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