298 lines
9.3 KiB
HTML
298 lines
9.3 KiB
HTML
<!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>Narrow-band companding</TITLE>
|
|
<META NAME="description" CONTENT="Narrow-band companding">
|
|
<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="node174.html">
|
|
<LINK REL="previous" HREF="node172.html">
|
|
<LINK REL="up" HREF="node172.html">
|
|
<LINK REL="next" HREF="node174.html">
|
|
</HEAD>
|
|
|
|
<BODY >
|
|
<!--Navigation Panel-->
|
|
<A NAME="tex2html3160"
|
|
HREF="node174.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="next.png"></A>
|
|
<A NAME="tex2html3154"
|
|
HREF="node172.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="up.png"></A>
|
|
<A NAME="tex2html3148"
|
|
HREF="node172.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="prev.png"></A>
|
|
<A NAME="tex2html3156"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="contents.png"></A>
|
|
<A NAME="tex2html3158"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A NAME="tex2html3161"
|
|
HREF="node174.html">Timbre stamping (classical vocoder)</A>
|
|
<B> Up:</B> <A NAME="tex2html3155"
|
|
HREF="node172.html">Fourier analysis and reconstruction</A>
|
|
<B> Previous:</B> <A NAME="tex2html3149"
|
|
HREF="node172.html">Fourier analysis and reconstruction</A>
|
|
<B> <A NAME="tex2html3157"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A NAME="tex2html3159"
|
|
HREF="node201.html">Index</A></B>
|
|
<BR>
|
|
<BR>
|
|
<!--End of Navigation Panel-->
|
|
|
|
<H2><A NAME="SECTION001341000000000000000">
|
|
Narrow-band companding</A>
|
|
</H2>
|
|
|
|
<P>
|
|
|
|
<DIV ALIGN="CENTER"><A NAME="fig09.08"></A><A NAME="12579"></A>
|
|
<TABLE>
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.8:</STRONG>
|
|
Block diagram for narrow-band noise suppression by companding.</CAPTION>
|
|
<TR><TD><IMG
|
|
WIDTH="270" HEIGHT="423" BORDER="0"
|
|
SRC="img1157.png"
|
|
ALT="\begin{figure}\psfig{file=figs/fig09.08.ps}\end{figure}"></TD></TR>
|
|
</TABLE>
|
|
</DIV>
|
|
|
|
<P>
|
|
A
|
|
<A NAME="12582"></A><I>compander</I>
|
|
is a tool that amplifies a signal with a variable gain, depending on the
|
|
signal's measured amplitude. The term is a contraction of "compressor" and
|
|
"expander". A compressor's gain decreases as the input level increases, so
|
|
that the
|
|
<A NAME="12584"></A><I>dynamic range</I>,
|
|
that is, the overall variation in signal level, is reduced. An expander does
|
|
the reverse, increasing the dynamic range. Frequently the gain depends not only
|
|
on the immediate signal level but on its history; for instance the rate of change
|
|
might be limited or there might be a time delay.
|
|
|
|
<P>
|
|
By using Fourier analysis and resynthesis, we can do companding individually on
|
|
narrow-band channels. If <IMG
|
|
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1158.png"
|
|
ALT="$C[m]$"> is one such band, we apply a gain <IMG
|
|
WIDTH="34" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1159.png"
|
|
ALT="$g[m]$"> to
|
|
it, to give <IMG
|
|
WIDTH="69" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1160.png"
|
|
ALT="$g[m]C[m]$">. Although <IMG
|
|
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1158.png"
|
|
ALT="$C[m]$"> is a complex number, the gain is a
|
|
non-negative real number. In general the gain could be a function not only of
|
|
<IMG
|
|
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1158.png"
|
|
ALT="$C[m]$"> but also of any or all the previous samples in the channel: <IMG
|
|
WIDTH="66" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1161.png"
|
|
ALT="$C[m-1]$">,
|
|
<IMG
|
|
WIDTH="66" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1162.png"
|
|
ALT="$C[m-2]$">, and so on. Here we'll consider the simplest situation where the gain is
|
|
simply a function of the magnitude of the current sample: <IMG
|
|
WIDTH="47" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1163.png"
|
|
ALT="$\vert C[m]\vert$">.
|
|
|
|
<P>
|
|
The patch diagrammed in Figure <A HREF="#fig09.08">9.8</A> shows one very useful
|
|
application of companding, called
|
|
a
|
|
<A NAME="12587"></A><I>noise gate</I>. Here the gain <IMG
|
|
WIDTH="34" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1159.png"
|
|
ALT="$g[m]$"> depends on the channel amplitude
|
|
<IMG
|
|
WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1158.png"
|
|
ALT="$C[m]$"> and a noise floor which is a function <IMG
|
|
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img112.png"
|
|
ALT="$f$"> of the channel number <IMG
|
|
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img58.png"
|
|
ALT="$k$">.
|
|
For clarity we will apply the frequency subscript <IMG
|
|
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img58.png"
|
|
ALT="$k$"> to the gain, now
|
|
written as <IMG
|
|
WIDTH="50" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1164.png"
|
|
ALT="$g[m, k]$">, and to the windowed Fourier transform <!-- MATH
|
|
$S[m, k] = C[m]$
|
|
-->
|
|
<IMG
|
|
WIDTH="109" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1165.png"
|
|
ALT="$S[m, k] = C[m]$">.
|
|
The gain is given by:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
g[m, k] =
|
|
\left \{
|
|
\begin{array}{ll}
|
|
{1 - f[k]/|S[m, k]|} & {|S[m, k]| > f[k]} \\
|
|
0 & \mbox{otherwise}
|
|
\end{array}
|
|
\right .
