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<H1><A NAME="SECTION00940000000000000000"></A>
<A NAME="sect5.FM"></A>
<BR>
Frequency and phase modulation
</H1>
<P>
If a sinusoid is given a frequency which varies slowly in time we hear it as
having a varying pitch. But if the pitch changes so quickly that our ears
can't track the change--for instance, if the change itself occurs at or
above the fundamental frequency of the sinusoid--we hear a timbral change.
The timbres so generated are rich and widely varying. The discovery by
John Chowning of this
possibility [<A
HREF="node202.html#r-chowning73">Cho73</A>] revolutionized the field of computer music.
Here we develop
<A NAME="5762"></A><A NAME="5763"></A><I>frequency modulation</I>,
usually called <I>FM</I>,
as a special case of waveshaping [<A
HREF="node202.html#r-lebrun79">Leb79</A>]
[<A
HREF="node202.html#r-dodge85">DJ85</A>, pp.155-158]; the
analysis given here is somewhat different [<A
HREF="node202.html#r-puckette01a">Puc01</A>].
<P>
The FM technique, in its simplest form, is shown in Figure <A HREF="#fig05.08">5.8</A>
(part a).
A frequency-modulated sinusoid is one whose frequency varies sinusoidally, at
some angular frequency <IMG
WIDTH="25" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img463.png"
ALT="$\omega_m$">, about a central frequency <IMG
WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img464.png"
ALT="$\omega_c$">, so
that the instantaneous frequencies vary between <IMG
WIDTH="67" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img465.png"
ALT="$(1-r)\omega_c $"> and
<!-- MATH
$(1+r) \omega_c$
-->
<IMG
WIDTH="67" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img466.png"
ALT="$(1+r) \omega_c$">, with parameters <IMG
WIDTH="25" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img463.png"
ALT="$\omega_m$"> controlling the frequency of
variation, and <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img467.png"
ALT="$r$"> controlling the depth of variation. The parameters
<IMG
WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img464.png"
ALT="$\omega_c$">, <IMG
WIDTH="25" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img463.png"
ALT="$\omega_m$">, and <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img467.png"
ALT="$r$"> are called the
<A NAME="5770"></A><A NAME="5771"></A><I>carrier frequency</I>, the
<A NAME="5773"></A><I>modulation frequency</I>, and the
<A NAME="5775"></A><I>index of modulation</I>, respectively.
<P>
It is customary to use a simpler, essentially equivalent formulation in
which the phase, instead of the frequency, of the carrier sinusoid is
modulated sinusoidally. (This gives an equivalent result since the
instantaneous frequency is the rate of change of phase, and since the
rate of change of a sinusoid is just another sinusoid.) The
phase modulation formulation is shown in part (b) of the figure.
<P>
We can analyze the result of phase modulation as follows, assuming that
the modulating oscillator and the wavetable are both sinusoidal, and that
the carrier and modulation frequencies don't themselves vary
in time. The resulting signal can then be written as
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
x[n] = \cos(a \cos(\omega_m n) + \omega_c n )
\end{displaymath}
-->
<IMG
WIDTH="202" HEIGHT="28" BORDER="0"
SRC="img468.png"
ALT="\begin{displaymath}
x[n] = \cos(a \cos(\omega_m n) + \omega_c n )
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
The
parameter <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img4.png"
ALT="$a$">, which takes the place of the earlier parameter <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img467.png"
ALT="$r$">, is
likewise called the index of modulation; it too
controls the extent of frequency variation relative to the carrier frequency
<IMG
WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img464.png"
ALT="$\omega_c$">. If <IMG
WIDTH="41" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img469.png"
ALT="$a=0$">, there
is no frequency variation and the expression reduces to the unmodified,
carrier sinusoid; as <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img4.png"
ALT="$a$"> increases the waveform becomes more complex.
<P>
<DIV ALIGN="CENTER"><A NAME="fig05.08"></A><A NAME="5779"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 5.8:</STRONG>
Block diagram for frequency modulation (FM) synthesis: (a) the classic
form; (b) realized as phase modulation.</CAPTION>
<TR><TD><IMG
WIDTH="507" HEIGHT="499" BORDER="0"
SRC="img470.png"
ALT="\begin{figure}\psfig{file=figs/fig05.08.ps}\end{figure}"></TD></TR>
</TABLE>
</DIV>
<P>
To analyse the resulting spectrum we can rewrite the signal as,
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
x[n] = \cos(\omega_c n) * \cos(a \cos(\omega_m n))
\end{displaymath}
-->
<IMG
WIDTH="231" HEIGHT="28" BORDER="0"
SRC="img471.png"
ALT="\begin{displaymath}
x[n] = \cos(\omega_c n) * \cos(a \cos(\omega_m n))
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
- \sin(\omega_c n) * \sin(a \cos(\omega_m n))
\end{displaymath}
-->
<IMG
WIDTH="194" HEIGHT="28" BORDER="0"
SRC="img472.png"
ALT="\begin{displaymath}
- \sin(\omega_c n) * \sin(a \cos(\omega_m n))
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
We can consider the result as a sum of two waveshaping
generators, each operating on a sinusoid of frequency <IMG
WIDTH="25" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img463.png"
ALT="$\omega_m$"> and
with a waveshaping index <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img4.png"
ALT="$a$">, and each ring modulated with a sinusoid of
frequency <IMG
WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img464.png"
ALT="$\omega_c$">. The waveshaping function <IMG
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img112.png"
ALT="$f$"> is given by
<!-- MATH
$f(x) = \cos(x)$
-->
<IMG
WIDTH="98" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img473.png"
ALT="$f(x) = \cos(x)$"> for the first term and by <!-- MATH
$f(x) = \sin(x)$
-->
<IMG
WIDTH="96" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img474.png"
ALT="$f(x) = \sin(x)$"> for the second.
