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492 lines
14 KiB
492 lines
14 KiB
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<H1><A ID="SECTION00940000000000000000"></A>
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<A ID="sect5.FM"></A>
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<BR>
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Frequency and phase modulation
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</H1>
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<P>
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If a sinusoid is given a frequency which varies slowly in time we hear it as
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having a varying pitch. But if the pitch changes so quickly that our ears
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can't track the change--for instance, if the change itself occurs at or
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above the fundamental frequency of the sinusoid--we hear a timbral change.
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The timbres so generated are rich and widely varying. The discovery by
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John Chowning of this
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possibility [<A
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HREF="node202.html#r-chowning73">Cho73</A>] revolutionized the field of computer music.
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Here we develop
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<A ID="5762"></A><A ID="5763"></A><I>frequency modulation</I>,
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usually called <I>FM</I>,
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as a special case of waveshaping [<A
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HREF="node202.html#r-lebrun79">Leb79</A>]
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[<A
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HREF="node202.html#r-dodge85">DJ85</A>, pp.155-158]; the
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analysis given here is somewhat different [<A
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HREF="node202.html#r-puckette01a">Puc01</A>].
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<P>
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The FM technique, in its simplest form, is shown in Figure <A HREF="#fig05.08">5.8</A>
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(part a).
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A frequency-modulated sinusoid is one whose frequency varies sinusoidally, at
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some angular 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="$\omega_m$">, about a central 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="$\omega_c$">, so
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that the instantaneous frequencies vary between <IMG
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WIDTH="67" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img465.png"
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ALT="$(1-r)\omega_c $"> and
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<!-- MATH
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$(1+r) \omega_c$
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-->
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<IMG
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WIDTH="67" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img466.png"
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ALT="$(1+r) \omega_c$">, with parameters <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="$\omega_m$"> controlling the frequency of
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variation, and <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img467.png"
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ALT="$r$"> controlling the depth of variation. The parameters
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<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="$\omega_c$">, <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="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img467.png"
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ALT="$r$"> are called the
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<A ID="5770"></A><A ID="5771"></A><I>carrier frequency</I>, the
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<A ID="5773"></A><I>modulation frequency</I>, and the
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<A ID="5775"></A><I>index of modulation</I>, respectively.
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<P>
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It is customary to use a simpler, essentially equivalent formulation in
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which the phase, instead of the frequency, of the carrier sinusoid is
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modulated sinusoidally. (This gives an equivalent result since the
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instantaneous frequency is the rate of change of phase, and since the
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rate of change of a sinusoid is just another sinusoid.) The
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phase modulation formulation is shown in part (b) of the figure.
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<P>
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We can analyze the result of phase modulation as follows, assuming that
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the modulating oscillator and the wavetable are both sinusoidal, and that
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the carrier and modulation frequencies don't themselves vary
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in time. The resulting signal can then be written as
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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(a \cos(\omega_m n) + \omega_c n )
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\end{displaymath}
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-->
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<IMG
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WIDTH="202" HEIGHT="28" BORDER="0"
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SRC="img468.png"
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ALT="\begin{displaymath}
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x[n] = \cos(a \cos(\omega_m n) + \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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The
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parameter <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="$a$">, which takes the place of the earlier parameter <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img467.png"
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ALT="$r$">, is
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likewise called the index of modulation; it too
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controls the extent of frequency variation relative to the carrier frequency
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<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="$\omega_c$">. If <IMG
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WIDTH="41" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img469.png"
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ALT="$a=0$">, there
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is no frequency variation and the expression reduces to the unmodified,
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carrier sinusoid; as <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="$a$"> increases the waveform becomes more complex.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig05.08"></A><A ID="5779"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 5.8:</STRONG>
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Block diagram for frequency modulation (FM) synthesis: (a) the classic
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form; (b) realized as phase modulation.</CAPTION>
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<TR><TD><IMG
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WIDTH="507" HEIGHT="499" BORDER="0"
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SRC="img470.png"
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ALT="\begin{figure}\psfig{file=figs/fig05.08.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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To analyse the resulting spectrum we can rewrite the signal as,
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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) * \cos(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="231" HEIGHT="28" BORDER="0"
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SRC="img471.png"
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ALT="\begin{displaymath}
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x[n] = \cos(\omega_c n) * \cos(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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<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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- \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="194" HEIGHT="28" BORDER="0"
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SRC="img472.png"
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ALT="\begin{displaymath}
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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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We can consider the result as a sum of two waveshaping
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generators, each operating on a sinusoid of 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="$\omega_m$"> and
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with a waveshaping 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="$a$">, and each ring modulated with a sinusoid of
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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="$\omega_c$">. The 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="$f$"> is given by
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<!-- MATH
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$f(x) = \cos(x)$
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-->
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<IMG
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WIDTH="98" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img473.png"
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ALT="$f(x) = \cos(x)$"> for the first term and by <!-- MATH
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$f(x) = \sin(x)$
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-->
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<IMG
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WIDTH="96" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img474.png"
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ALT="$f(x) = \sin(x)$"> for the second.
