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<H2><A ID="SECTION00955000000000000000">
Waveshaping using an exponential function</A>
</H2>
<A ID="sect5.example.expon"></A>
<P>
We return again to the spectra computed on Page <A HREF="node78.html#eq-waveshaping"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
SRC="crossref.png"></A>,
corresponding to waveshaping functions of the form <IMG
WIDTH="72" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
SRC="img504.png"
ALT="$f(x) = x^k$">. We note
with pleasure that not only are they all in phase (so that they can
be superposed with easily predictable results) but also that the spectra
spread out as <IMG
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img58.png"
ALT="$k$"> increases. Also, in a series
of the form,
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
f(x) = {f_0} + {f_1} x + {f_2} {x^2} + \cdots,
\end{displaymath}
-->
<IMG
WIDTH="205" HEIGHT="28" BORDER="0"
SRC="img505.png"
ALT="\begin{displaymath}
f(x) = {f_0} + {f_1} x + {f_2} {x^2} + \cdots,
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
a higher index of modulation will lend more relative weight to the higher
power terms in the expansion; as we saw seen earlier, if the index of
modulation is <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img4.png"
ALT="$a$">, the various <IMG
WIDTH="20" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
SRC="img506.png"
ALT="$x^k$"> terms are multiplied by <IMG
WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img507.png"
ALT="$f_0$">,
<IMG
WIDTH="26" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img508.png"
ALT="$af_1$">, <IMG
WIDTH="34" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img509.png"
ALT="${a^2}{f_2}$">, and so on.
<P>
Now suppose we wish to arrange for different terms in the above expansion
to dominate the result in a predictable way as a function of the index <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img4.png"
ALT="$a$">.
To choose the simplest possible example, suppose we wish <IMG
WIDTH="18" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img507.png"
ALT="$f_0$"> to be the largest
term for <IMG
WIDTH="70" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img510.png"
ALT="$0&lt;a&lt;1$">, then for it to be overtaken by the more quickly growing
<IMG
WIDTH="26" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img508.png"
ALT="$af_1$"> term for <IMG
WIDTH="70" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img511.png"
ALT="$1&lt;a&lt;2$">, which is then overtaken by the <IMG
WIDTH="34" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img509.png"
ALT="${a^2}{f_2}$"> term for
<IMG
WIDTH="70" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img512.png"
ALT="$2&lt;a&lt;3$"> and so on, so that each <IMG
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img75.png"
ALT="$n$">th term takes over at index <IMG
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img75.png"
ALT="$n$">.
To make this happen we just require that
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
{f_1} = {f_0} , 2 {f_2} = {f_1}, 3 {f_3} = {f_2} , \ldots
\end{displaymath}
-->
<IMG
WIDTH="207" HEIGHT="27" BORDER="0"
SRC="img513.png"
ALT="\begin{displaymath}
{f_1} = {f_0} , 2 {f_2} = {f_1}, 3 {f_3} = {f_2} , \ldots
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
and so choosing <IMG
WIDTH="47" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img514.png"
ALT="${f_0}=0$">, we get <IMG
WIDTH="47" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img515.png"
ALT="${f_1}=1$">, <IMG
WIDTH="63" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img516.png"
ALT="${f_2}=1/2$">, <IMG
WIDTH="63" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img517.png"
ALT="${f_3}=1/6$">, and in
general,
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
{f_k} = {1 \over {1 \cdot 2 \cdot 3 \cdot ... \cdot k}}
\end{displaymath}
-->
<IMG
WIDTH="129" HEIGHT="38" BORDER="0"
SRC="img518.png"
ALT="\begin{displaymath}
{f_k} = {1 \over {1 \cdot 2 \cdot 3 \cdot ... \cdot k}}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
These happen to be the coefficients of the power series for the function
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
f(x) = {e ^ x}
\end{displaymath}
-->
<IMG
WIDTH="65" HEIGHT="28" BORDER="0"
SRC="img519.png"
ALT="\begin{displaymath}
f(x) = {e ^ x}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where <IMG
WIDTH="52" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img520.png"
ALT="$e \approx 2.7$"> is Euler's constant.
<P>
Before plugging in <IMG
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img521.png"
ALT="$e^x$"> as a transfer function it's wise to plan how we
will deal with signal amplitude, since <IMG
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img521.png"
ALT="$e^x$"> grows quickly as
<IMG
WIDTH="12" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img243.png"
ALT="$x$"> increases. If we're going to plug in a sinusoid of amplitude <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img4.png"
ALT="$a$">, the maximum output
will be <IMG
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img522.png"
ALT="$e^a$">, occurring whenever the phase is zero. A simple and natural
choice is simply to divide by <IMG
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img522.png"
ALT="$e^a$"> to reduce the peak to one, giving:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
f(a \cos(\omega n)) =
{{{e^{a \cos(\omega n)}}} \over {e^a}} = {e^{a (\cos(\omega n) - 1)}}
\end{displaymath}
-->
<IMG
WIDTH="275" HEIGHT="42" BORDER="0"
SRC="img523.png"
ALT="\begin{displaymath}
f(a \cos(\omega n)) =
{{{e^{a \cos(\omega n)}}} \over {e^a}} = {e^{a (\cos(\omega n) - 1)}}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
This is realized in Example E06.exponential.pd. Resulting spectra for
<IMG
WIDTH="28" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img524.png"
ALT="$a=$"> 0, 4, and 16 are shown in Figure <A HREF="#fig05.13">5.13</A>. As the waveshaping index
rises, progressively less energy is present in the fundamental; the energy
is increasingly spread over the partials.
<P>
<DIV ALIGN="CENTER"><A ID="fig05.13"></A><A ID="5876"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 5.13:</STRONG>
Spectra of waveshaping output using an exponential transfer function.
Indices of modulation of 0, 4, and 16 are shown; note the different vertical
scales.</CAPTION>
<TR><TD><IMG
WIDTH="421" HEIGHT="466" BORDER="0"
SRC="img525.png"
ALT="\begin{figure}\psfig{file=figs/fig05.13.ps}\end{figure}"></TD></TR>
</TABLE>
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<ADDRESS>
Miller Puckette
2006-12-30
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