<P>
Returning to Figure <A HREF="node77.html#fig05.04">5.4</A>, we can predict what the
spectrum will look like. The two harmonic spectra, of the waveshaping outputs
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\cos(a \cos(\omega_m n))
\end{displaymath}
-->
<IMG
WIDTH="108" HEIGHT="28" BORDER="0"
SRC="img475.png"
ALT="\begin{displaymath}
\cos(a \cos(\omega_m n))
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
and
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\sin(a \cos(\omega_m n))
\end{displaymath}
-->
<IMG
WIDTH="107" HEIGHT="28" BORDER="0"
SRC="img476.png"
ALT="\begin{displaymath}
\sin(a \cos(\omega_m n))
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
have, respectively, harmonics tuned to
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
0, 2\omega_m, 4\omega_m, \ldots
\end{displaymath}
-->
<IMG
WIDTH="105" HEIGHT="27" BORDER="0"
SRC="img477.png"
ALT="\begin{displaymath}
0, 2\omega_m, 4\omega_m, \ldots
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
and
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\omega_m, 3\omega_m, 5\omega_m, \ldots
\end{displaymath}
-->
<IMG
WIDTH="120" HEIGHT="27" BORDER="0"
SRC="img478.png"
ALT="\begin{displaymath}
\omega_m, 3\omega_m, 5\omega_m, \ldots
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
and each is multiplied by a sinusoid at the carrier frequency. So there
will be a spectrum centered at the carrier frequency <IMG
WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img464.png"
ALT="$\omega_c$">, with
sidebands at both even and odd multiples of the modulation frequency <IMG
WIDTH="25" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img463.png"
ALT="$\omega_m$">,
contributed respectively by the sine and cosine waveshaping terms above.
The index of modulation <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img4.png"
ALT="$a$">, as it changes, controls the relative strength of
the various partials. The partials themselves are situated at the frequencies
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\omega_c + m \omega_m
\end{displaymath}
-->
<IMG
WIDTH="71" HEIGHT="26" BORDER="0"
SRC="img479.png"
ALT="\begin{displaymath}
\omega_c + m \omega_m
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
m = \ldots -2, -1, 0, 1, 2, \ldots
\end{displaymath}
-->
<IMG
WIDTH="178" HEIGHT="27" BORDER="0"
SRC="img480.png"
ALT="\begin{displaymath}
m = \ldots -2, -1, 0, 1, 2, \ldots
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
As with any situation where two periodic signals are multiplied, if there is
some common supermultiple of the two periods, the resulting product will repeat
at that longer period. So if the two periods are <IMG
WIDTH="21" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img481.png"
ALT="$k \tau$"> and <IMG
WIDTH="26" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img482.png"
ALT="$m \tau$">, where
<IMG
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img58.png"
ALT="$k$"> and <IMG
WIDTH="17" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img111.png"
ALT="$m$"> are relatively prime, they both repeat after a time interval of
<IMG
WIDTH="35" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img483.png"
ALT="$km\tau$">. In other words, if the two have frequencies which are both multiples
of some common frequency, so that <!-- MATH
$\omega_m=k\omega$
-->
<IMG
WIDTH="65" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img484.png"
ALT="$\omega_m=k\omega$"> and <!-- MATH
$\omega_c=m\omega$
-->
<IMG
WIDTH="65" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img485.png"
ALT="$\omega_c=m\omega$">,
again with <IMG
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img58.png"
ALT="$k$"> and <IMG
WIDTH="17" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img111.png"
ALT="$m$"> relatively prime, the result will repeat at a frequency
of the common submultiple <IMG
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img27.png"
ALT="$\omega $">. On the other hand, if no common
submultiple <IMG
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img27.png"
ALT="$\omega $"> can be found, or if the only submultiples are lower than
any discernible pitch, then the result will be inharmonic.
<P>
Much more about FM can be found in textbooks [<A
HREF="node202.html#r-moore90">Moo90</A>, p. 316]
[<A
HREF="node202.html#r-dodge85">DJ85</A>, pp.115-139] [<A
HREF="node202.html#r-boulanger00">Bou00</A>] and the research literature.
Some of the
possibilities are shown in the following examples.
<P>
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Miller Puckette
2006-12-30
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