You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
405 lines
11 KiB
405 lines
11 KiB
<!DOCTYPE html>
|
|
|
|
<!--Converted with LaTeX2HTML 2002-2-1 (1.71)
|
|
original version by: Nikos Drakos, CBLU, University of Leeds
|
|
* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
|
|
* with significant contributions from:
|
|
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
|
|
<HTML>
|
|
<HEAD>
|
|
|
|
<meta name="viewport" content="width=device-width, initial-scale=1.0">
|
|
|
|
|
|
<TITLE>Phase modulation and FM</TITLE>
|
|
<META NAME="description" CONTENT="Phase modulation and FM">
|
|
<META NAME="keywords" CONTENT="book">
|
|
<META NAME="resource-type" CONTENT="document">
|
|
<META NAME="distribution" CONTENT="global">
|
|
|
|
<META NAME="Generator" CONTENT="LaTeX2HTML v2002-2-1">
|
|
<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
|
|
|
|
<LINK REL="STYLESHEET" HREF="book.css">
|
|
|
|
<LINK REL="previous" HREF="node86.html">
|
|
<LINK REL="up" HREF="node80.html">
|
|
<LINK REL="next" HREF="node88.html">
|
|
</HEAD>
|
|
|
|
<BODY >
|
|
<!--Navigation Panel-->
|
|
<A ID="tex2html1842"
|
|
HREF="node88.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="next.png"></A>
|
|
<A ID="tex2html1836"
|
|
HREF="node80.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="up.png"></A>
|
|
<A ID="tex2html1832"
|
|
HREF="node86.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="prev.png"></A>
|
|
<A ID="tex2html1838"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="contents.png"></A>
|
|
<A ID="tex2html1840"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A ID="tex2html1843"
|
|
HREF="node88.html">Exercises</A>
|
|
<B> Up:</B> <A ID="tex2html1837"
|
|
HREF="node80.html">Examples</A>
|
|
<B> Previous:</B> <A ID="tex2html1833"
|
|
HREF="node86.html">Sinusoidal waveshaping: evenness and</A>
|
|
<B> <A ID="tex2html1839"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A ID="tex2html1841"
|
|
HREF="node201.html">Index</A></B>
|
|
<BR>
|
|
<BR>
|
|
<!--End of Navigation Panel-->
|
|
|
|
<H2><A ID="SECTION00957000000000000000">
|
|
Phase modulation and FM</A>
|
|
</H2>
|
|
<A ID="sect5.example.fm"></A>
|
|
<P>
|
|
Example E08.phase.mod.pd, shown in Figure <A HREF="#fig05.15">5.15</A>, shows how to use Pd to
|
|
realize true frequency
|
|
modulation (part a) and phase modulation (part b). These correspond to the
|
|
block diagrams of Figure <A HREF="node79.html#fig05.08">5.8</A>. To
|
|
accomplish phase modulation, the carrier oscillator is split into its phase and
|
|
cosine lookup components. The signal is of the form
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
x[t] = \cos( \omega_c n + a \cos(\omega_m n))
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="198" HEIGHT="28" BORDER="0"
|
|
SRC="img533.png"
|
|
ALT="\begin{displaymath}
|
|
x[t] = \cos( \omega_c n + a \cos(\omega_m n))
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
where <IMG
|
|
WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img464.png"
|
|
ALT="$\omega_c$"> is the carrier frequency, <IMG
|
|
WIDTH="25" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img463.png"
|
|
ALT="$\omega_m$"> is the modulation
|
|
frequency, and <IMG
|
|
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img4.png"
|
|
ALT="$a$"> is the index of modulation--all in angular units.
