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194 lines
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194 lines
5.5 KiB
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original version by: Nikos Drakos, CBLU, University of Leeds
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<H1><A ID="SECTION001380000000000000000">
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Exercises</A>
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</H1>
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<P>
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<OL>
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<LI>A signal <IMG
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WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img80.png"
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ALT="$x[n]$"> is 1 for <IMG
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WIDTH="42" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img79.png"
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ALT="$n=0$"> and <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img179.png"
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ALT="$0$"> otherwise (an impulse). What is
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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="$N$">-point) Fourier transform as a function of <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img58.png"
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ALT="$k$">?
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<P>
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</LI>
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<LI>Assuming further that <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="$N$"> is an even number, what does
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the Fourier transform become if <IMG
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WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img80.png"
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ALT="$x[n]$"> is 1 at <IMG
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WIDTH="63" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1246.png"
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ALT="$n=N/2$"> instead of at <IMG
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WIDTH="42" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img79.png"
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ALT="$n=0$">?
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<P>
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</LI>
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<LI>For what integer values of <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img58.png"
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ALT="$k$"> is the Fourier transform of the <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="$N$">-point
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Hann window function nonzero?
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<P>
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</LI>
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<LI>In order to Fourier analyze a 100-Hertz periodic tone (at a sample rate
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of 44100 Hertz), using a Hann window, what value of <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="$N$"> would be needed to
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completely resolve all the partials of the tone (in the sense of having
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non-overlapping peaks in the spectrum)?
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<P>
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</LI>
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<LI>Suppose an N-point Fourier transform is done on a complex sinusoid of frequency
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<IMG
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WIDTH="34" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img1247.png"
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ALT="$2.5\omega$"> where <IMG
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WIDTH="74" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1047.png"
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ALT="$\omega=2\pi/N$"> is the fundamental frequency. What percentage
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of the signal energy lands in the main lobe, channels <IMG
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WIDTH="41" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img1248.png"
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ALT="$k=2$"> and <IMG
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WIDTH="41" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img1178.png"
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ALT="$k=3$">? If the signal
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is Hann windowed, what percentage of the energy is now in the main lobe (which
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is then channels 1 through 4)?
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<P>
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</LI>
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</OL>
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<BR>
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<B> Next:</B> <A ID="tex2html3298"
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HREF="node184.html">Classical waveforms</A>
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<B> Up:</B> <A ID="tex2html3292"
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HREF="node163.html">Fourier analysis and resynthesis</A>
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<B> Previous:</B> <A ID="tex2html3288"
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HREF="node182.html">Phase vocoder time bender</A>
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<B> <A ID="tex2html3294"
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HREF="node4.html">Contents</A></B>
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<B> <A ID="tex2html3296"
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HREF="node201.html">Index</A></B>
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<ADDRESS>
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Miller Puckette
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2006-12-30
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