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<H2><A NAME="SECTION001022000000000000000">
Pulse trains via wavetable stretching</A>
</H2>
<P>
In the wavetable formulation, a pulse train can be made by a stretched
wavetable:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
{M_a}(\phi) = W (a \phi),
\end{displaymath}
-->
<IMG
WIDTH="117" HEIGHT="28" BORDER="0"
SRC="img574.png"
ALT="\begin{displaymath}
{M_a}(\phi) = W (a \phi),
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where <!-- MATH
$-\pi \le \phi \le \pi$
-->
<IMG
WIDTH="87" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img556.png"
ALT="$-\pi \le \phi \le \pi$"> is the phase, i.e., the value <IMG
WIDTH="23" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img575.png"
ALT="$\omega n$"> wrapped
to lie between <IMG
WIDTH="25" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img576.png"
ALT="$-\pi$"> and <IMG
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img41.png"
ALT="$\pi $">. The function <IMG
WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img31.png"
ALT="$W$"> should be zero at and beyond
the points <IMG
WIDTH="25" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img576.png"
ALT="$-\pi$"> and <IMG
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img41.png"
ALT="$\pi $">, and rise to a maximum at 0. A possible choice for
the function <IMG
WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img31.png"
ALT="$W$"> is
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
W(\phi) = {1\over2} \left ( \cos(\phi) + 1 \right )
\end{displaymath}
-->
<IMG
WIDTH="156" HEIGHT="38" BORDER="0"
SRC="img577.png"
ALT="\begin{displaymath}
W(\phi) = {1\over2} \left ( \cos(\phi) + 1 \right )
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
which is graphed in part (a) of Figure <A HREF="#fig06.04">6.4</A>. This is known as the
<A NAME="6857"></A><A NAME="6858"></A><I>Hann window function</I>;
it will come up again
in Chapter <A HREF="node163.html#chapter-fourier">9</A>.
<P>
<DIV ALIGN="CENTER"><A NAME="fig06.04"></A><A NAME="6863"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure:</STRONG>
Pulse width modulation using the von Hann window function: (a) the
function <!-- MATH
$W(\phi)=(1+\cos(\phi))/2$
-->
<IMG
WIDTH="162" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img578.png"
ALT="$W(\phi)=(1+\cos(\phi))/2$">; (b) the function as a waveform,
repeated at a duty
cycle of 100% (modulation index <IMG
WIDTH="41" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img18.png"
ALT="$a=1$">); (c) the waveform at a 50% duty
cycle (<IMG
WIDTH="41" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img19.png"
ALT="$a=2$">).</CAPTION>
<TR><TD><IMG
WIDTH="407" HEIGHT="403" BORDER="0"
SRC="img579.png"
ALT="\begin{figure}\psfig{file=figs/fig06.04.ps}\end{figure}"></TD></TR>
</TABLE>
</DIV>
<P>
Realizing this as a repeating waveform, we get a succession of (appropriately
sampled) copies of the function <IMG
WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img31.png"
ALT="$W$">, whose duty cycle is
<IMG
WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img580.png"
ALT="$1/a$"> (parts b and c of the figure).
If you don't wish the copies to overlap the index <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img4.png"
ALT="$a$"> must be
at least 1. If you want to allow overlap the simplest strategy is to
duplicate the block diagram (Figure <A HREF="node90.html#fig06.03">6.3</A>) out of phase, as described
in Section <A HREF="node30.html#sect2.stretching">2.4</A> and realized in Section
<A HREF="node37.html#sect2.example.overlap">2.6</A>.
<P>
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