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<H2><A NAME="SECTION001021000000000000000">
Pulse trains via waveshaping</A>
</H2>
<P>
When we use waveshaping the shape of the formant is determined by
a modulation term
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
{m_a}[n] = f (a \cos(\omega n))
\end{displaymath}
-->
<IMG
WIDTH="147" HEIGHT="28" BORDER="0"
SRC="img561.png"
ALT="\begin{displaymath}
{m_a}[n] = f (a \cos(\omega n))
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
For small values of the index <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img4.png"
ALT="$a$">, the modulation term varies only slightly from
the constant value <IMG
WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img562.png"
ALT="$f(0)$">, so most of the energy is concentrated at DC.
As <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img4.png"
ALT="$a$"> increases, the energy spreads out among progressively higher harmonics
of the fundamental <IMG
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img27.png"
ALT="$\omega $">. Depending on the function <IMG
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img112.png"
ALT="$f$">, this spread
may be orderly or disorderly. An orderly spread may be desirable and
then again may not, depending on whether our goal is a predictable spectrum or
a wide range of different (and perhaps hard-to-predict) spectra.
<P>
The waveshaping function <IMG
WIDTH="71" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img563.png"
ALT="$f(x) = {e^x}$">, analyzed on
Page <A HREF="node85.html#sect5.example.expon"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
SRC="file:/usr/local/share/lib/latex2html/icons/crossref.png"></A>,
gives well-behaved, simple and predictable results. After normalizing suitably,
we got the spectra shown in Figure <A HREF="node85.html#fig05.13">5.13</A>. A slight rewriting of the
waveshaping modulator for this choice of <IMG
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img112.png"
ALT="$f$"> (and taking the renormalization
into account) gives:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
{m_a}[n] = {e^{a \cdot (\cos(\omega n) - 1))}}
\end{displaymath}
-->
<IMG
WIDTH="153" HEIGHT="28" BORDER="0"
SRC="img564.png"
ALT="\begin{displaymath}
{m_a}[n] = {e^{a \cdot (\cos(\omega n) - 1))}}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
= e ^ {
{ -\left [
b \sin {\omega \over 2}
\right ] }
^2
}
\end{displaymath}
-->
<IMG
WIDTH="84" HEIGHT="24" BORDER="0"
SRC="img565.png"
ALT="\begin{displaymath}
= e ^ {
{ -\left [
b \sin {\omega \over 2}
\right ] }
^2
}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where <IMG
WIDTH="55" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img566.png"
ALT="${b^2}=2a$"> so that <IMG
WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img21.png"
ALT="$b$"> is proportional to the bandwidth. This can
be rewritten as
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
{m_a}[n] = g ( b \sin {\omega \over 2} n )
\end{displaymath}
-->
<IMG
WIDTH="136" HEIGHT="35" BORDER="0"
SRC="img567.png"
ALT="\begin{displaymath}
{m_a}[n] = g ( b \sin {\omega \over 2} n )
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
with
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
g(x) = e ^ {- x ^ 2}
\end{displaymath}
-->
<IMG
WIDTH="80" HEIGHT="28" BORDER="0"
SRC="img568.png"
ALT="\begin{displaymath}
g(x) = e ^ {- x ^ 2}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Except for a missing normalization factor, this is a Gaussian distribution,
sometimes called a ``bell curve". The amplitudes of the harmonics are
given by Bessel ``I" type functions.
<P>
Another fine choice is the (again unnormalized) Cauchy distribution:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
h(x) = {1\over{1 + {x^2}}}
\end{displaymath}
-->
<IMG
WIDTH="97" HEIGHT="40" BORDER="0"
SRC="img569.png"
ALT="\begin{displaymath}
h(x) = {1\over{1 + {x^2}}}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
which gives rise to a spectrum of exponentially falling harmonics:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
h(b \sin({\omega \over 2} n)) =
G \cdot \left (
{1\over 2} + H \cos(\omega n) + {H^2} \cos(2 \omega n)
+ \cdots
\right )
\end{displaymath}
-->
<IMG
WIDTH="397" HEIGHT="45" BORDER="0"
SRC="img570.png"
ALT="\begin{displaymath}
h(b \sin({\omega \over 2} n)) =
G \cdot \left (
{1\over 2...
...H \cos(\omega n) + {H^2} \cos(2 \omega n)
+ \cdots
\right )
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where <IMG
WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img571.png"
ALT="$G$"> and <IMG
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img25.png"
ALT="$H$"> are functions of the index <IMG
WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img21.png"
ALT="$b$">
(explicit formulas are given in [<A
HREF="node202.html#r-puckette95a">Puc95a</A>]).
<P>
In both this and the Gaussian case above, the bandwidth (counted in peaks,
i.e., units of <IMG
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img27.png"
ALT="$\omega $">) is roughly proportional to the index <IMG
WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img21.png"
ALT="$b$">, and the
amplitude of the DC term (the apex of the spectrum) is roughly proportional
to <IMG
WIDTH="66" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img572.png"
ALT="$1/(1+b)$"> .
For either waveshaping function (<IMG
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img29.png"
ALT="$g$"> or <IMG
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img194.png"
ALT="$h$">), if <IMG
WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img21.png"
ALT="$b$"> is larger than about 2,
the waveshape of <!-- MATH
${m_a}(\omega n)$
-->
<IMG
WIDTH="57" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img573.png"
ALT="${m_a}(\omega n)$"> is
approximately a (forward or backward) scan of the transfer function, so
the resulting waveform looks
like pulses whose widths decrease as the specified bandwidth increases.
<P>
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<ADDRESS>
Miller Puckette
2006-12-30
</ADDRESS>
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