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<H1><A ID="SECTION00550000000000000000"></A>
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<A ID="sect1.synth"></A>
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<BR>
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Synthesizing a sinusoid
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</H1>
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<P>
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In most widely used audio synthesis and processing packages (Csound, Max/MSP,
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and Pd, for instance), the audio operations are specified as
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networks of
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<A ID="1112"></A><I>unit generators</I>[<A
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HREF="node202.html#r-mathews69">Mat69</A>] which pass audio signals among themselves. The user of
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the software package specifies the network, sometimes called a
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<A ID="1115"></A><I>patch</I>,
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which essentially corresponds to the synthesis algorithm to be used, and then
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worries about how to control the various unit generators in time. In this
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section, we'll use abstract block diagrams to describe patches, but in the
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"examples" section (Page <A HREF="node18.html#sect1.examples"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="crossref.png"></A>), we'll choose a
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specific implementation environment and show some of the software-dependent
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details.
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<P>
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To show how to produce a sinusoid with time-varying amplitude we'll need to
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introduce two unit generators. First we need a pure sinusoid which is
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made with an
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<A ID="1118"></A><I>oscillator</I>. Figure <A HREF="#fig01.05">1.5</A> (part a) shows a pictorial
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representation of
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a sinusoidal oscillator as an icon. The input is a frequency (in cycles per second), and
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the output is a sinusoid of peak amplitude one.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig01.05"></A><A ID="1123"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1.5:</STRONG>
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Block diagrams for (a) a sinusoidal oscillator; (b) controlling the
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amplitude using a multiplier and an amplitude signal <IMG
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WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img2.png"
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ALT="$y[n]$">.</CAPTION>
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<TR><TD><IMG
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WIDTH="311" HEIGHT="266" BORDER="0"
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SRC="img117.png"
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ALT="\begin{figure}\psfig{file=figs/fig01.05.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Figure <A HREF="#fig01.05">1.5</A> (part b) shows how to multiply the output of a sinusoidal
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oscillator by an appropriate scale factor <IMG
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WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img2.png"
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ALT="$y[n]$"> to control its amplitude.
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Since the oscillator's peak amplitude is 1, the peak amplitude of the product
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is about <IMG
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WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img2.png"
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ALT="$y[n]$">, assuming <IMG
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WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img2.png"
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ALT="$y[n]$"> changes slowly enough and doesn't become
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negative in value.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig01.06"></A><A ID="1351"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1.6:</STRONG>
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Two amplitude functions (parts a, c), and (parts b, d), the result of
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multiplying them by
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the pure sinusoid of Figure <A HREF="node7.html#fig01.01">1.1</A>.</CAPTION>
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<TR><TD><IMG
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WIDTH="474" HEIGHT="528" BORDER="0"
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SRC="img118.png"
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ALT="\begin{figure}\psfig{file=figs/fig01.06.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Figure <A HREF="#fig01.06">1.6</A> shows how the sinusoid of Figure <A HREF="node7.html#fig01.01">1.1</A> is
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affected by amplitude change by two different controlling signals <IMG
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WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img2.png"
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ALT="$y[n]$">.
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The controlling signal shown in part (a) has a discontinuity, and so
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therefore does the resulting amplitude-controlled sinusoid shown in (b).
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Parts (c) and (d) show a more gently-varying possibility for <IMG
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WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img2.png"
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ALT="$y[n]$"> and the
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result. Intuition suggests that the result shown in (b) won't sound like an
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amplitude-varying sinusoid, but instead like a sinusoid interrupted by
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an audible "pop" after which it continues more quietly. In general, for
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reasons that can't be explained in this chapter, amplitude control signals
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<IMG
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WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img2.png"
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ALT="$y[n]$"> which ramp smoothly from one value to another are less likely to give
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rise to parasitic results (such as that "pop") than are abruptly changing
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ones.
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<P>
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For now we can state two general rules without justifying them. First, pure
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sinusoids are the signals most sensitive to the parasitic effects of
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quick amplitude change. So when you want to test an amplitude transition, if
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it works for sinusoids it will probably work for other signals as well.
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Second, depending on the signal whose amplitude you are changing, the amplitude
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control will need between 0 and 30 milliseconds of "ramp" time--zero for the
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most forgiving signals (such as white noise), and 30 for the least (such as a
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sinusoid). All this also depends in a complicated way on listening levels and
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the acoustic context.
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<P>
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Suitable amplitude control functions <IMG
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WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img2.png"
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ALT="$y[n]$"> may be made using an
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<A ID="1134"></A><I>envelope generator</I>.
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Figure <A HREF="#fig01.07">1.7</A> shows a network in which an envelope generator is used to
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control the amplitude of an oscillator. Envelope generators vary widely in
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design, but we will focus on the simplest
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kind, which generates line segments as shown in Figure <A HREF="#fig01.06">1.6</A> (part c).
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If a
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line segment is specified to ramp between two output values <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="$a$"> and <IMG
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WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img21.png"
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ALT="$b$"> over
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<IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$"> samples starting at sample number <IMG
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WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img86.png"
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ALT="$M$">, the output is:
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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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y[n] = a + (b - a) {{n-M} \over N}, \hspace{0.1in} M \le n \le M+N-1.
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\end{displaymath}
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-->
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<IMG
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WIDTH="342" HEIGHT="39" BORDER="0"
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SRC="img119.png"
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ALT="\begin{displaymath}
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y[n] = a + (b - a) {{n-M} \over N}, \hspace{0.1in} M \le n \le M+N-1.
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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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The output may have any number of segments such as this, laid end to end,
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over the entire range of sample numbers <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img75.png"
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ALT="$n$">; flat, horizontal segments can be
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made by setting <IMG
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WIDTH="39" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img120.png"
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ALT="$a=b$">.
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<P>
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In addition to changing amplitudes of sounds, amplitude control is often used,
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especially in real-time applications, simply to turn sounds on and off: to turn
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one off, ramp the amplitude smoothly to zero. Most software synthesis packages
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also provide ways to actually stop modules from computing samples at all, but
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here we'll use amplitude control instead.
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<P>
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The envelope generator dates from the analog era
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[<A
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HREF="node202.html#r-strange72">Str95</A>, p.64] [<A
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HREF="node202.html#r-chamberlin80">Cha80</A>, p.90], as does the rest of
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Figure <A HREF="#fig01.07">1.7</A>; oscillators with controllable frequency were called
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voltage-controlled oscillators or VCOs, and the multiplication step was
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done using a voltage-controlled amplifier or VCA
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[<A
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HREF="node202.html#r-strange72">Str95</A>, pp.34-35] [<A
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HREF="node202.html#r-chamberlin80">Cha80</A>, pp.84-89].
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Envelope generators are described in more detail in Section <A HREF="node59.html#sect4.adsr">4.1</A>.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig01.07"></A><A ID="1148"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1.7:</STRONG>
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Using an envelope generator to control amplitude.</CAPTION>
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<TR><TD><IMG
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WIDTH="173" HEIGHT="225" BORDER="0"
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SRC="img121.png"
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ALT="\begin{figure}\psfig{file=figs/fig01.07.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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2006-12-30
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