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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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<!--End of Navigation Panel-->
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<H1><A NAME="SECTION001010000000000000000"></A>
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<A NAME="sect6.model"></A>
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<BR>
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Carrier/modulator model
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</H1>
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<P>
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Earlier we saw how to
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use ring modulation
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to modify the spectrum of a periodic signal, placing spectral peaks in
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specified locations (see Figure
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<A HREF="node77.html#fig05.04">5.4</A>, Page <A HREF="node77.html#fig05.04"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="file:/usr/local/share/lib/latex2html/icons/crossref.png"></A>). To do
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so we need to be able to generate periodic signals whose spectra have maxima
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at DC and fall off monotonically with increasing frequency.
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If we can make a signal with a formant at frequency zero--and no other
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formants besides that one--we can use ring modulation to displace the formant
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to any desired harmonic. If we use waveshaping to generate the initial formant,
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the ring modulation product will be of the form
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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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x[n] = \cos(\omega_c n) f (a \cos(\omega_m n))
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\end{displaymath}
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-->
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<IMG
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WIDTH="204" HEIGHT="28" BORDER="0"
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SRC="img552.png"
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ALT="\begin{displaymath}
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x[n] = \cos(\omega_c n) f (a \cos(\omega_m 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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where <IMG
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WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img464.png"
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ALT="$\omega_c$"> (the
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<A NAME="6813"></A><I>carrier frequency</I>) is set to the formant center frequency and
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<!-- MATH
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$f (a \cdot \cos(\omega_m n))$
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-->
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<IMG
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WIDTH="110" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img553.png"
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ALT="$f (a \cdot \cos(\omega_m n))$"> is a signal with fundamental frequency
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determined by
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<IMG
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WIDTH="25" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img463.png"
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ALT="$\omega_m$">,
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produced using a waveshaping function <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img112.png"
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ALT="$f$"> and 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="$a$">.
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This second term is the signal we wish to give a formant at DC with a
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controllable bandwidth. A block diagram for synthesizing this signal is
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shown in Figure <A HREF="#fig06.02">6.2</A>.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig06.02"></A><A NAME="6818"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 6.2:</STRONG>
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Ring modulated waveshaping for formant generation</CAPTION>
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<TR><TD><IMG
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WIDTH="251" HEIGHT="382" BORDER="0"
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SRC="img554.png"
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ALT="\begin{figure}\psfig{file=figs/fig06.02.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Much earlier in Section <A HREF="node30.html#sect2.stretching">2.4</A> we introduced the
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technique of
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<A NAME="6822"></A><I>timbre stretching</I>,
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as part of the discussion of wavetable synthesis. This technique, which is
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capable of generating complex, variable timbres, can be fit into the
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same framework. The enveloped wavetable output for one cycle 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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x(\phi) = T (c \phi) * W (a \phi),
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\end{displaymath}
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-->
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<IMG
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WIDTH="158" HEIGHT="28" BORDER="0"
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SRC="img555.png"
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ALT="\begin{displaymath}
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x(\phi) = T (c \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 <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img77.png"
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ALT="$\phi$">, the phase, satisfies <!-- 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="$-\pi \le \phi \le \pi$">. Here
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<IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img557.png"
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ALT="$T$"> is a function stored in a wavetable, <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 a windowing function,
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and <IMG
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WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img293.png"
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ALT="$c$"> and <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="$a$"> are the wavetable stretching and a modulation index for
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the windowing function. Figure <A HREF="#fig06.03">6.3</A> shows how to realize
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this in block
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diagram form. Comparing this to Figure <A HREF="node29.html#fig02.07">2.7</A>, we see that the
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only significant new feature is the addition of 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="$a$">.
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<P>
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In this setup, as in the previous one, the first term specifies the placement of
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energy in the spectrum--in this case, with the parameter <IMG
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WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img293.png"
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ALT="$c$"> acting to stretch
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out the wavetable spectrum. This is the role that was previously carried out
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by the choice of ring modulation carrier frequency <IMG
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WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img464.png"
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ALT="$\omega_c$">.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig06.03"></A><A NAME="6828"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 6.3:</STRONG>
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Wavetable synthesis generalized as a variable spectrum generator.</CAPTION>
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<TR><TD><IMG
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WIDTH="325" HEIGHT="396" BORDER="0"
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SRC="img558.png"
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ALT="\begin{figure}\psfig{file=figs/fig06.03.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Both of these (ring modulated waveshaping and stretched wavetable synthesis)
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can be considered as particular cases of a more general approach
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which is to compute functions of the form,
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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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x[n] = c(\omega n) {m_a}(\omega n)
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\end{displaymath}
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-->
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<IMG
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WIDTH="140" HEIGHT="28" BORDER="0"
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SRC="img559.png"
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ALT="\begin{displaymath}
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x[n] = c(\omega n) {m_a}(\omega 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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where <IMG
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WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img293.png"
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ALT="$c$"> is a periodic function describing the carrier signal, and <IMG
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WIDTH="25" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img560.png"
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ALT="${m_a}$">
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is a periodic modulator function which
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depends on an 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="$a$">.
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The modulation functions we're interested in will
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usually take the form of pulse trains, and 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="$a$"> will control the
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width of the pulse; higher values of <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="$a$"> will give narrower pulses.
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In the wavetable case, the modulation function must
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reach zero at phase wraparound points to suppress any discontinuities in the
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carrier function when the phase wraps around. The carrier signal will give
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rise to a single spectral peak (a formant) in the ring modulated waveshaping
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case; for wavetables, it may have a more complicated spectrum.
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<P>
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In the next section we will further develop the two forms of modulating
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signal we've introduced here, and in the following one we'll
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look more closely at the carrier signal.
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<B> Up:</B> <A NAME="tex2html1891"
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HREF="node89.html">Designer spectra</A>
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<B> Previous:</B> <A NAME="tex2html1885"
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HREF="node89.html">Designer spectra</A>
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<B> <A NAME="tex2html1893"
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HREF="node4.html">Contents</A></B>
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<B> <A NAME="tex2html1895"
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<ADDRESS>
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Miller Puckette
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2006-12-30
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