|
|
|
|
<P>
|
|
|
|
<DIV ALIGN="CENTER"><A ID="fig05.15"></A><A ID="5903"></A>
|
|
<TABLE>
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 5.15:</STRONG>
|
|
Pd patches for: (a) frequency modulation; (b) phase modulation.</CAPTION>
|
|
<TR><TD><IMG
|
|
WIDTH="480" HEIGHT="265" BORDER="0"
|
|
SRC="img534.png"
|
|
ALT="\begin{figure}\psfig{file=figs/fig05.15.ps}\end{figure}"></TD></TR>
|
|
</TABLE>
|
|
</DIV>
|
|
|
|
<P>
|
|
We can predict the spectrum by expanding the outer cosine:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
x[t] = \cos( \omega_c n ) \cos (a \cos(\omega_m n))
|
|
- \sin( \omega_c n ) \sin (a \cos(\omega_m n))
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="403" HEIGHT="28" BORDER="0"
|
|
SRC="img535.png"
|
|
ALT="\begin{displaymath}
|
|
x[t] = \cos( \omega_c n ) \cos (a \cos(\omega_m n))
|
|
- \sin( \omega_c n ) \sin (a \cos(\omega_m n))
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
Plugging in the expansions from Page <A HREF="node86.html#sect5.bessel"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
|
SRC="crossref.png"></A> and simplifying
|
|
yields:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
x[t] = {J_0}(a) \cos( \omega_c n )
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="142" HEIGHT="28" BORDER="0"
|
|
SRC="img536.png"
|
|
ALT="\begin{displaymath}
|
|
x[t] = {J_0}(a) \cos( \omega_c n )
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
+ {J_1}(a) \cos( (\omega_c+\omega_m) n + {\pi\over2})
|
|
+ {J_1}(a) \cos( (\omega_c-\omega_m) n + {\pi\over2})
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="401" HEIGHT="35" BORDER="0"
|
|
SRC="img537.png"
|
|
ALT="\begin{displaymath}
|
|
+ {J_1}(a) \cos( (\omega_c+\omega_m) n + {\pi\over2})
|
|
+ {J_1}(a) \cos( (\omega_c-\omega_m) n + {\pi\over2})
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
+ {J_2}(a) \cos( (\omega_c+2\omega_m) n + \pi)
|
|
+ {J_2}(a) \cos( (\omega_c-2\omega_m) n + \pi)
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="409" HEIGHT="28" BORDER="0"
|
|
SRC="img538.png"
|
|
ALT="\begin{displaymath}
|
|
+ {J_2}(a) \cos( (\omega_c+2\omega_m) n + \pi)
|
|
+ {J_2}(a) \cos( (\omega_c-2\omega_m) n + \pi)
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
+ {J_3}(a) \cos( (\omega_c+3\omega_m) n + {{3\pi}\over2})
|
|
+ {J_3}(a) \cos( (\omega_c-3\omega_m) n + {{3\pi}\over2})
|
|
+ \cdots
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="471" HEIGHT="38" BORDER="0"
|
|
SRC="img539.png"
|
|
ALT="\begin{displaymath}
|
|
+ {J_3}(a) \cos( (\omega_c+3\omega_m) n + {{3\pi}\over2})
|
|
...
|
|
...}(a) \cos( (\omega_c-3\omega_m) n + {{3\pi}\over2})
|
|
+ \cdots
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
So the components are centered about the carrier frequency <IMG
|
|
WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img464.png"
|
|
ALT="$\omega_c$"> with
|
|
sidebands extending in either direction, each spaced <IMG
|
|
WIDTH="25" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img463.png"
|
|
ALT="$\omega_m$"> from the
|
|
next. The amplitudes are functions of the index of
|
|
modulation, and don't depend on the frequencies.
|
|
Figure <A HREF="#fig05.16">5.16</A> shows some two-operator phase modulation spectra,
|
|
measured using Example E09.FM.spectrum.pd.
|
|
|
|
<P>
|
|
|
|
<DIV ALIGN="CENTER"><A ID="fig05.16"></A><A ID="5921"></A>
|
|
<TABLE>
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 5.16:</STRONG>
|
|
Spectra from phase modulation at three different indices. The indices
|
|
are given as multiples of <IMG
|
|
WIDTH="21" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img16.png"
|
|
ALT="$2\pi $"> radians.</CAPTION>
|
|
<TR><TD><IMG
|
|
WIDTH="441" HEIGHT="438" BORDER="0"
|
|
SRC="img540.png"
|
|
ALT="\begin{figure}\psfig{file=figs/fig05.16.ps}\end{figure}"></TD></TR>
|
|
</TABLE>
|
|
</DIV>
|
|
|
|
<P>
|
|
Phase modulation can thus be seen simply as a form of ring modulated
|
|
waveshaping. So
|
|
we can use the
|
|
strategies described in Section <A HREF="node77.html#sect5.ringmod">5.2</A> to generate particular
|
|
combinations of frequencies. For example, if the carrier frequency is
|
|
half the modulation frequency, you get a sound with odd harmonics exactly
|
|
as in the octave dividing example (Figure <A HREF="node82.html#fig05.10">5.10</A>).
