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<H2><A ID="SECTION00954000000000000000">
Waveshaping using Chebychev polynomials</A>
</H2>
<A ID="sect5.chebychev"></A>
<P>
Example E05.chebychev.pd (Figure <A HREF="#fig05.12">5.12</A>) demonstrates how you can use waveshaping
to generate pure harmonics. We'll limit ourselves to a specific example here
in which we would like
to generate the pure fifth harmonic,
<DIV ALIGN="CENTER"><A ID="fig05.12"></A><A ID="5829"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 5.12:</STRONG>
Using Chebychev polynomials as waveshaping transfer functions.</CAPTION>
<TR><TD><IMG
WIDTH="411" HEIGHT="313" BORDER="0"
SRC="img494.png"
ALT="\begin{figure}\psfig{file=figs/fig05.12.ps}\end{figure}"></TD></TR>
</TABLE>
</DIV>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\cos(5 \omega n)
\end{displaymath}
-->
<IMG
WIDTH="60" HEIGHT="28" BORDER="0"
SRC="img495.png"
ALT="\begin{displaymath}
\cos(5 \omega n)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
by waveshaping a sinusoid
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
x[n] = \cos(\omega n)
\end{displaymath}
-->
<IMG
WIDTH="100" HEIGHT="28" BORDER="0"
SRC="img453.png"
ALT="\begin{displaymath}
x[n] = \cos (\omega n)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
We
need to find a suitable transfer function <IMG
WIDTH="34" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img460.png"
ALT="$f(x)$">. First we recall
the formula for the waveshaping function <IMG
WIDTH="71" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img496.png"
ALT="$f(x) = x^5$">
(Page <A HREF="node78.html#eq-waveshaping"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
SRC="crossref.png"></A>), which gives first,
third and fifth harmonics:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
16 {x^5} = \cos (5 \omega n) + 5 \cos(3 \omega n) + 10 \cos(\omega n)
\end{displaymath}
-->
<IMG
WIDTH="296" HEIGHT="28" BORDER="0"
SRC="img497.png"
ALT="\begin{displaymath}
16 {x^5} = \cos (5 \omega n) + 5 \cos(3 \omega n) + 10 \cos(\omega n)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Next we add a suitable multiple of <IMG
WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
SRC="img498.png"
ALT="$x^3$"> to
cancel the third harmonic:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
16 {x^5} - 20 {x^3} = \cos (5 \omega n) - 5 \cos(\omega n)
\end{displaymath}
-->
<IMG
WIDTH="248" HEIGHT="28" BORDER="0"
SRC="img499.png"
ALT="\begin{displaymath}
16 {x^5} - 20 {x^3} = \cos (5 \omega n) - 5 \cos(\omega n)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
and then a multiple of <IMG
WIDTH="12" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img243.png"
ALT="$x$"> to cancel the first harmonic:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
16 {x^5} - 20 {x^3} + 5 x = \cos (5 \omega n)
\end{displaymath}
-->
<IMG
WIDTH="201" HEIGHT="28" BORDER="0"
SRC="img500.png"
ALT="\begin{displaymath}
16 {x^5} - 20 {x^3} + 5 x = \cos (5 \omega n)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
So for our waveshaping function we choose
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
f(x) = 16 {x^5} - 20 {x^3} + 5 x
\end{displaymath}
-->
<IMG
WIDTH="170" HEIGHT="28" BORDER="0"
SRC="img501.png"
ALT="\begin{displaymath}
f(x) = 16 {x^5} - 20 {x^3} + 5 x
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
This procedure allows us to isolate any desired harmonic; the resulting
functions <IMG
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img112.png"
ALT="$f$"> are known as
<A ID="5840"></A><I>Chebychev polynomials</I> [<A
HREF="node202.html#r-lebrun79">Leb79</A>].
<P>
To incorporate this in a waveshaping instrument, we simply build a patch
that works as in Figure <A HREF="node78.html#fig05.05">5.5</A>, computing the expression
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
x[n] = f( a[n] \cos(\omega n))
\end{displaymath}
-->
<IMG
WIDTH="151" HEIGHT="28" BORDER="0"
SRC="img502.png"
ALT="\begin{displaymath}
x[n] = f( a[n] \cos(\omega n))
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where <IMG
WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img503.png"
ALT="$a[n]$"> is a suitable
<A ID="5844"></A><I>index</I>
which may vary as a function of the sample number <IMG
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img75.png"
ALT="$n$">. When <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img4.png"
ALT="$a$"> happens to
be one in value, out comes the pure fifth harmonic. Other values of <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img4.png"
ALT="$a$">
give varying spectra which, in general, have first and third harmonics as
well as the fifth.
<P>
By suitably combining Chebychev polynomials we can fix any desired
superposition of components in the output waveform (again, as long as
the waveshaping index is one). But the real
promise of waveshaping--that by simply changing the index we can
manufacture spectra that evolve in interesting but controllable ways--is
not addressed, at least directly, in the Chebychev picture.
<P>
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<ADDRESS>
Miller Puckette
2006-12-30
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