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