<!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>Resulting spectra</TITLE>
<META NAME="description" CONTENT="Resulting spectra">
<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="node93.html">
<LINK REL="up" HREF="node91.html">
<LINK REL="next" HREF="node95.html">
</HEAD>
<BODY >
<!--Navigation Panel-->
<A NAME="tex2html1953"
HREF="node95.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
SRC="next.png"></A>
<A NAME="tex2html1947"
HREF="node91.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
SRC="up.png"></A>
<A NAME="tex2html1943"
HREF="node93.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
SRC="prev.png"></A>
<A NAME="tex2html1949"
HREF="node4.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
SRC="contents.png"></A>
<A NAME="tex2html1951"
HREF="node201.html">
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
SRC="index.png"></A>
<BR>
<B> Next:</B> <A NAME="tex2html1954"
HREF="node95.html">Movable ring modulation</A>
<B> Up:</B> <A NAME="tex2html1948"
HREF="node91.html">Pulse trains</A>
<B> Previous:</B> <A NAME="tex2html1944"
HREF="node93.html">Pulse trains via wavetable</A>
<B> <A NAME="tex2html1950"
HREF="node4.html">Contents</A></B>
<B> <A NAME="tex2html1952"
HREF="node201.html">Index</A></B>
<BR>
<BR>
<!--End of Navigation Panel-->
<H2><A NAME="SECTION001023000000000000000">
Resulting spectra</A>
</H2>
<P>
Before considering more complicated carrier signals to go with the modulators
we've seen so far, it is instructive to see what multiplication by a pure
sinusoid gives us as waveforms and spectra. Figure <A HREF="#fig06.05">6.5</A> shows
the result of multiplying two different pulse trains by a sinusoid at the
sixth partial:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\cos(6 \omega n) {M_a}(\omega n)
\end{displaymath}
-->
<IMG
WIDTH="115" HEIGHT="28" BORDER="0"
SRC="img581.png"
ALT="\begin{displaymath}
\cos(6 \omega n) {M_a}(\omega n)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where the index of modulation <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img4.png"
ALT="$a$"> is two in both cases. In part (a) <IMG
WIDTH="26" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img582.png"
ALT="$M_a$"> is
the stretched Hann windowing function; part (b)
shows waveshaping via the unnormalized Cauchy distribution. One period
of each waveform is shown.
<P>
<DIV ALIGN="CENTER"><A NAME="fig06.05"></A><A NAME="6874"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 6.5:</STRONG>
Audio signals resulting from multiplying a cosine (partial number
6) by pulse trains: (a) windowing function from the wavetable formulation;
(b) waveshaping output using the Cauchy lookup function.</CAPTION>
<TR><TD><IMG
WIDTH="354" HEIGHT="242" BORDER="0"
SRC="img583.png"
ALT="\begin{figure}\psfig{file=figs/fig06.05.ps}\end{figure}"></TD></TR>
</TABLE>
</DIV>
<P>
In both situations we see, in effect, the sixth harmonic (the carrier signal)
enveloped into a
<A NAME="6877"></A><I>wave packet</I>
centered at the middle of the cycle, where the phase of the sinusoid is zero.
Changing the frequency of the sinusoid changes the center frequency of the
formant; changing the width of the packet (the proportion of the waveform
during which the sinusoid is strong) changes the bandwidth. Note that
the stretched Hann window function is zero at the beginning and end
of the period, unlike the waveshaping packet.
<P>
Figure <A HREF="#fig06.06">6.6</A> shows how the
shape of the formant depends on the method of production.
The stretched wavetable form (part (a) of the
figure) behaves well in the neighborhood of the peak, but somewhat oddly
starting at four partials' distance from the peak, past which we see what
are called
<A NAME="6880"></A><I>sidelobes</I>:
spurious extra peaks at lower amplitude than the central peak. As the analysis
of Section <A HREF="node30.html#sect2.stretching">2.4</A> predicts, the entire formant, sidelobes and
all, stretches or contracts inversely as the pulse train is contracted or
stretched in time.
<P>
<DIV ALIGN="CENTER"><A NAME="fig06.06"></A><A NAME="6885"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 6.6:</STRONG>
Spectra of three ring-modulated pulse trains: (a) the von Hann
window function, 50% duty cycle (corresponding to an index of 2); (b)
a waveshaping pulse train using a Gaussian transfer function; (c) the
same, with a Cauchy transfer function. Amplitudes are in decibels.</CAPTION>
<TR><TD><IMG
WIDTH="488" HEIGHT="675" BORDER="0"
SRC="img584.png"
ALT="\begin{figure}\psfig{file=figs/fig06.06.ps}\end{figure}"></TD></TR>
</TABLE>
</DIV>
<P>
The first, strongest sidelobes on either side are about 32 dB lower in amplitude
than the main peak. Further sidelobes drop off slowly when expressed in decibels;
the amplitudes decrease as the square of the distance from the center peak so
that the sixth sidelobe to the right, three times further than the first one
from the center frequency, is about twenty decibels further down. The effect
of these sidelobes is often audible as a slight buzziness in the sound.
<P>
This formant shape may be made arbitrarily fat (i.e., high bandwidth), but
there is a limit on how thin it can be made, since the duty cycle of the
waveform cannot exceed 100%. At this maximum duty cycle the formant strength
drops to zero at two harmonics' distance from the center peak. If a still
lower bandwidth is needed, waveforms may be made to overlap as described in
Section <A HREF="node37.html#sect2.example.overlap">2.6</A>.
<P>
Parts (b) and (c) of the figure show formants generated using ring modulated
waveshaping, with Gaussian and Cauchy transfer functions. The index of
modulation is two in both cases (the same as for the Hann window of part
a), and the bandwidth is comparable to that of the Hann example. In these
examples there are no sidelobes, and moreover, the index of modulation may be
dropped all the way to zero, giving a pure sinusoid; there is no lower limit on
bandwidth. On the other hand, since the waveform does not reach zero at the
ends of a cycle, this type of pulse train cannot be used to window an arbitrary
wavetable, as the Hann pulse train could.
<P>
The Cauchy example is particularly handy for designing spectra, since
the shape of the formant is a perfect isosceles triangle, when graphed in
decibels. On the other hand, the Gaussian example gathers more energy toward
the formant, and drops off faster at the tails, and so
has a cleaner sound and offers better protection against foldover.
<P>
<HR>
<!--Navigation Panel-->
<A NAME="tex2html1953"
HREF="node95.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
SRC="next.png"></A>
<A NAME="tex2html1947"
HREF="node91.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
SRC="up.png"></A>
<A NAME="tex2html1943"
HREF="node93.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
SRC="prev.png"></A>
<A NAME="tex2html1949"
HREF="node4.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
SRC="contents.png"></A>
<A NAME="tex2html1951"
HREF="node201.html">
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
SRC="index.png"></A>
<BR>
<B> Next:</B> <A NAME="tex2html1954"
HREF="node95.html">Movable ring modulation</A>
<B> Up:</B> <A NAME="tex2html1948"
HREF="node91.html">Pulse trains</A>
<B> Previous:</B> <A NAME="tex2html1944"
HREF="node93.html">Pulse trains via wavetable</A>
<B> <A NAME="tex2html1950"
HREF="node4.html">Contents</A></B>
<B> <A NAME="tex2html1952"
HREF="node201.html">Index</A></B>
<!--End of Navigation Panel-->
<ADDRESS>
Miller Puckette
2006-12-30
</ADDRESS>
</BODY>
</HTML>