourier analysis and resynthesis</A>
|
|
<B> Previous:</B> <A NAME="tex2html3288"
|
|
HREF="node182.html">Phase vocoder time bender</A>
|
|
<B> <A NAME="tex2html3294"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A NAME="tex2html3296"
|
|
HREF="node201.html">Index</A></B>
|
|
<!--End of Navigation Panel-->
|
|
<ADDRESS>
|
|
Miller Puckette
|
|
2006-12-30
|
|
</ADDRESS>
|
|
</BODY>
|
|
</HTML>
|