|
|
|
|
<P>
|
|
Frequency modulation need not be restricted to purely sinusoidal carrier or
|
|
modulation oscillators. One well-trodden path is to effect phase modulation
|
|
on the phase modulation spectrum itself. There are then two indices of
|
|
modulation (call
|
|
them <IMG
|
|
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img4.png"
|
|
ALT="$a$"> and <IMG
|
|
WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img21.png"
|
|
ALT="$b$">) and two frequencies of modulation (<IMG
|
|
WIDTH="25" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img463.png"
|
|
ALT="$\omega_m$"> and <IMG
|
|
WIDTH="20" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img541.png"
|
|
ALT="$\omega_p$">)
|
|
and the waveform is:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
x[n] = \cos(\omega_c n + a \cos(\omega_m n) + b \cos(\omega_p n))
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="291" HEIGHT="29" BORDER="0"
|
|
SRC="img542.png"
|
|
ALT="\begin{displaymath}
|
|
x[n] = \cos(\omega_c n + a \cos(\omega_m n) + b \cos(\omega_p n))
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
To analyze the result, just rewrite the original FM series above, replacing
|
|
<IMG
|
|
WIDTH="29" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img543.png"
|
|
ALT="$\omega_c n$"> everywhere with <!-- MATH
|
|
$\omega_c n + b \cos(\omega_p n)$
|
|
-->
|
|
<IMG
|
|
WIDTH="118" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img544.png"
|
|
ALT="$\omega_c n + b \cos(\omega_p n)$">. The third
|
|
positive sideband becomes for instance:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
{J_3}(a) \cos( (\omega_c+3\omega_m) n + {{3\pi}\over2} + b \cos(\omega_p n))
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="289" HEIGHT="38" BORDER="0"
|
|
SRC="img545.png"
|
|
ALT="\begin{displaymath}
|
|
{J_3}(a) \cos( (\omega_c+3\omega_m) n + {{3\pi}\over2} + b \cos(\omega_p n))
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
This is itself just another FM spectrum, with its own sidebands of frequency
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
\omega_c+3\omega_m + k \omega_p , k = 0, \pm 1, \pm 2, \ldots
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="236" HEIGHT="29" BORDER="0"
|
|
SRC="img546.png"
|
|
ALT="\begin{displaymath}
|
|
\omega_c+3\omega_m + k \omega_p , k = 0, \pm 1, \pm 2, \ldots
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
having amplitude <!-- MATH
|
|
${J_3}(a) {J_k}(b)$
|
|
-->
|
|
<IMG
|
|
WIDTH="76" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img547.png"
|
|
ALT="${J_3}(a) {J_k}(b)$"> and phase <IMG
|
|
WIDTH="77" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img548.png"
|
|
ALT="$(3+k)\pi / 2$">
|
|
[<A
|
|
HREF="node202.html#r-lebrun77">Leb77</A>]. Example E10.complex.FM.pd (not shown here) illustrates this by
|
|
graphing spectra from a two-modulator FM instrument.
|
|
|
|
<P>
|
|
Since early times [<A
|
|
HREF="node202.html#r-schottstaedt87">Sch77</A>] researchers have sought combinations
|
|
of phases, frequencies, and modulation indices, for simple and compact phase
|
|
modulation instruments, that manage to imitate familiar instrumental sounds.
|
|
This became a major industry with the introduction of commercial FM
|
|
synthesizers.
|
|
|
|
<P>
|
|
<HR>
|
|
<!--Navigation Panel-->
|
|
<A ID="tex2html1842"
|
|
HREF="node88.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="next.png"></A>
|
|
<A ID="tex2html1836"
|
|
HREF="node80.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="up.png"></A>
|
|
<A ID="tex2html1832"
|
|
HREF="node86.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="prev.png"></A>
|
|
<A ID="tex2html1838"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="contents.png"></A>
|
|
<A ID="tex2html1840"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A ID="tex2html1843"
|
|
HREF="node88.html">Exercises</A>
|
|
<B> Up:</B> <A ID="tex2html1837"
|
|
HREF="node80.html">Examples</A>
|
|
<B> Previous:</B> <A ID="tex2html1833"
|
|
HREF="node86.html">Sinusoidal waveshaping: evenness and</A>
|
|
<B> <A ID="tex2html1839"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A ID="tex2html1841"
|
|
HREF="node201.html">Index</A></B>
|
|
<!--End of Navigation Panel-->
|
|
<ADDRESS>
|
|
Miller Puckette
|
|
2006-12-30
|
|
</ADDRESS>
|
|
</BODY>
|
|
</HTML>
|
|
|