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HREF="node86.html">Sinusoidal waveshaping: evenness and</A>
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<B> <A ID="tex2html1841"
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HREF="node201.html">Index</A></B>
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<H2><A ID="SECTION00957000000000000000">
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Phase modulation and FM</A>
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</H2>
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<A ID="sect5.example.fm"></A>
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<P>
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Example E08.phase.mod.pd, shown in Figure <A HREF="#fig05.15">5.15</A>, shows how to use Pd to
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realize true frequency
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modulation (part a) and phase modulation (part b). These correspond to the
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block diagrams of Figure <A HREF="node79.html#fig05.08">5.8</A>. To
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accomplish phase modulation, the carrier oscillator is split into its phase and
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cosine lookup components. The signal is 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[t] = \cos( \omega_c n + 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="198" HEIGHT="28" BORDER="0"
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SRC="img533.png"
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ALT="\begin{displaymath}
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x[t] = \cos( \omega_c n + 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="$\omega_c$"> is the carrier frequency, <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="$\omega_m$"> is the modulation
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frequency, 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="$a$"> is the index of modulation--all in angular units.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig05.15"></A><A ID="5903"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 5.15:</STRONG>
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Pd patches for: (a) frequency modulation; (b) phase modulation.</CAPTION>
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<TR><TD><IMG
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WIDTH="480" HEIGHT="265" BORDER="0"
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SRC="img534.png"
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ALT="\begin{figure}\psfig{file=figs/fig05.15.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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We can predict the spectrum by expanding the outer cosine:
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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[t] = \cos( \omega_c n ) \cos (a \cos(\omega_m n))
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- \sin( \omega_c n ) \sin (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="403" HEIGHT="28" BORDER="0"
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SRC="img535.png"
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ALT="\begin{displaymath}
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x[t] = \cos( \omega_c n ) \cos (a \cos(\omega_m n))
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- \sin( \omega_c n ) \sin (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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Plugging in the expansions from Page <A HREF="node86.html#sect5.bessel"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="crossref.png"></A> and simplifying
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yields:
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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[t] = {J_0}(a) \cos( \omega_c n )
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\end{displaymath}
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-->
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<IMG
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WIDTH="142" HEIGHT="28" BORDER="0"
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SRC="img536.png"
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ALT="\begin{displaymath}
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x[t] = {J_0}(a) \cos( \omega_c 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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+ {J_1}(a) \cos( (\omega_c+\omega_m) n + {\pi\over2})
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+ {J_1}(a) \cos( (\omega_c-\omega_m) n + {\pi\over2})
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\end{displaymath}
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-->
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<IMG
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WIDTH="401" HEIGHT="35" BORDER="0"
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SRC="img537.png"
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ALT="\begin{displaymath}
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+ {J_1}(a) \cos( (\omega_c+\omega_m) n + {\pi\over2})
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+ {J_1}(a) \cos( (\omega_c-\omega_m) n + {\pi\over2})
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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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+ {J_2}(a) \cos( (\omega_c+2\omega_m) n + \pi)
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+ {J_2}(a) \cos( (\omega_c-2\omega_m) n + \pi)
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\end{displaymath}
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-->
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<IMG
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WIDTH="409" HEIGHT="28" BORDER="0"
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SRC="img538.png"
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ALT="\begin{displaymath}
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+ {J_2}(a) \cos( (\omega_c+2\omega_m) n + \pi)
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+ {J_2}(a) \cos( (\omega_c-2\omega_m) n + \pi)
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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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+ {J_3}(a) \cos( (\omega_c+3\omega_m) n + {{3\pi}\over2})
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+ {J_3}(a) \cos( (\omega_c-3\omega_m) n + {{3\pi}\over2})
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+ \cdots
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\end{displaymath}
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-->
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<IMG
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WIDTH="471" HEIGHT="38" BORDER="0"
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SRC="img539.png"
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ALT="\begin{displaymath}
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+ {J_3}(a) \cos( (\omega_c+3\omega_m) n + {{3\pi}\over2})
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...
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...}(a) \cos( (\omega_c-3\omega_m) n + {{3\pi}\over2})
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+ \cdots
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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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So the components are centered about the 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="$\omega_c$"> with
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sidebands extending in either direction, each spaced <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="$\omega_m$"> from the
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next. The amplitudes are functions of the index of
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modulation, and don't depend on the frequencies.
