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<H1><A NAME="SECTION001030000000000000000"></A>
<A NAME="sect6.carrier"></A>
<BR>
Movable ring modulation
</H1>
<P>
We turn now to the carrier signal, seeking ways to make it more controllable.
We would particularly like to be able to slide the spectral energy continuously
up and
down in frequency. Simply ramping the frequency of the carrier oscillator
will not accomplish this, since the spectra won't be harmonic except when the
carrier is an integer multiple of the fundamental frequency.
<P>
In the stretched wavetable approach we can accomplish this simply by sampling
a sinusoid and transposing it to the desired "pitch". The transposed pitch
isn't heard as a periodicity since the wavetable itself is read periodically at
the fundamental frequency. Instead, the sinusoid is transposed as a spectral
envelope.
<P>
Figure <A HREF="#fig06.07">6.7</A> shows a carrier signal produced in this way, tuned to
produce a formant centered at 1.5 times the fundamental frequency. The signal
has no outright discontinuity at the phase wraparound frequency, but it does
have a discontinuity in slope, which, if not removed by applying a suitable
modulation signal, would have very audible high-frequency components.
<P>
<DIV ALIGN="CENTER"><A NAME="fig06.07"></A><A NAME="6894"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 6.7:</STRONG>
Waveform for a wavetable-based carrier signal tuned to 1.5 times
the fundamental. Two periods are shown.</CAPTION>
<TR><TD><IMG
WIDTH="353" HEIGHT="109" BORDER="0"
SRC="img585.png"
ALT="\begin{figure}\psfig{file=figs/fig06.07.ps}\end{figure}"></TD></TR>
</TABLE>
</DIV>
<P>
Using this idea we can make a complete description of how to use the block
diagram of Figure <A HREF="node90.html#fig06.03">6.3</A> to produce a desired formant. The wavetable
lookup on the left hand side would hold a sinusoid (placed symmetrically so
that the phase is zero at the center of the wavetable). The right-hand-side
wavetable would hold a Hann or other appropriate window function.
If we desire the fundamental frequency to be <IMG
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img27.png"
ALT="$\omega $">, the
formant center frequency to be <IMG
WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img464.png"
ALT="$\omega_c$">, and the bandwidth to be <IMG
WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img586.png"
ALT="$\omega_b$">,
we set the "stretch" parameter to the <I>center frequency quotient</I>
defined as <!-- MATH
${\omega_c}/\omega$
-->
<IMG
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img587.png"
ALT="${\omega_c}/\omega$">, and the index of
modulation to the <I>bandwidth quotient</I>, <!-- MATH
${\omega_b}/\omega$
-->
<IMG
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img588.png"
ALT="${\omega_b}/\omega$">.
<P>
The output signal is simply a sample of a cosine wave at the
desired center frequency, repeated at the (unrelated in general) desired
period, and windowed to take out the discontinuities at period boundaries.
<P>
Although we aren't able to derive this result yet (we will need Fourier
analysis), it will turn out that, in the main lobe of the formant, the phases
are all zero at the center of the waveform (i.e., the components are all
cosines if we consider the phase to be zero at the center of
the waveform). This means we may
superpose any number of these formants to build a more complex spectrum and the
amplitudes of the partials will combine by addition. (The sidelobes don't
behave so well: they are alternately of opposite sign and will produce
cancellation patterns; but we can often just shrug them off as a small,
uncontrollable, residual signal.)
<P>
This method leads to an interesting generalization, which is to take a sequence
of recorded wavetables, align all their component phases to those of
cosines, and use them in place of the cosine function as the carrier signal.
The phase alignment is necessary to allow coherent cross-fading between
samples so that the spectral envelope can change smoothly. If, for example, we
use successive snippets of a vocal sample as input, we get a strikingly
effective vocoder; see Section <A HREF="node177.html#sect9-bash">9.6</A>.
