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