207 lines
6.5 KiB
HTML
207 lines
6.5 KiB
HTML
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<H1><A NAME="SECTION00530000000000000000">
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Controlling Amplitude</A>
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</H1>
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<P>
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Perhaps the most frequently used operation on electronic sounds is to change
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their amplitudes. For example, a simple strategy for synthesizing sounds is by
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combining sinusoids, which can be generated by evaluating the formula on Page
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<A HREF="node7.html#eq-realsinusoid"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="crossref.png"></A>, sample by sample. But the sinusoid has a constant
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nominal amplitude <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 we would like to be able to vary that in time.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig01.04"></A><A NAME="1090"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1.4:</STRONG>
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The relationship between ``MIDI" pitch and frequency in cycles per
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second (Hertz). The span of 24 MIDI values on the horizontal axis represents
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two octaves, over which the frequency increases by a factor of four.</CAPTION>
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<TR><TD><IMG
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WIDTH="445" HEIGHT="274" BORDER="0"
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SRC="img104.png"
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ALT="\begin{figure}\psfig{file=figs/fig01.04.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 general, to multiply the amplitude of a signal <IMG
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WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img80.png"
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ALT="$x[n]$"> by a factor <IMG
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WIDTH="41" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img105.png"
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ALT="$y \ge
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0$">, you can just multiply each sample by <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img106.png"
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ALT="$y$">, giving a new signal <IMG
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WIDTH="50" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img107.png"
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ALT="$y \cdot
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x[n]$">. Any measurement of the RMS or peak amplitude of <IMG
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WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img80.png"
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ALT="$x[n]$"> will be greater
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or less by the factor <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img106.png"
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ALT="$y$">. More generally, you can change the amplitude by an
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amount <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 varies sample by sample. If <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="$y[n]$"> is nonnegative and if
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it varies slowly enough, the amplitude of the product <!-- MATH
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$y[n] \cdot x[n]$
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-->
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<IMG
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WIDTH="69" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img108.png"
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ALT="$y[n] \cdot x[n]$"> (in a
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fixed window from <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="$M$"> to <IMG
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WIDTH="82" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img109.png"
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ALT="$M+N-1$">) will be that of <IMG
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WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img80.png"
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ALT="$x[n]$">, multiplied by the
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value of <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="$y[n]$"> in the window (which we assume doesn't change much over the <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="$N$">
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samples in the window).
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<P>
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In the more general case where both <IMG
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WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img80.png"
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ALT="$x[n]$"> and <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="$y[n]$"> are allowed to take
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negative and positive values and/or to change quickly, the effect of multiplying
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them can't be described as simply changing the amplitude of one of them; this is
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considered later in Chapter <A HREF="node75.html#chapter-modulation">5</A>.
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<P>
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<B> Next:</B> <A NAME="tex2html694"
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HREF="node11.html">Frequency</A>
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<B> Up:</B> <A NAME="tex2html688"
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HREF="node7.html">Sinusoids, amplitude and frequency</A>
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<B> Previous:</B> <A NAME="tex2html682"
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HREF="node9.html">Units of Amplitude</A>
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<B> <A NAME="tex2html690"
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HREF="node4.html">Contents</A></B>
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<B> <A NAME="tex2html692"
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HREF="node201.html">Index</A></B>
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<ADDRESS>
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Miller Puckette
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2006-12-30
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