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original version by: Nikos Drakos, CBLU, University of Leeds
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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<TITLE>Exercises</TITLE>
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<B> Next:</B> <A ID="tex2html2428"
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HREF="node127.html">Filters</A>
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<B> Up:</B> <A ID="tex2html2422"
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HREF="node104.html">Time shifts and delays</A>
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<B> Previous:</B> <A ID="tex2html2418"
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HREF="node125.html">Pitch shifter</A>
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<B> <A ID="tex2html2424"
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HREF="node4.html">Contents</A></B>
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<B> <A ID="tex2html2426"
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HREF="node201.html">Index</A></B>
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<BR>
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<BR>
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<!--End of Navigation Panel-->
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2022-04-12 23:32:40 -03:00
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<H1><A ID="SECTION0011110000000000000000">
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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 complex number has magnitude one and argument <IMG
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WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img52.png"
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ALT="$\pi /4$">. What are
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its real and imaginary parts?
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<P>
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</LI>
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<LI>A complex number has magnitude one and real part <IMG
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WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img98.png"
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ALT="$1/2$">. What is its
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imaginary part? (There are two possible values.)
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<P>
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</LI>
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<LI>What delay time would you give a comb filter so that its first frequency
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response peak is at 440 Hertz? If the sample rate is 44100, what frequency
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would correspond to the nearest integer delay?
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<P>
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</LI>
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<LI>Suppose you made a variation on the non-recirculating comb filter so that
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the delayed signal was subtracted from the original instead of adding. What
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would the new frequency response be?
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<P>
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</LI>
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<LI>If you want to make a 6-Hertz vibrato with a sinusoidally varying
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delay line, and if you want the vibrato to change the frequency by 5%, how
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big a delay variation would you need? How would this change if the same
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depth of vibrato was desired at 12 Hertz?
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<P>
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</LI>
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<LI>A complex sinusoid <IMG
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WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img669.png"
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ALT="$X[n]$"> has frequency 11025 Hertz, amplitude
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50 and initial phase 135 degrees. Another one, <IMG
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WIDTH="34" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img717.png"
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ALT="$Y[n]$">, has the same frequency,
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but amplitude 20 and initial phase 45 degrees. What are the amplitude and
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initial phase of the sum of <IMG
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WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img670.png"
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ALT="$X$"> and <IMG
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WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img672.png"
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ALT="$Y$">?
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<P>
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</LI>
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<LI>What are the frequency, initial phase, and amplitude of the signal
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obtained when <IMG
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WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img669.png"
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ALT="$X[n]$"> (above) is delayed 4 samples?
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<P>
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</LI>
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<LI>Show that the frequency response of
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a recirculating comb filter with delay time <IMG
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WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img28.png"
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ALT="$d$"> and feedback gain <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img29.png"
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ALT="$g$">, as a
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function of angular frequency <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $">, is equal to:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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{[ {{(1-g \cos (\omega d))}^2} + {{(g \sin (\omega d))}^2} ]}
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^
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{-1/2}
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\end{displaymath}
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-->
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<IMG
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WIDTH="246" HEIGHT="30" BORDER="0"
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SRC="img851.png"
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ALT="\begin{displaymath}
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{[ {{(1-g \cos (\omega d))}^2} + {{(g \sin (\omega d))}^2} ]}
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^
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{-1/2}
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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<P>
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</LI>
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</OL>
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<P>
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<B> Next:</B> <A ID="tex2html2428"
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HREF="node127.html">Filters</A>
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<B> Up:</B> <A ID="tex2html2422"
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HREF="node104.html">Time shifts and delays</A>
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<B> Previous:</B> <A ID="tex2html2418"
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HREF="node125.html">Pitch shifter</A>
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<B> <A ID="tex2html2424"
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HREF="node4.html">Contents</A></B>
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<B> <A ID="tex2html2426"
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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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