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303 lines
8.2 KiB
303 lines
8.2 KiB
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original version by: Nikos Drakos, CBLU, University of Leeds
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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HREF="node189.html">Sawtooth wave</A>
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HREF="node184.html">Classical waveforms</A>
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HREF="node187.html">Dissecting classical waveforms</A>
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HREF="node4.html">Contents</A></B>
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<H1><A ID="SECTION001430000000000000000"></A>
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<A ID="sect10.spectra"></A>
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<BR>
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Fourier series of the elementary waveforms
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</H1>
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<P>
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In general, given a repeating waveform <IMG
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WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img669.png"
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ALT="$X[n]$">, we can evaluate its Fourier
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series coefficients <IMG
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WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1251.png"
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ALT="$A[k]$"> by directly evaluating the Fourier transform:
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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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A[k] = {1 \over N} {\cal FT}\{X[n]\}(k)
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\end{displaymath}
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-->
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<IMG
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WIDTH="164" HEIGHT="38" BORDER="0"
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SRC="img1301.png"
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ALT="\begin{displaymath}
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A[k] = {1 \over N} {\cal FT}\{X[n]\}(k)
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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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<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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= {1 \over N} \left [ X[0] +
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{U^{-k}} X[1] + \cdots +
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{U^{-(N-1)k}} X[N-1] \right ]
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\end{displaymath}
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-->
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<IMG
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WIDTH="351" HEIGHT="38" BORDER="0"
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SRC="img1302.png"
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ALT="\begin{displaymath}
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= {1 \over N} \left [ X[0] +
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{U^{-k}} X[1] + \cdots +
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{U^{-(N-1)k}} X[N-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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but doing this directly for sawtooth and parabolic waves will require pages of
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algebra (somewhat less if we were willing resort to differential calculus).
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Instead, we rely on properties of the Fourier transform to relate the transform
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of a signal <IMG
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WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img80.png"
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ALT="$x[n]$"> with its
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<A ID="14332"></A><I>first difference</I>,
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defined as <IMG
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WIDTH="105" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1303.png"
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ALT="$x[n] - x[n-1]$">. The first difference of the parabolic wave will
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turn out to be a sawtooth, and that of a sawtooth will be simple enough to
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evaluate directly, and thus we'll get the desired Fourier series.
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<P>
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In general, to evaluate the strength of the <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img58.png"
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ALT="$k$">th harmonic, we'll make the
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assumption that <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$"> is much larger than <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img58.png"
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ALT="$k$">, or equivalently, that <IMG
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WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1304.png"
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ALT="$k/N$"> is
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negligible.
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<P>
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We start from the Time Shift Formula for Fourier Transforms
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(Page <A HREF="node169.html#sect9.shift"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="crossref.png"></A>) setting the time shift to one sample:
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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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{\cal FT}\{ x[n-1] \} =
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\left [ \cos(k \omega) - i \sin (k \omega) \right ]
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{\cal FT}\{ x[n] \}
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\end{displaymath}
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-->
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<IMG
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WIDTH="329" HEIGHT="28" BORDER="0"
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SRC="img1305.png"
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ALT="\begin{displaymath}
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{\cal FT}\{ x[n-1] \} =
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\left [ \cos(k \omega) - i \sin (k \omega) \right ]
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{\cal FT}\{ x[n] \}
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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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<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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\approx (1 - i \omega k) {\cal FT}\{ x[n] \}
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\end{displaymath}
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-->
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<IMG
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WIDTH="148" HEIGHT="28" BORDER="0"
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SRC="img1306.png"
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ALT="\begin{displaymath}
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\approx (1 - i \omega k) {\cal FT}\{ x[n] \}
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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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Here we're using the assumption that, because <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$"> is much larger than <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img58.png"
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ALT="$k$">,
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<!-- MATH
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$k \omega = 2\pi k / N$
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-->
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<IMG
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WIDTH="91" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1307.png"
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ALT="$k \omega = 2\pi k / N$"> is much smaller than unity and we can make
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approximations:
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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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\cos(k \omega) \approx 1 \; , \; \sin(k \omega) \approx k \omega
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\end{displaymath}
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-->
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<IMG
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WIDTH="187" HEIGHT="28" BORDER="0"
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SRC="img1308.png"
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ALT="\begin{displaymath}
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\cos(k \omega) \approx 1 \; , \; \sin(k \omega) \approx k \omega
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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 are good to within a small error, on the order of <IMG
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WIDTH="53" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img1309.png"
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ALT="$(k/N)^2$">.
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Now we plug this result in to evaluate:
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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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{\cal FT}\{ x[n] - x[n-1] \} \approx i \omega k {\cal FT}\{ x[n] \}
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\end{displaymath}
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-->
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<IMG
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WIDTH="257" HEIGHT="28" BORDER="0"
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SRC="img1310.png"
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ALT="\begin{displaymath}
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{\cal FT}\{ x[n] - x[n-1] \} \approx i \omega k {\cal FT}\{ x[n] \}
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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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<BR><HR>
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<!--Table of Child-Links-->
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<A ID="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<UL>
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<LI><A ID="tex2html3384"
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HREF="node189.html">Sawtooth wave</A>
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<LI><A ID="tex2html3385"
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HREF="node190.html">Parabolic wave</A>
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<LI><A ID="tex2html3386"
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HREF="node191.html">Square and symmetric triangle waves</A>
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<LI><A ID="tex2html3387"
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HREF="node192.html">General (non-symmetric) triangle wave</A>
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</UL>
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<B> Next:</B> <A ID="tex2html3383"
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HREF="node189.html">Sawtooth wave</A>
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<B> Up:</B> <A ID="tex2html3377"
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HREF="node184.html">Classical waveforms</A>
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<B> Previous:</B> <A ID="tex2html3371"
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HREF="node187.html">Dissecting classical waveforms</A>
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<B> <A ID="tex2html3379"
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HREF="node4.html">Contents</A></B>
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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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