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<H1><A NAME="SECTION00510000000000000000"></A>
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<A NAME="sect1.amplitude"></A>
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<BR>
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Measures of Amplitude
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</H1>
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<P>
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The most fundamental property of a digital audio signal is its
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<A NAME="1038"></A>amplitude. Unfortunately, a signal's amplitude has no one canonical
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definition.
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Strictly speaking, all the samples in a digital audio signal are themselves
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amplitudes, and we also spoke of the 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="$a$"> of the sinusoid as a whole.
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It is useful to have measures
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of amplitude for digital audio signals in general. Amplitude
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is best thought of as applying to a
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<A NAME="1039"></A><I>window</I>, a fixed range of samples of the signal. For instance, the
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window starting at sample <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="$M$"> of length <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="$N$"> of an audio 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="$x[n]$"> consists of the
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samples,
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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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x[M], x[M+1], \ldots, x[M+N-1]
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\end{displaymath}
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-->
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<IMG
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WIDTH="234" HEIGHT="28" BORDER="0"
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SRC="img87.png"
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ALT="\begin{displaymath}
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x[M], x[M+1], \ldots, x[M+N-1]
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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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The two most frequently used measures of amplitude are the
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<A NAME="1041"></A><A NAME="1042"></A><I>peak amplitude</I>, which is simply the greatest sample (in absolute value)
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over the window:
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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_{\mathrm{peak}}} \{x[n]\} = \max | x[n] | ,
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\hspace{0.3in}n = M, \ldots, M+N-1
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\end{displaymath}
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-->
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<IMG
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WIDTH="373" HEIGHT="29" BORDER="0"
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SRC="img88.png"
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ALT="\begin{displaymath}
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{A_{\mathrm{peak}}} \{x[n]\} = \max \vert x[n] \vert ,
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\hspace{0.3in}n = M, \ldots, M+N-1
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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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and the
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<A NAME="1046"></A><I>root mean square</I> (RMS) amplitude:
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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_{\mathrm{RMS}}} \{x[n]\} = \sqrt{P\{x[n]\}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="177" HEIGHT="28" BORDER="0"
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SRC="img89.png"
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ALT="\begin{displaymath}
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{A_{\mathrm{RMS}}} \{x[n]\} = \sqrt{P\{x[n]\}}
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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="59" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img90.png"
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ALT="$P\{x[n]\}$"> is the mean
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<A NAME="1050"></A><I>power</I>, defined 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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{P\{x[n]\}} = {1 \over N} \left (
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{{|x[M]|} ^2} + \cdots + {{|x[M+N-1]|} ^2}
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\right )
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\end{displaymath}
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-->
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<IMG
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WIDTH="335" HEIGHT="38" BORDER="0"
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SRC="img91.png"
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ALT="\begin{displaymath}
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{P\{x[n]\}} = {1 \over N} \left (
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{{\vert x[M]\vert} ^2} + \cdots + {{\vert x[M+N-1]\vert} ^2}
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\right )
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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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(In this last formula, the absolute value signs aren't necessary at the moment
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since we're working on real-valued signals, but they will become important
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later when we consider complex-valued signals.) Neither the peak nor the RMS
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amplitude of any signal can be negative, and either one can be exactly zero
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only if the signal itself is zero for all <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img75.png"
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ALT="$n$"> in the window.
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<P>
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The RMS amplitude of a signal may equal the peak amplitude but never exceeds
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it; and it may be as little as <IMG
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WIDTH="47" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
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SRC="img92.png"
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ALT="$1 / {\sqrt N}$"> times the peak amplitude, but
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never less than that.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig01.02"></A><A NAME="1059"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1.2:</STRONG>
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Root mean square (RMS) and peak amplitudes of signals compared.
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For a sinusoid (part a), the peak amplitude is higher than RMS by a factor of
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<IMG
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WIDTH="24" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
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SRC="img1.png"
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ALT="$\sqrt 2$">.</CAPTION>
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<TR><TD><IMG
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WIDTH="362" HEIGHT="306" BORDER="0"
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SRC="img93.png"
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ALT="\begin{figure}\psfig{file=figs/fig01.02.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Under reasonable conditions--if the window contains at least several periods and
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if the angular frequency is well under one radian per sample--the peak
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amplitude of the sinusoid of Page <A HREF="node7.html#eq-realsinusoid"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
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SRC="crossref.png"></A>
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is approximately <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 its RMS amplitude
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about <IMG
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WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
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SRC="img94.png"
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ALT="$a / {\sqrt 2}$">. Figure <A HREF="#fig01.02">1.2</A> shows the peak and RMS amplitudes
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of two digital audio signals.
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<ADDRESS>
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Miller Puckette
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2006-12-30
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