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>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 NAME="tex2html679"
|
|
|
|
|
HREF="node10.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 21:54:18 -03:00
|
|
|
<A NAME="tex2html673"
|
|
|
|
|
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 21:54:18 -03:00
|
|
|
<A NAME="tex2html667"
|
|
|
|
|
HREF="node8.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 21:54:18 -03:00
|
|
|
<A NAME="tex2html675"
|
|
|
|
|
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 21:54:18 -03:00
|
|
|
<A NAME="tex2html677"
|
|
|
|
|
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>
|
|
|
|
|
<B> Next:</B> <A NAME="tex2html680"
|
|
|
|
|
HREF="node10.html">Controlling Amplitude</A>
|
|
|
|
|
<B> Up:</B> <A NAME="tex2html674"
|
|
|
|
|
HREF="node7.html">Sinusoids, amplitude and frequency</A>
|
|
|
|
|
<B> Previous:</B> <A NAME="tex2html668"
|
|
|
|
|
HREF="node8.html">Measures of Amplitude</A>
|
|
|
|
|
<B> <A NAME="tex2html676"
|
|
|
|
|
HREF="node4.html">Contents</A></B>
|
|
|
|
|
<B> <A NAME="tex2html678"
|
|
|
|
|
HREF="node201.html">Index</A></B>
|
|
|
|
|
<BR>
|
|
|
|
|
<BR>
|
|
|
|
|
<!--End of Navigation Panel-->
|
|
|
|
|
|
|
|
|
|
<H1><A NAME="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 NAME="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 NAME="fig01.03"></A><A NAME="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 NAME="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 NAME="tex2html679"
|
|
|
|
|
HREF="node10.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 21:54:18 -03:00
|
|
|
<A NAME="tex2html673"
|
|
|
|
|
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 21:54:18 -03:00
|
|
|
<A NAME="tex2html667"
|
|
|
|
|
HREF="node8.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 21:54:18 -03:00
|
|
|
<A NAME="tex2html675"
|
|
|
|
|
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 21:54:18 -03:00
|
|
|
<A NAME="tex2html677"
|
|
|
|
|
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>
|
|
|
|
|
<B> Next:</B> <A NAME="tex2html680"
|
|
|
|
|
HREF="node10.html">Controlling Amplitude</A>
|
|
|
|
|
<B> Up:</B> <A NAME="tex2html674"
|
|
|
|
|
HREF="node7.html">Sinusoids, amplitude and frequency</A>
|
|
|
|
|
<B> Previous:</B> <A NAME="tex2html668"
|
|
|
|
|
HREF="node8.html">Measures of Amplitude</A>
|
|
|
|
|
<B> <A NAME="tex2html676"
|
|
|
|
|
HREF="node4.html">Contents</A></B>
|
|
|
|
|
<B> <A NAME="tex2html678"
|
|
|
|
|
HREF="node201.html">Index</A></B>
|
|
|
|
|
<!--End of Navigation Panel-->
|
|
|
|
|
<ADDRESS>
|
|
|
|
|
Miller Puckette
|
|
|
|
|
2006-12-30
|
|
|
|
|
</ADDRESS>
|
|
|
|
|
</BODY>
|
|
|
|
|
</HTML>
|
