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<H2><A NAME="SECTION001053000000000000000">
Two-cosine carrier signal</A>
</H2>
<P>
Example F08.two.cosines.pd (Figure <A HREF="#fig06.16">6.17</A>) shows how to make a carrier signal that
cross-fades between harmonics to make continuously variable center frequencies.
The center frequency quotient
appears as the output of a <TT>line~</TT> object. This is separated into its
fractional part (using the <TT>wrap~</TT> object) and its integer part (by
subtracting the fractional part from the original). These are labeled as <IMG
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img592.png"
ALT="$q$">
and <IMG
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img58.png"
ALT="$k$"> to agree with the treatment in Section <A HREF="node95.html#sect6.carrier">6.3</A>.
<P>
<DIV ALIGN="CENTER"><A NAME="fig06.16"></A><A NAME="6994"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 6.17:</STRONG>
Cross-fading between sinusoids to make movable
center frequencies.</CAPTION>
<TR><TD><IMG
WIDTH="548" HEIGHT="323" BORDER="0"
SRC="img616.png"
ALT="\begin{figure}\psfig{file=figs/fig06.16.ps}\end{figure}"></TD></TR>
</TABLE>
</DIV>
<P>
The phase--a sawtooth wave at the fundamental frequency--is multiplied by
both <IMG
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img58.png"
ALT="$k$"> and <IMG
WIDTH="39" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img602.png"
ALT="$k+1$"> (the latter by adding the original sawtooth into the
former), and the cosines of both are taken; they are therefore at <IMG
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img58.png"
ALT="$k$"> and <IMG
WIDTH="39" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img602.png"
ALT="$k+1$">
times the fundamental frequency and have no discontinuities at phase wrapping
points. The next several objects in the patch compute the weighted sum <!-- MATH
$p
{c_1} + q {c_2}$
-->
<IMG
WIDTH="66" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img617.png"
ALT="$p
{c_1} + q {c_2}$">, where <IMG
WIDTH="17" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img238.png"
ALT="$c_1$">, <IMG
WIDTH="17" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img618.png"
ALT="$c_2$"> are the two sinusoids and <IMG
WIDTH="67" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img619.png"
ALT="$p=1-q$">, by
evaluating an equivalent expression, <!-- MATH
${c_1} + q ({c_2} - {c_1})$
-->
<IMG
WIDTH="104" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img620.png"
ALT="${c_1} + q ({c_2} - {c_1})$">. This gives
us the desired movable-frequency carrier signal.
<P>
Example F09.declickit.pd (not shown here) shows how, by adding a
<TT>samphold~</TT> object after the <TT>line~</TT> object controlling center
frequency, you can avoid discontinuities in the output signal even if
the desired center frequency changes discontinuously. In the
example the center frequency quotient
alternates between 4 and 13.5. At ramp times below about 20 msec there are
audible artifacts when using the <TT>line~</TT> object alone which disappear
when the <TT>samphold~</TT> object is added.
(A disadvantage of sample-and-holding the frequency quotient is that, for
very low fundamental frequencies, the changes can be heard as discrete
steps. So in situations where the fundamental frequency is low and the
center frequency need not change very quickly, it may be better to omit the
sample-and-hold step.)
<P>
The next two examples demonstrate using the crossfading-oscillators
carrier as part of the classic two-operator phase modulation technique. The
same modulating oscillator is added separately to the phases of the two
cosines. The resulting spectra can be made to travel up and down in frequency,
but because of the complicated phase relationships between neighboring peaks in
the phase modulation spectrum, no matter how you align two such spectra you can
never avoid getting phase cancellations where they overlap.
<P>
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<ADDRESS>
Miller Puckette
2006-12-30
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