<P>
Another technique for making carrier signals that can be slid continuously
up and down in frequency while maintaining a fundamental frequency
is simply to cross-fade between harmonics. The carrier signal is then:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
c(\phi) = c(\omega n) = p \cos( k \omega n) + q \cos( (k+1) \omega n)
\end{displaymath}
-->
<IMG
WIDTH="313" HEIGHT="28" BORDER="0"
SRC="img589.png"
ALT="\begin{displaymath}
c(\phi) = c(\omega n) = p \cos( k \omega n) + q \cos( (k+1) \omega n)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where <IMG
WIDTH="67" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img590.png"
ALT="$p + q = 1$"> and <IMG
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img58.png"
ALT="$k$"> is an integer, all three chosen so that
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
(k + q) * {\omega} = {\omega_c}
\end{displaymath}
-->
<IMG
WIDTH="109" HEIGHT="28" BORDER="0"
SRC="img591.png"
ALT="\begin{displaymath}
(k + q) * {\omega} = {\omega_c}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
so that the spectral center of mass of the two cosines is placed at
<IMG
WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img464.png"
ALT="$\omega_c$">. (Note that we make the amplitudes of the two cosines add to one
instead of setting the total power to one; we do this because the modulator
will operate phase-coherently on them.) To accomplish this we simply
set <IMG
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img58.png"
ALT="$k$"> and <IMG
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img592.png"
ALT="$q$"> to be the integer and fractional part, respectively,
of the center frequency quotient <!-- MATH
$\omega_c/\omega$
-->
<IMG
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img593.png"
ALT="$\omega_c/\omega$">.
<P>
The simplest way of making a control interface for this synthesis technique
would be to use
ramps to update <IMG
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img27.png"
ALT="$\omega $"> and <IMG
WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img464.png"
ALT="$\omega_c$">, and then to compute <IMG
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img592.png"
ALT="$q$"> and <IMG
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img58.png"
ALT="$k$">
as audio signals from the ramped, smoothly varying <IMG
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img27.png"
ALT="$\omega $"> and <IMG
WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img464.png"
ALT="$\omega_c$">.
Oddly enough, despite the fact that <IMG
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img58.png"
ALT="$k$">, <IMG
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img57.png"
ALT="$p$">, and <IMG
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img592.png"
ALT="$q$"> are discontinuous
functions of <!-- MATH
$\omega_c/\omega$
-->
<IMG
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img593.png"
ALT="$\omega_c/\omega$">, the carrier <IMG
WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img594.png"
ALT="$c(\phi)$"> turns out to vary
continuously with <!-- MATH
$\omega_c/\omega$
-->
<IMG
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img593.png"
ALT="$\omega_c/\omega$">, and so if the desired center frequency
<IMG
WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img464.png"
ALT="$\omega_c$"> is ramped from value to value the result is a continuous sweep in
center frequency. However, more work is needed if discontinuous changes
in center frequency are needed. This is not an unreasonable thing to wish
for, being analogous to changing the frequency of an oscillator discontinuously.
<P>
There turns out to be a good way to accomodate this.
The trick to updating <IMG
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img58.png"
ALT="$k$"> and <IMG
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img592.png"
ALT="$q$"> is to note that <IMG
WIDTH="61" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img595.png"
ALT="$c(\phi) = 1$"> whenever <IMG
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img77.png"
ALT="$\phi$">
is a multiple of <IMG
WIDTH="21" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img16.png"
ALT="$2\pi $">, regardless of the choice of <IMG
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img58.png"
ALT="$k$">, <IMG
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img57.png"
ALT="$p$">, and <IMG
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img592.png"
ALT="$q$"> as long
as <IMG
WIDTH="67" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img590.png"
ALT="$p + q = 1$">. Hence, we may make discontinuous changes in <IMG
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img58.png"
ALT="$k$">, <IMG
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img57.png"
ALT="$p$">, and <IMG
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img592.png"
ALT="$q$"> once
per period (right when the phase is a multiple of <IMG
WIDTH="21" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img16.png"
ALT="$2\pi $">), without making
discontinuities in the carrier signal.
<P>
In the specific case of FM, if we wish we can now go back and modify the
original formulation to:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
p \cos ( n {\omega_2} t + r \cos ({\omega_1} t)) +
\end{displaymath}
-->
<IMG
WIDTH="176" HEIGHT="28" BORDER="0"
SRC="img596.png"
ALT="\begin{displaymath}
p \cos ( n {\omega_2} t + r \cos ({\omega_1} t)) +
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P> <BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
+ q \cos ( (n+1) {\omega_2} t + r \cos ({\omega_1} t))
\end{displaymath}
-->
<IMG
WIDTH="212" HEIGHT="28" BORDER="0"
SRC="img597.png"
ALT="\begin{displaymath}
+ q \cos ( (n+1) {\omega_2} t + r \cos ({\omega_1} t))
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
This allows us to add glissandi (which are heard as dipthongs) to Chowning's
original phase-modulation-based vocal synthesis technique.
<P>
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<ADDRESS>
Miller Puckette
2006-12-30
</ADDRESS>
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