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239 lines
6.6 KiB
239 lines
6.6 KiB
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<H2><A ID="SECTION00956000000000000000">
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Sinusoidal waveshaping: evenness and oddness</A>
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</H2>
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<A ID="sect5.bessel"></A>
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<P>
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Another interesting class of waveshaping transfer functions is the sinusoids:
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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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f(x) = \cos(x + \phi)
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\end{displaymath}
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-->
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<IMG
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WIDTH="122" HEIGHT="28" BORDER="0"
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SRC="img526.png"
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ALT="\begin{displaymath}
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f(x) = \cos(x + \phi)
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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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which include the cosine and sine functions (got by choosing <IMG
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WIDTH="42" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img162.png"
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ALT="$\phi=0$"> and
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<IMG
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WIDTH="72" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img527.png"
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ALT="$\phi=-\pi/2$">, respectively). These functions, one being even and the
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other odd, give rise to even and odd harmonic spectra, which turn out to be:
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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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\cos(a \cos(\omega n)) = {J_0}(a)
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- 2 {J_2}(a) \cos(2 \omega n)
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+ 2 {J_4}(a) \cos(4 \omega n)
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- 2 {J_6}(a) \cos(6 \omega n) \pm \cdots
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\end{displaymath}
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-->
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<IMG
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WIDTH="553" HEIGHT="28" BORDER="0"
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SRC="img528.png"
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ALT="\begin{displaymath}
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\cos(a \cos(\omega n)) = {J_0}(a)
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- 2 {J_2}(a) \cos(2 \om...
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...a) \cos(4 \omega n)
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- 2 {J_6}(a) \cos(6 \omega n) \pm \cdots
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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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<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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\sin(a \cos(\omega n)) =
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2 {J_1}(a) \cos(\omega n)
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- 2{J_3}(a) \cos(3 \omega n)
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+ 2{J_5}(a) \cos(5 \omega n) \mp \cdots
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\end{displaymath}
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-->
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<IMG
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WIDTH="512" HEIGHT="28" BORDER="0"
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SRC="img529.png"
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ALT="\begin{displaymath}
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\sin(a \cos(\omega n)) =
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2 {J_1}(a) \cos(\omega n)
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- 2{J_3}(a) \cos(3 \omega n)
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+ 2{J_5}(a) \cos(5 \omega n) \mp \cdots
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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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The functions <IMG
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WIDTH="40" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img530.png"
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ALT="${J_k}(a)$"> are the
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<A ID="5889"></A><I>Bessel functions</I>
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of the first kind, which
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engineers sometimes use to solve problems about vibrations or heat flow on
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discs. For other values of <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img77.png"
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ALT="$\phi$">, we can expand the expression for <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img112.png"
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ALT="$f$">:
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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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f(x) = \cos(x) \cos(\phi) - \sin(x) \sin(\phi)
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\end{displaymath}
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-->
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<IMG
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WIDTH="243" HEIGHT="28" BORDER="0"
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SRC="img531.png"
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ALT="\begin{displaymath}
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f(x) = \cos(x) \cos(\phi) - \sin(x) \sin(\phi)
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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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so the result is a mix between the even and the odd harmonics, with <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img77.png"
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ALT="$\phi$">
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controlling the relative amplitudes of the two. This is demonstrated in Patch
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E07.evenodd.pd, shown in Figure <A HREF="#fig05.14">5.14</A>.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig05.14"></A><A ID="5894"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 5.14:</STRONG>
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Using an additive offset to a cosine transfer function to alter the
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symmetry between even and odd. With no offset the symmetry is even. For odd symmetry, a quarter cycle is added
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to the phase. Smaller offsets give a mixture of even and odd.</CAPTION>
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<TR><TD><IMG
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WIDTH="234" HEIGHT="171" BORDER="0"
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SRC="img532.png"
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ALT="\begin{figure}\psfig{file=figs/fig05.14.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<B> Next:</B> <A ID="tex2html1831"
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HREF="node87.html">Phase modulation and FM</A>
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<B> Up:</B> <A ID="tex2html1825"
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HREF="node80.html">Examples</A>
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<B> Previous:</B> <A ID="tex2html1819"
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HREF="node85.html">Waveshaping using an exponential</A>
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<B> <A ID="tex2html1827"
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HREF="node4.html">Contents</A></B>
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<B> <A ID="tex2html1829"
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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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