<!DOCTYPE html>
<!--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>
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<TITLE>Units of Amplitude</TITLE>
<META NAME="description" CONTENT="Units of 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="node10.html">
<LINK REL="previous" HREF="node8.html">
<LINK REL="up" HREF="node7.html">
<LINK REL="next" HREF="node10.html">
</HEAD>
<BODY >
<!--Navigation Panel-->
<A ID="tex2html679"
HREF="node10.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
SRC="next.png"></A>
<A ID="tex2html673"
HREF="node7.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
SRC="up.png"></A>
<A ID="tex2html667"
HREF="node8.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
SRC="prev.png"></A>
<A ID="tex2html675"
HREF="node4.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
SRC="contents.png"></A>
<A ID="tex2html677"
HREF="node201.html">
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
SRC="index.png"></A>
<BR>
<B> Next:</B> <A ID="tex2html680"
HREF="node10.html">Controlling Amplitude</A>
<B> Up:</B> <A ID="tex2html674"
HREF="node7.html">Sinusoids, amplitude and frequency</A>
<B> Previous:</B> <A ID="tex2html668"
HREF="node8.html">Measures of Amplitude</A>
<B> <A ID="tex2html676"
HREF="node4.html">Contents</A></B>
<B> <A ID="tex2html678"
HREF="node201.html">Index</A></B>
<BR>
<BR>
<!--End of Navigation Panel-->
<H1><A ID="SECTION00520000000000000000">
Units of Amplitude</A>
</H1>
<P>
Two amplitudes are often better compared using their ratio than their
difference. Saying that one signal's amplitude is greater than
another's by a factor of two might be more informative than saying it is greater by
30 millivolts. This is true for any measure of amplitude (RMS or peak, for
instance). To facilitate comparisons, we often express amplitudes in
logarithmic units called
<A ID="1066"></A><I>decibels</I>. If <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img4.png"
ALT="$a$"> is the amplitude of a signal (either peak or RMS),
then we can define the decibel (dB) level <IMG
WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img28.png"
ALT="$d$"> as:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
d = 20 \cdot {{{\log}_{10}} ( {a / {a_0}} )}
\end{displaymath}
-->
<IMG
WIDTH="133" HEIGHT="28" BORDER="0"
SRC="img95.png"
ALT="\begin{displaymath}
d = 20 \cdot {{{\log}_{10}} ( {a / {a_0}} )}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where <IMG
WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img96.png"
ALT="$a_0$"> is a reference amplitude. This definition is set up so that, if we
increase the signal power by a factor of ten (so that the amplitude increases
by a factor of <IMG
WIDTH="32" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
SRC="img97.png"
ALT="$\sqrt {10}$">), the logarithm will increase by <IMG
WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img98.png"
ALT="$1/2$">, and so the
value in decibels goes up (additively) by ten.
An increase in amplitude by a
factor of two corresponds to an increase of about 6.02 decibels; doubling power
is an increase of 3.01 dB. The relationship between linear
amplitude and amplitude in decibels is graphed in Figure <A HREF="#fig01.03">1.3</A>.
<P>
<DIV ALIGN="CENTER"><A ID="fig01.03"></A><A ID="1075"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1.3:</STRONG>
The relationship between decibel and linear scales of amplitude.
The linear amplitude 1 is assigned to 0 dB.</CAPTION>
<TR><TD><IMG
WIDTH="419" HEIGHT="267" BORDER="0"
SRC="img99.png"
ALT="\begin{figure}\psfig{file=figs/fig01.03.ps}\end{figure}"></TD></TR>
</TABLE>
</DIV>
<P>
Still using <IMG
WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img96.png"
ALT="$a_0$"> to denote the reference amplitude, a signal with linear
amplitude smaller than <IMG
WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img96.png"
ALT="$a_0$"> will have a negative amplitude in decibels:
<IMG
WIDTH="42" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img100.png"
ALT="${a_0}/10$"> gives -20 dB, <IMG
WIDTH="50" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img101.png"
ALT="${a_0}/100$"> gives -40, and so on. A linear amplitude
of zero is smaller than that of any value in dB, so we give it a dB level of
<IMG
WIDTH="31" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img102.png"
ALT="$-\infty$">.
<P>
In digital audio a convenient choice of reference, assuming the hardware
has a maximum amplitude of one, is
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
{a_0} = {10^{-5}} = 0.00001
\end{displaymath}
-->
<IMG
WIDTH="141" HEIGHT="26" BORDER="0"
SRC="img103.png"
ALT="\begin{displaymath}
{a_0} = {10^{-5}} = 0.00001
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
so that the maximum amplitude possible is 100 dB, and 0 dB is likely to be
inaudibly quiet at any reasonable listening level. Conveniently enough, the
dynamic range of human hearing--the ratio between a damagingly loud sound and
an inaudibly quiet one--is about 100 dB.
<P>
Amplitude is related in an inexact way to the perceived loudness of a sound.
In general, two signals with the same peak or RMS amplitude won't necessarily
have the same loudness at all. But amplifying a signal by 3 dB, say, will
fairly reliably make it sound about one "step" louder. Much has been made of
the supposedly logarithmic nature of human hearing (and other senses), which
may partially explain why decibels are such a useful scale of
amplitude [<A
HREF="node202.html#r-rossing02">RMW02</A>, p. 99].
<P>
Amplitude is also related in an inexact way to musical
<A ID="1083"></A><I>dynamic</I>. Dynamic is better thought of as a measure of effort than of
loudness or power. It ranges over nine values: rest,
ppp, pp, p, mp, mf, f, ff, fff. These correlate in an even looser way with the amplitude
of a signal than does loudness [<A
HREF="node202.html#r-rossing02">RMW02</A>, pp. 110-111].
<P>
<HR>
<!--Navigation Panel-->
<A ID="tex2html679"
HREF="node10.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
SRC="next.png"></A>
<A ID="tex2html673"
HREF="node7.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
SRC="up.png"></A>
<A ID="tex2html667"
HREF="node8.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
SRC="prev.png"></A>
<A ID="tex2html675"
HREF="node4.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
SRC="contents.png"></A>
<A ID="tex2html677"
HREF="node201.html">
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
SRC="index.png"></A>
<BR>
<B> Next:</B> <A ID="tex2html680"
HREF="node10.html">Controlling Amplitude</A>
<B> Up:</B> <A ID="tex2html674"
HREF="node7.html">Sinusoids, amplitude and frequency</A>
<B> Previous:</B> <A ID="tex2html668"
HREF="node8.html">Measures of Amplitude</A>
<B> <A ID="tex2html676"
HREF="node4.html">Contents</A></B>
<B> <A ID="tex2html678"
HREF="node201.html">Index</A></B>
<!--End of Navigation Panel-->
<ADDRESS>
Miller Puckette
2006-12-30
</ADDRESS>
</BODY>
</HTML>