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<H1><A NAME="SECTION00510000000000000000"></A>
<A NAME="sect1.amplitude"></A>
<BR>
Measures of Amplitude
</H1>

<P>
The most fundamental property of a digital audio signal is its
<A NAME="1038"></A>amplitude.  Unfortunately, a signal's amplitude has no one canonical
definition. 
Strictly speaking, all the samples in a digital audio signal are themselves
amplitudes, and we also spoke of the amplitude <IMG
 WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="img4.png"
 ALT="$a$"> of the sinusoid as a whole.
It is useful to have measures
of amplitude for digital audio signals in general.  Amplitude
is best thought of as applying to a
<A NAME="1039"></A><I>window</I>, a fixed range of samples of the signal.  For instance, the 
window starting at sample <IMG
 WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img86.png"
 ALT="$M$"> of length <IMG
 WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img3.png"
 ALT="$N$"> of an audio signal <IMG
 WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img80.png"
 ALT="$x[n]$"> consists of the
samples,
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
x[M], x[M+1], \ldots, x[M+N-1]
\end{displaymath}
 -->

<IMG
 WIDTH="234" HEIGHT="28" BORDER="0"
 SRC="img87.png"
 ALT="\begin{displaymath}
x[M], x[M+1], \ldots, x[M+N-1]
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
The two most frequently used measures of amplitude are the
<A NAME="1041"></A><A NAME="1042"></A><I>peak amplitude</I>, which is simply the greatest sample (in absolute value)
over the window:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
{A_{\mathrm{peak}}} \{x[n]\} = \max | x[n] | , 
    \hspace{0.3in}n = M, \ldots, M+N-1
\end{displaymath}
 -->

<IMG
 WIDTH="373" HEIGHT="29" BORDER="0"
 SRC="img88.png"
 ALT="\begin{displaymath}
{A_{\mathrm{peak}}} \{x[n]\} = \max \vert x[n] \vert ,
\hspace{0.3in}n = M, \ldots, M+N-1
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
and the
<A NAME="1046"></A><I>root mean square</I> (RMS) amplitude:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
{A_{\mathrm{RMS}}} \{x[n]\} = \sqrt{P\{x[n]\}}
\end{displaymath}
 -->

<IMG
 WIDTH="177" HEIGHT="28" BORDER="0"
 SRC="img89.png"
 ALT="\begin{displaymath}
{A_{\mathrm{RMS}}} \{x[n]\} = \sqrt{P\{x[n]\}}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where <IMG
 WIDTH="59" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img90.png"
 ALT="$P\{x[n]\}$"> is the mean
<A NAME="1050"></A><I>power</I>, defined as:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
{P\{x[n]\}} = {1 \over N} \left (
    	{{|x[M]|} ^2} + \cdots + {{|x[M+N-1]|} ^2}
    \right )
\end{displaymath}
 -->

<IMG
 WIDTH="335" HEIGHT="38" BORDER="0"
 SRC="img91.png"
 ALT="\begin{displaymath}
{P\{x[n]\}} = {1 \over N} \left (
{{\vert x[M]\vert} ^2} + \cdots + {{\vert x[M+N-1]\vert} ^2}
\right )
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
(In this last formula, the absolute value signs aren't necessary at the moment
since we're working on real-valued signals, but they will become important
later when we consider complex-valued signals.)  Neither the peak nor the RMS
amplitude of any signal can be negative, and either one can be exactly zero
only if the signal itself is zero for all <IMG
 WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="img75.png"
 ALT="$n$"> in the window.

<P>
The RMS amplitude of a signal may equal the peak amplitude but never exceeds
it; and it may be as little as <IMG
 WIDTH="47" HEIGHT="38" ALIGN="MIDDLE" BORDER="0"
 SRC="img92.png"
 ALT="$1 / {\sqrt N}$"> times the peak amplitude, but
never less than that.

<P>

<DIV ALIGN="CENTER"><A NAME="fig01.02"></A><A NAME="1059"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 1.2:</STRONG>
Root mean square (RMS) and peak amplitudes of signals compared.
For a sinusoid (part a), the peak amplitude is higher than RMS by a factor of
<IMG
 WIDTH="24" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img1.png"
 ALT="$\sqrt 2$">.</CAPTION>
<TR><TD><IMG
 WIDTH="362" HEIGHT="306" BORDER="0"
 SRC="img93.png"
 ALT="\begin{figure}\psfig{file=figs/fig01.02.ps}\end{figure}"></TD></TR>
</TABLE>
</DIV>

<P>
Under reasonable conditions--if the window contains at least several periods and
if the angular frequency is well under one radian per sample--the peak
amplitude of the sinusoid of Page <A HREF="node7.html#eq-realsinusoid"><IMG  ALIGN="BOTTOM" BORDER="1" ALT="[*]"
 SRC="crossref.png"></A>
is approximately <IMG
 WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="img4.png"
 ALT="$a$"> and its RMS amplitude
about <IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img94.png"
 ALT="$a / {\sqrt 2}$">.  Figure <A HREF="#fig01.02">1.2</A> shows the peak and RMS amplitudes
of two digital audio signals.

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<ADDRESS>
Miller Puckette
2006-12-30
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