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<TITLE>Two-cosine carrier signal</TITLE>
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<B> <A ID="tex2html2039"
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HREF="node4.html">Contents</A></B>
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<B> <A ID="tex2html2041"
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HREF="node201.html">Index</A></B>
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<H2><A ID="SECTION001053000000000000000">
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Two-cosine carrier signal</A>
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</H2>
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<P>
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Example F08.two.cosines.pd (Figure <A HREF="#fig06.16">6.17</A>) shows how to make a carrier signal that
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cross-fades between harmonics to make continuously variable center frequencies.
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The center frequency quotient
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appears as the output of a <TT>line~</TT> object. This is separated into its
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fractional part (using the <TT>wrap~</TT> object) and its integer part (by
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subtracting the fractional part from the original). These are labeled as <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img592.png"
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ALT="$q$">
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and <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="$k$"> to agree with the treatment in Section <A HREF="node95.html#sect6.carrier">6.3</A>.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig06.16"></A><A ID="6994"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 6.17:</STRONG>
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Cross-fading between sinusoids to make movable
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center frequencies.</CAPTION>
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<TR><TD><IMG
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WIDTH="548" HEIGHT="323" BORDER="0"
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SRC="img616.png"
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ALT="\begin{figure}\psfig{file=figs/fig06.16.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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The phase--a sawtooth wave at the fundamental frequency--is multiplied by
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both <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="39" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img602.png"
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ALT="$k+1$"> (the latter by adding the original sawtooth into the
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former), and the cosines of both are taken; they are therefore at <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="39" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img602.png"
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ALT="$k+1$">
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times the fundamental frequency and have no discontinuities at phase wrapping
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points. The next several objects in the patch compute the weighted sum <!-- MATH
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$p
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{c_1} + q {c_2}$
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-->
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<IMG
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WIDTH="66" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img617.png"
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ALT="$p
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{c_1} + q {c_2}$">, where <IMG
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WIDTH="17" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img238.png"
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ALT="$c_1$">, <IMG
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WIDTH="17" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img618.png"
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ALT="$c_2$"> are the two sinusoids and <IMG
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WIDTH="67" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img619.png"
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ALT="$p=1-q$">, by
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evaluating an equivalent expression, <!-- MATH
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${c_1} + q ({c_2} - {c_1})$
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-->
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<IMG
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WIDTH="104" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img620.png"
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ALT="${c_1} + q ({c_2} - {c_1})$">. This gives
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us the desired movable-frequency carrier signal.
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<P>
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Example F09.declickit.pd (not shown here) shows how, by adding a
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<TT>samphold~</TT> object after the <TT>line~</TT> object controlling center
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frequency, you can avoid discontinuities in the output signal even if
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the desired center frequency changes discontinuously. In the
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example the center frequency quotient
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alternates between 4 and 13.5. At ramp times below about 20 msec there are
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audible artifacts when using the <TT>line~</TT> object alone which disappear
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when the <TT>samphold~</TT> object is added.
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(A disadvantage of sample-and-holding the frequency quotient is that, for
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very low fundamental frequencies, the changes can be heard as discrete
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steps. So in situations where the fundamental frequency is low and the
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center frequency need not change very quickly, it may be better to omit the
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sample-and-hold step.)
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<P>
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The next two examples demonstrate using the crossfading-oscillators
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carrier as part of the classic two-operator phase modulation technique. The
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same modulating oscillator is added separately to the phases of the two
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cosines. The resulting spectra can be made to travel up and down in frequency,
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but because of the complicated phase relationships between neighboring peaks in
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the phase modulation spectrum, no matter how you align two such spectra you can
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never avoid getting phase cancellations where they overlap.
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<P>
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