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<H1><A NAME="SECTION00530000000000000000">
Controlling Amplitude</A>
</H1>
<P>
Perhaps the most frequently used operation on electronic sounds is to change
their amplitudes. For example, a simple strategy for synthesizing sounds is by
combining sinusoids, which can be generated by evaluating the formula on Page
<A HREF="node7.html#eq-realsinusoid"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
SRC="file:/usr/local/share/lib/latex2html/icons/crossref.png"></A>, sample by sample. But the sinusoid has a constant
nominal amplitude <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img4.png"
ALT="$a$">, and we would like to be able to vary that in time.
<P>
<DIV ALIGN="CENTER"><A NAME="fig01.04"></A><A NAME="1090"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1.4:</STRONG>
The relationship between ``MIDI" pitch and frequency in cycles per
second (Hertz). The span of 24 MIDI values on the horizontal axis represents
two octaves, over which the frequency increases by a factor of four.</CAPTION>
<TR><TD><IMG
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ALT="\begin{figure}\psfig{file=figs/fig01.04.ps}\end{figure}"></TD></TR>
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<P>
In general, to multiply the amplitude of a signal <IMG
WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img80.png"
ALT="$x[n]$"> by a factor <IMG
WIDTH="41" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img105.png"
ALT="$y \ge
0$">, you can just multiply each sample by <IMG
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img106.png"
ALT="$y$">, giving a new signal <IMG
WIDTH="50" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img107.png"
ALT="$y \cdot
x[n]$">. Any measurement of the RMS or peak amplitude of <IMG
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ALT="$x[n]$"> will be greater
or less by the factor <IMG
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ALT="$y$">. More generally, you can change the amplitude by an
amount <IMG
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ALT="$y[n]$"> which varies sample by sample. If <IMG
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SRC="img2.png"
ALT="$y[n]$"> is nonnegative and if
it varies slowly enough, the amplitude of the product <!-- MATH
$y[n] \cdot x[n]$
-->
<IMG
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SRC="img108.png"
ALT="$y[n] \cdot x[n]$"> (in a
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ALT="$M+N-1$">) will be that of <IMG
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SRC="img80.png"
ALT="$x[n]$">, multiplied by the
value of <IMG
WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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ALT="$y[n]$"> in the window (which we assume doesn't change much over the <IMG
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SRC="img3.png"
ALT="$N$">
samples in the window).
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In the more general case where both <IMG
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SRC="img80.png"
ALT="$x[n]$"> and <IMG
WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img2.png"
ALT="$y[n]$"> are allowed to take
negative and positive values and/or to change quickly, the effect of multiplying
them can't be described as simply changing the amplitude of one of them; this is
considered later in Chapter <A HREF="node75.html#chapter-modulation">5</A>.
<P>
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<B> Next:</B> <A NAME="tex2html694"
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<ADDRESS>
Miller Puckette
2006-12-30
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