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<H1><A ID="SECTION0011110000000000000000">
Exercises</A>
</H1>
<P>
<OL>
<LI>A complex number has magnitude one and argument <IMG
WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img52.png"
ALT="$\pi /4$">. What are
its real and imaginary parts?
<P>
</LI>
<LI>A complex number has magnitude one and real part <IMG
WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img98.png"
ALT="$1/2$">. What is its
imaginary part? (There are two possible values.)
<P>
</LI>
<LI>What delay time would you give a comb filter so that its first frequency
response peak is at 440 Hertz? If the sample rate is 44100, what frequency
would correspond to the nearest integer delay?
<P>
</LI>
<LI>Suppose you made a variation on the non-recirculating comb filter so that
the delayed signal was subtracted from the original instead of adding. What
would the new frequency response be?
<P>
</LI>
<LI>If you want to make a 6-Hertz vibrato with a sinusoidally varying
delay line, and if you want the vibrato to change the frequency by 5%, how
big a delay variation would you need? How would this change if the same
depth of vibrato was desired at 12 Hertz?
<P>
</LI>
<LI>A complex sinusoid <IMG
WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img669.png"
ALT="$X[n]$"> has frequency 11025 Hertz, amplitude
50 and initial phase 135 degrees. Another one, <IMG
WIDTH="34" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img717.png"
ALT="$Y[n]$">, has the same frequency,
but amplitude 20 and initial phase 45 degrees. What are the amplitude and
initial phase of the sum of <IMG
WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img670.png"
ALT="$X$"> and <IMG
WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img672.png"
ALT="$Y$">?
<P>
</LI>
<LI>What are the frequency, initial phase, and amplitude of the signal
obtained when <IMG
WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img669.png"
ALT="$X[n]$"> (above) is delayed 4 samples?
<P>
</LI>
<LI>Show that the frequency response of
a recirculating comb filter with delay time <IMG
WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img28.png"
ALT="$d$"> and feedback gain <IMG
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img29.png"
ALT="$g$">, as a
function of angular frequency <IMG
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img27.png"
ALT="$\omega $">, is equal to:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
{[ {{(1-g \cos (\omega d))}^2} + {{(g \sin (\omega d))}^2} ]}
^
{-1/2}
\end{displaymath}
-->
<IMG
WIDTH="246" HEIGHT="30" BORDER="0"
SRC="img851.png"
ALT="\begin{displaymath}
{[ {{(1-g \cos (\omega d))}^2} + {{(g \sin (\omega d))}^2} ]}
^
{-1/2}
\end{displaymath}">
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<ADDRESS>
Miller Puckette
2006-12-30
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