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<P>
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Returning to Figure <A HREF="node77.html#fig05.04">5.4</A>, we can predict what the
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spectrum will look like. The two harmonic spectra, of the waveshaping outputs
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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(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="108" HEIGHT="28" BORDER="0"
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SRC="img475.png"
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ALT="\begin{displaymath}
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\cos(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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and
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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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\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="107" HEIGHT="28" BORDER="0"
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SRC="img476.png"
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ALT="\begin{displaymath}
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\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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have, respectively, harmonics tuned to
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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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0, 2\omega_m, 4\omega_m, \ldots
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\end{displaymath}
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-->
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<IMG
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WIDTH="105" HEIGHT="27" BORDER="0"
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SRC="img477.png"
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ALT="\begin{displaymath}
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0, 2\omega_m, 4\omega_m, \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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and
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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_m, 3\omega_m, 5\omega_m, \ldots
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\end{displaymath}
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-->
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<IMG
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WIDTH="120" HEIGHT="27" BORDER="0"
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SRC="img478.png"
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ALT="\begin{displaymath}
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\omega_m, 3\omega_m, 5\omega_m, \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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and each is multiplied by a sinusoid at the carrier frequency. So there
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will be a spectrum centered at 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 at both even and odd multiples of the modulation 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="$\omega_m$">,
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contributed respectively by the sine and cosine waveshaping terms above.
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The index of modulation <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="$a$">, as it changes, controls the relative strength of
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the various partials. The partials themselves are situated at the frequencies
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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 + m \omega_m
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\end{displaymath}
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-->
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<IMG
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WIDTH="71" HEIGHT="26" BORDER="0"
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SRC="img479.png"
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ALT="\begin{displaymath}
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\omega_c + m \omega_m
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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
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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 = \ldots -2, -1, 0, 1, 2, \ldots
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\end{displaymath}
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-->
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<IMG
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WIDTH="178" HEIGHT="27" BORDER="0"
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SRC="img480.png"
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ALT="\begin{displaymath}
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m = \ldots -2, -1, 0, 1, 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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As with any situation where two periodic signals are multiplied, if there is
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some common supermultiple of the two periods, the resulting product will repeat
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at that longer period. So if the two periods are <IMG
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WIDTH="21" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img481.png"
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ALT="$k \tau$"> and <IMG
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WIDTH="26" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img482.png"
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ALT="$m \tau$">, where
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<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="$k$"> and <IMG
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WIDTH="17" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img111.png"
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ALT="$m$"> are relatively prime, they both repeat after a time interval of
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<IMG
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WIDTH="35" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img483.png"
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ALT="$km\tau$">. In other words, if the two have frequencies which are both multiples
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of some common frequency, so that <!-- MATH
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$\omega_m=k\omega$
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-->
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<IMG
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WIDTH="65" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img484.png"
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ALT="$\omega_m=k\omega$"> and <!-- MATH
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$\omega_c=m\omega$
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-->
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<IMG
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WIDTH="65" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img485.png"
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ALT="$\omega_c=m\omega$">,
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again with <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="$k$"> and <IMG
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WIDTH="17" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img111.png"
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ALT="$m$"> relatively prime, the result will repeat at a frequency
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of the common submultiple <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $">. On the other hand, if no common
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submultiple <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $"> can be found, or if the only submultiples are lower than
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any discernible pitch, then the result will be inharmonic.
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<P>
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Much more about FM can be found in textbooks [<A
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HREF="node202.html#r-moore90">Moo90</A>, p. 316]
|
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[<A
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HREF="node202.html#r-dodge85">DJ85</A>, pp.115-139] [<A
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HREF="node202.html#r-boulanger00">Bou00</A>] and the research literature.
|
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Some of the
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possibilities are shown in the following examples.
|
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|
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<P>
|
|
<HR>
|
|
<!--Navigation Panel-->
|
|
<A ID="tex2html1725"
|
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HREF="node80.html">
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<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
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SRC="next.png"></A>
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<A ID="tex2html1719"
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<A ID="tex2html1713"
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<A ID="tex2html1721"
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HREF="node201.html">
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SRC="index.png"></A>
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<BR>
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<B> Next:</B> <A ID="tex2html1726"
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HREF="node80.html">Examples</A>
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<B> Up:</B> <A ID="tex2html1720"
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HREF="node75.html">Modulation</A>
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<B> Previous:</B> <A ID="tex2html1714"
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HREF="node78.html">Waveshaping</A>
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<B> <A ID="tex2html1722"
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HREF="node4.html">Contents</A></B>
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<B> <A ID="tex2html1724"
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HREF="node201.html">Index</A></B>
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<!--End of Navigation Panel-->
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<ADDRESS>
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Miller Puckette
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2006-12-30
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</ADDRESS>
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</BODY>
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</HTML>
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