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233 lines
6.7 KiB
233 lines
6.7 KiB
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<H2><A ID="SECTION001022000000000000000">
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Pulse trains via wavetable stretching</A>
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</H2>
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<P>
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In the wavetable formulation, a pulse train can be made by a stretched
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wavetable:
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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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{M_a}(\phi) = W (a \phi),
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\end{displaymath}
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-->
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<IMG
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WIDTH="117" HEIGHT="28" BORDER="0"
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SRC="img574.png"
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ALT="\begin{displaymath}
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{M_a}(\phi) = W (a \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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where <!-- MATH
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$-\pi \le \phi \le \pi$
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-->
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<IMG
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WIDTH="87" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img556.png"
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ALT="$-\pi \le \phi \le \pi$"> is the phase, i.e., the value <IMG
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WIDTH="23" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img575.png"
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ALT="$\omega n$"> wrapped
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to lie between <IMG
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WIDTH="25" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img576.png"
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ALT="$-\pi$"> and <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img41.png"
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ALT="$\pi $">. The function <IMG
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WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img31.png"
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ALT="$W$"> should be zero at and beyond
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the points <IMG
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WIDTH="25" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img576.png"
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ALT="$-\pi$"> and <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img41.png"
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ALT="$\pi $">, and rise to a maximum at 0. A possible choice for
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the function <IMG
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WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img31.png"
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ALT="$W$"> is
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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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W(\phi) = {1\over2} \left ( \cos(\phi) + 1 \right )
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\end{displaymath}
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-->
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<IMG
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WIDTH="156" HEIGHT="38" BORDER="0"
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SRC="img577.png"
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ALT="\begin{displaymath}
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W(\phi) = {1\over2} \left ( \cos(\phi) + 1 \right )
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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 is graphed in part (a) of Figure <A HREF="#fig06.04">6.4</A>. This is known as the
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<A ID="6857"></A><A ID="6858"></A><I>Hann window function</I>;
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it will come up again
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in Chapter <A HREF="node163.html#chapter-fourier">9</A>.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig06.04"></A><A ID="6863"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure:</STRONG>
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Pulse width modulation using the von Hann window function: (a) the
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function <!-- MATH
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$W(\phi)=(1+\cos(\phi))/2$
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-->
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<IMG
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WIDTH="162" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img578.png"
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ALT="$W(\phi)=(1+\cos(\phi))/2$">; (b) the function as a waveform,
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repeated at a duty
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cycle of 100% (modulation index <IMG
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WIDTH="41" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img18.png"
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ALT="$a=1$">); (c) the waveform at a 50% duty
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cycle (<IMG
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WIDTH="41" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img19.png"
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ALT="$a=2$">).</CAPTION>
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<TR><TD><IMG
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WIDTH="407" HEIGHT="403" BORDER="0"
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SRC="img579.png"
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ALT="\begin{figure}\psfig{file=figs/fig06.04.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Realizing this as a repeating waveform, we get a succession of (appropriately
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sampled) copies of the function <IMG
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WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img31.png"
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ALT="$W$">, whose duty cycle is
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<IMG
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WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img580.png"
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ALT="$1/a$"> (parts b and c of the figure).
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If you don't wish the copies to overlap the index <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img4.png"
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ALT="$a$"> must be
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at least 1. If you want to allow overlap the simplest strategy is to
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duplicate the block diagram (Figure <A HREF="node90.html#fig06.03">6.3</A>) out of phase, as described
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in Section <A HREF="node30.html#sect2.stretching">2.4</A> and realized in Section
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<A HREF="node37.html#sect2.example.overlap">2.6</A>.
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<P>
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<BR>
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<B> Next:</B> <A ID="tex2html1942"
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HREF="node94.html">Resulting spectra</A>
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<B> Up:</B> <A ID="tex2html1936"
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HREF="node91.html">Pulse trains</A>
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<B> Previous:</B> <A ID="tex2html1930"
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HREF="node92.html">Pulse trains via waveshaping</A>
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<B> <A ID="tex2html1938"
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HREF="node4.html">Contents</A></B>
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<B> <A ID="tex2html1940"
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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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