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="330" HEIGHT="45" BORDER="0"
|
|
SRC="img1166.png"
|
|
ALT="\begin{displaymath}
|
|
g[m, k] =
|
|
\left \{
|
|
\begin{array}{ll}
|
|
{1 - f[k]/\vert S[...
|
|
...]\vert > f[k]} \\
|
|
0 & \mbox{otherwise}
|
|
\end{array} \right .
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
Whenever the magnitude <IMG
|
|
WIDTH="52" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img64.png"
|
|
ALT="$S[m, k]$"> is less than the threshold <IMG
|
|
WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1167.png"
|
|
ALT="$f[k]$"> the
|
|
gain is zero and so the amplitude <IMG
|
|
WIDTH="52" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img64.png"
|
|
ALT="$S[m, k]$"> is replaced
|
|
by zero. Otherwise, multiplying the amplitude by <IMG
|
|
WIDTH="50" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1164.png"
|
|
ALT="$g[m, k]$"> reduces
|
|
the the magnitude downward to <!-- MATH
|
|
$|S[m, k]|-f[k]$
|
|
-->
|
|
<IMG
|
|
WIDTH="108" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1168.png"
|
|
ALT="$\vert S[m, k]\vert-f[k]$">. Since the gain is a
|
|
non-negative real number, the phase is preserved.
|
|
|
|
<P>
|
|
In the figure, the gain is computed as a thresholding function of the
|
|
ratio <!-- MATH
|
|
$x = |S[m, k]|/f[k]$
|
|
-->
|
|
<IMG
|
|
WIDTH="127" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1169.png"
|
|
ALT="$x = \vert S[m, k]\vert/f[k]$"> of the signal magnitude to the noise floor; the
|
|
threshold is <IMG
|
|
WIDTH="106" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1170.png"
|
|
ALT="$g(x) = 1-1/x$"> when <IMG
|
|
WIDTH="41" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1171.png"
|
|
ALT="$x<1$"> and zero otherwise, although other
|
|
thresholding functions could easily be substituted.
|
|
|
|
<P>
|
|
This technique is useful for removing noise from a recorded sound. We either
|
|
measure or guess values of the noise floor <IMG
|
|
WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1167.png"
|
|
ALT="$f[k]$">. Because of the
|
|
design of the gain function <IMG
|
|
WIDTH="50" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1164.png"
|
|
ALT="$g[m, k]$">, only amplitudes which are above the
|
|
noise floor reach the output. Since this is done on narrow frequency bands, it
|
|
is sometimes possible to remove most of the noise even while the signal itself,
|
|
in the frequency ranges where it is louder than the noise floor, is mostly
|
|
preserved.
|
|
|
|
<P>
|
|
The technique is also useful as preparation before applying a non-linear
|
|
operation, such as distortion, to a sound. It is often best to distort only
|
|
the most salient frequencies of the sound. Subtracting the noise-gated sound
|
|
from the original then gives a residual signal which can be passed through
|
|
undistorted.
|
|
|
|
<P>
|
|
<HR>
|
|
<!--Navigation Panel-->
|
|
<A NAME="tex2html3160"
|
|
HREF="node174.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="next.png"></A>
|
|
<A NAME="tex2html3154"
|
|
HREF="node172.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="up.png"></A>
|
|
<A NAME="tex2html3148"
|
|
HREF="node172.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="prev.png"></A>
|
|
<A NAME="tex2html3156"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="contents.png"></A>
|
|
<A NAME="tex2html3158"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A NAME="tex2html3161"
|
|
HREF="node174.html">Timbre stamping (classical vocoder)</A>
|
|
<B> Up:</B> <A NAME="tex2html3155"
|
|
HREF="node172.html">Fourier analysis and reconstruction</A>
|
|
<B> Previous:</B> <A NAME="tex2html3149"
|
|
HREF="node172.html">Fourier analysis and reconstruction</A>
|
|
<B> <A NAME="tex2html3157"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A NAME="tex2html3159"
|
|
HREF="node201.html">Index</A></B>
|
|
<!--End of Navigation Panel-->
|
|
<ADDRESS>
|
|
Miller Puckette
|
|
2006-12-30
|
|
</ADDRESS>
|
|
</BODY>
|
|
</HTML>
|