2022-04-12 22:02:59 -03:00
|
|
|
<!DOCTYPE html>
|
2022-04-12 21:54:18 -03:00
|
|
|
|
|
|
|
|
<!--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>
|
2022-04-12 22:02:59 -03:00
|
|
|
|
|
|
|
|
<meta name="viewport" content="width=device-width, initial-scale=1.0">
|
|
|
|
|
|
|
|
|
|
|
2022-04-12 21:54:18 -03:00
|
|
|
<TITLE>Controlling Amplitude</TITLE>
|
|
|
|
|
<META NAME="description" CONTENT="Controlling Amplitude">
|
|
|
|
|
<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="next" HREF="node11.html">
|
|
|
|
|
<LINK REL="previous" HREF="node9.html">
|
|
|
|
|
<LINK REL="up" HREF="node7.html">
|
|
|
|
|
<LINK REL="next" HREF="node11.html">
|
|
|
|
|
</HEAD>
|
|
|
|
|
|
|
|
|
|
<BODY >
|
|
|
|
|
<!--Navigation Panel-->
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html693"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node11.html">
|
|
|
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="next.png"></A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html687"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node7.html">
|
|
|
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="up.png"></A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html681"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node9.html">
|
|
|
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="prev.png"></A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html689"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node4.html">
|
|
|
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="contents.png"></A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html691"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node201.html">
|
|
|
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="index.png"></A>
|
2022-04-12 21:54:18 -03:00
|
|
|
<BR>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> Next:</B> <A ID="tex2html694"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node11.html">Frequency</A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> Up:</B> <A ID="tex2html688"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node7.html">Sinusoids, amplitude and frequency</A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> Previous:</B> <A ID="tex2html682"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node9.html">Units of Amplitude</A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> <A ID="tex2html690"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node4.html">Contents</A></B>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> <A ID="tex2html692"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node201.html">Index</A></B>
|
|
|
|
|
<BR>
|
|
|
|
|
<BR>
|
|
|
|
|
<!--End of Navigation Panel-->
|
|
|
|
|
|
2022-04-12 23:32:40 -03:00
|
|
|
<H1><A ID="SECTION00530000000000000000">
|
2022-04-12 21:54:18 -03:00
|
|
|
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="[*]"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="crossref.png"></A>, sample by sample. But the sinusoid has a constant
|
2022-04-12 21:54:18 -03:00
|
|
|
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>
|
|
|
|
|
|
2022-04-12 23:32:40 -03:00
|
|
|
<DIV ALIGN="CENTER"><A ID="fig01.04"></A><A ID="1090"></A>
|
2022-04-12 21:54:18 -03:00
|
|
|
<TABLE>
|
|
|
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1.4:</STRONG>
|
2022-04-12 23:17:03 -03:00
|
|
|
The relationship between "MIDI" pitch and frequency in cycles per
|
2022-04-12 21:54:18 -03:00
|
|
|
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
|
|
|
|
|
WIDTH="445" HEIGHT="274" BORDER="0"
|
|
|
|
|
SRC="img104.png"
|
|
|
|
|
ALT="\begin{figure}\psfig{file=figs/fig01.04.ps}\end{figure}"></TD></TR>
|
|
|
|
|
</TABLE>
|
|
|
|
|
</DIV>
|
|
|
|
|
|
|
|
|
|
<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
|
|
|
|
|
WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img80.png"
|
|
|
|
|
ALT="$x[n]$"> will be greater
|
|
|
|
|
or less by the factor <IMG
|
|
|
|
|
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img106.png"
|
|
|
|
|
ALT="$y$">. More generally, you can change the amplitude by an
|
|
|
|
|
amount <IMG
|
|
|
|
|
WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img2.png"
|
|
|
|
|
ALT="$y[n]$"> which varies sample by sample. If <IMG
|
|
|
|
|
WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
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
|
|
|
|
|
WIDTH="69" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img108.png"
|
|
|
|
|
ALT="$y[n] \cdot x[n]$"> (in a
|
|
|
|
|
fixed window from <IMG
|
|
|
|
|
WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img86.png"
|
|
|
|
|
ALT="$M$"> to <IMG
|
|
|
|
|
WIDTH="82" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img109.png"
|
|
|
|
|
ALT="$M+N-1$">) will be that of <IMG
|
|
|
|
|
WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img80.png"
|
|
|
|
|
ALT="$x[n]$">, multiplied by the
|
|
|
|
|
value of <IMG
|
|
|
|
|
WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
SRC="img2.png"
|
|
|
|
|
ALT="$y[n]$"> in the window (which we assume doesn't change much over the <IMG
|
|
|
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
|
|
|
SRC="img3.png"
|
|
|
|
|
ALT="$N$">
|
|
|
|
|
samples in the window).
|
|
|
|
|
|
|
|
|
|
<P>
|
|
|
|
|
In the more general case where both <IMG
|
|
|
|
|
WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
|
|
|
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>
|
|
|
|
|
<HR>
|
|
|
|
|
<!--Navigation Panel-->
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html693"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node11.html">
|
|
|
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="next.png"></A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html687"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node7.html">
|
|
|
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="up.png"></A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html681"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node9.html">
|
|
|
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="prev.png"></A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html689"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node4.html">
|
|
|
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="contents.png"></A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html691"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node201.html">
|
|
|
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="index.png"></A>
|
2022-04-12 21:54:18 -03:00
|
|
|
<BR>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> Next:</B> <A ID="tex2html694"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node11.html">Frequency</A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> Up:</B> <A ID="tex2html688"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node7.html">Sinusoids, amplitude and frequency</A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> Previous:</B> <A ID="tex2html682"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node9.html">Units of Amplitude</A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> <A ID="tex2html690"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node4.html">Contents</A></B>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> <A ID="tex2html692"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node201.html">Index</A></B>
|
|
|
|
|
<!--End of Navigation Panel-->
|
|
|
|
|
<ADDRESS>
|
|
|
|
|
Miller Puckette
|
|
|
|
|
2006-12-30
|
|
|
|
|
</ADDRESS>
|
|
|
|
|
</BODY>
|
|
|
|
|
</HTML>
|