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Figure <A HREF="#fig05.16">5.16</A> shows some two-operator phase modulation spectra,
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measured using Example E09.FM.spectrum.pd.
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<P>
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2022-04-12 23:32:40 -03:00
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<DIV ALIGN="CENTER"><A ID="fig05.16"></A><A ID="5921"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 5.16:</STRONG>
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Spectra from phase modulation at three different indices. The indices
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are given as multiples of <IMG
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WIDTH="21" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img16.png"
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ALT="$2\pi $"> radians.</CAPTION>
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<TR><TD><IMG
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WIDTH="441" HEIGHT="438" BORDER="0"
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SRC="img540.png"
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ALT="\begin{figure}\psfig{file=figs/fig05.16.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Phase modulation can thus be seen simply as a form of ring modulated
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waveshaping. So
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we can use the
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strategies described in Section <A HREF="node77.html#sect5.ringmod">5.2</A> to generate particular
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combinations of frequencies. For example, if the carrier frequency is
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half the modulation frequency, you get a sound with odd harmonics exactly
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as in the octave dividing example (Figure <A HREF="node82.html#fig05.10">5.10</A>).
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<P>
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Frequency modulation need not be restricted to purely sinusoidal carrier or
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modulation oscillators. One well-trodden path is to effect phase modulation
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on the phase modulation spectrum itself. There are then two indices of
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modulation (call
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them <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$"> and <IMG
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WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img21.png"
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ALT="$b$">) and two frequencies of modulation (<IMG
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WIDTH="25" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img463.png"
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ALT="$\omega_m$"> and <IMG
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WIDTH="20" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img541.png"
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ALT="$\omega_p$">)
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and the waveform is:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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x[n] = \cos(\omega_c n + a \cos(\omega_m n) + b \cos(\omega_p n))
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\end{displaymath}
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-->
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<IMG
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WIDTH="291" HEIGHT="29" BORDER="0"
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SRC="img542.png"
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ALT="\begin{displaymath}
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x[n] = \cos(\omega_c n + a \cos(\omega_m n) + b \cos(\omega_p n))
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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To analyze the result, just rewrite the original FM series above, replacing
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<IMG
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WIDTH="29" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img543.png"
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ALT="$\omega_c n$"> everywhere with <!-- MATH
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$\omega_c n + b \cos(\omega_p n)$
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-->
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<IMG
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WIDTH="118" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img544.png"
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ALT="$\omega_c n + b \cos(\omega_p n)$">. The third
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positive sideband becomes for instance:
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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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{J_3}(a) \cos( (\omega_c+3\omega_m) n + {{3\pi}\over2} + b \cos(\omega_p n))
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\end{displaymath}
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-->
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<IMG
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WIDTH="289" HEIGHT="38" BORDER="0"
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SRC="img545.png"
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ALT="\begin{displaymath}
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{J_3}(a) \cos( (\omega_c+3\omega_m) n + {{3\pi}\over2} + b \cos(\omega_p 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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This is itself just another FM spectrum, with its own sidebands of frequency
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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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\omega_c+3\omega_m + k \omega_p , k = 0, \pm 1, \pm 2, \ldots
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\end{displaymath}
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-->
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<IMG
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WIDTH="236" HEIGHT="29" BORDER="0"
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SRC="img546.png"
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ALT="\begin{displaymath}
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\omega_c+3\omega_m + k \omega_p , k = 0, \pm 1, \pm 2, \ldots
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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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having amplitude <!-- MATH
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${J_3}(a) {J_k}(b)$
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-->
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<IMG
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WIDTH="76" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img547.png"
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ALT="${J_3}(a) {J_k}(b)$"> and phase <IMG
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WIDTH="77" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img548.png"
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ALT="$(3+k)\pi / 2$">
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[<A
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HREF="node202.html#r-lebrun77">Leb77</A>]. Example E10.complex.FM.pd (not shown here) illustrates this by
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graphing spectra from a two-modulator FM instrument.
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<P>
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Since early times [<A
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HREF="node202.html#r-schottstaedt87">Sch77</A>] researchers have sought combinations
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of phases, frequencies, and modulation indices, for simple and compact phase
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modulation instruments, that manage to imitate familiar instrumental sounds.
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This became a major industry with the introduction of commercial FM
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synthesizers.
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<P>
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<HR>
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