301 lines
9.2 KiB
HTML
301 lines
9.2 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>Envelope following</TITLE>
|
|
<META NAME="description" CONTENT="Envelope following">
|
|
<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="node154.html">
|
|
<LINK REL="previous" HREF="node152.html">
|
|
<LINK REL="up" HREF="node151.html">
|
|
<LINK REL="next" HREF="node154.html">
|
|
</HEAD>
|
|
|
|
<BODY >
|
|
<!--Navigation Panel-->
|
|
<A ID="tex2html2857"
|
|
HREF="node154.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="next.png"></A>
|
|
<A ID="tex2html2851"
|
|
HREF="node151.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="up.png"></A>
|
|
<A ID="tex2html2845"
|
|
HREF="node152.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="prev.png"></A>
|
|
<A ID="tex2html2853"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="contents.png"></A>
|
|
<A ID="tex2html2855"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A ID="tex2html2858"
|
|
HREF="node154.html">Single Sideband Modulation</A>
|
|
<B> Up:</B> <A ID="tex2html2852"
|
|
HREF="node151.html">Applications</A>
|
|
<B> Previous:</B> <A ID="tex2html2846"
|
|
HREF="node152.html">Subtractive synthesis</A>
|
|
<B> <A ID="tex2html2854"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A ID="tex2html2856"
|
|
HREF="node201.html">Index</A></B>
|
|
<BR>
|
|
<BR>
|
|
<!--End of Navigation Panel-->
|
|
|
|
<H2><A ID="SECTION001242000000000000000"></A>
|
|
<A ID="sect8.envelopefollower"></A>
|
|
<BR>
|
|
Envelope following
|
|
</H2>
|
|
|
|
<P>
|
|
It is frequently desirable to use the time-varying power of an incoming signal
|
|
to trigger or control a musical process. To do this, we will need a procedure
|
|
for measuring the power of an audio signal. Since most audio signals pass
|
|
through zero many times per second, it won't suffice to take instantaneous values
|
|
of the signal to measure its power; instead, we must calculate the average
|
|
power over an interval of time long enough that its variations won't
|
|
show up in the power estimate, but short enough that changes in signal level
|
|
are quickly reported. A computation that provides a
|
|
time-varying power estimate of a signal is called an
|
|
<A ID="10533"></A><I>envelope follower</I>.
|
|
|
|
<P>
|
|
The output of a low-pass filter can be viewed as a moving average of its input.
|
|
For example, suppose we apply a normalized one-pole low-pass filter with
|
|
coefficient <IMG
|
|
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img57.png"
|
|
ALT="$p$">, as in Figure <A HREF="node148.html#fig08.21">8.21</A>, to an incoming signal <IMG
|
|
WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img80.png"
|
|
ALT="$x[n]$">.
|
|
The output (call it y[n]) is the sum of the delay output times <IMG
|
|
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img57.png"
|
|
ALT="$p$">, with
|
|
the input times <IMG
|
|
WIDTH="38" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img910.png"
|
|
ALT="$1-p$">:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
y[n] = p \cdot y[n-1] + (1-p) \cdot x[n]
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="226" HEIGHT="28" BORDER="0"
|
|
SRC="img988.png"
|
|
ALT="\begin{displaymath}
|
|
y[n] = p \cdot y[n-1] + (1-p) \cdot x[n]
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
so each input is averaged, with weight <IMG
|
|
WIDTH="38" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img910.png"
|
|
ALT="$1-p$">, into the previous output to
|
|
produce a new output. So we can make a moving average of the square of an
|
|
audio signal using the diagram of Figure <A HREF="#fig08.26">8.26</A>. The output is
|
|
a time-varying average of the instantaneous power <IMG
|
|
WIDTH="38" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img989.png"
|
|
ALT="$x[n]^2$">, and the design
|
|
of the low-pass filter controls, among other things, the settling time of
|
|
the moving average.
|
|
|
|
<P>
|
|
|
|
<DIV ALIGN="CENTER"><A ID="fig08.26"></A><A ID="10539"></A>
|
|
<TABLE>
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.26:</STRONG>
|
|
Envelope follower. The output is the average power of the input
|
|
signal.</CAPTION>
|
|
<TR><TD><IMG
|
|
WIDTH="82" HEIGHT="238" BORDER="0"
|
|
SRC="img990.png"
|
|
ALT="\begin{figure}\psfig{file=figs/fig08.26.ps}\end{figure}"></TD></TR>
|
|
</TABLE>
|
|
</DIV>
|
|
|
|
<P>
|
|
For more insight into the design of a suitable low-pass filter for an envelope
|
|
follower, we analyze it from the point of view of signal spectra. If, for
|
|
instance, we put in a real-valued sinusoid:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
x[n] = a \cdot \cos(\alpha n)
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="120" HEIGHT="28" BORDER="0"
|
|
SRC="img991.png"
|
|
ALT="\begin{displaymath}
|
|
x[n] = a \cdot \cos(\alpha n)
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
the result of squaring is:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
{{x[n]}^2} = {{a^2}\over 2} \left ( \cos(2 \alpha n) + 1 \right )
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="177" HEIGHT="41" BORDER="0"
|
|
SRC="img992.png"
|
|
ALT="\begin{displaymath}
|
|
{{x[n]}^2} = {{a^2}\over 2} \left ( \cos(2 \alpha n) + 1 \right )
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
and so if the low-pass filter effectively stops the component of frequency
|
|
<IMG
|
|
WIDTH="21" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img993.png"
|
|
ALT="$2 \alpha$"> we will get out approximately the constant <IMG
|
|
WIDTH="34" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img994.png"
|
|
ALT="${{a^2} / 2}$">, which
|
|
is indeed the average power.
|
|
|
|
<P>
|
|
The situation for a signal with several components is similar. Suppose the
|
|
input signal is now,
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
x[n] = a \cdot \cos(\alpha n) + b \cdot \cos(\beta n)
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="211" HEIGHT="28" BORDER="0"
|
|
SRC="img995.png"
|
|
ALT="\begin{displaymath}
|
|
x[n] = a \cdot \cos(\alpha n) + b \cdot \cos(\beta n)
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
whose spectrum is plotted in Figure <A HREF="#fig08.27">8.27</A> (part a). (We have omitted
|
|
the two phase terms but they will have no effect on the outcome.) Squaring the
|
|
signal produces the spectrum shown in part (b) (see Section
|
|
<A HREF="node77.html#sect5.ringmod">5.2</A>).) We can get the desired fixed value of <!-- MATH
|
|
$({a^2} +
|
|
{b^2})/2$
|
|
-->
|
|
<IMG
|
|
WIDTH="80" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img996.png"
|
|
ALT="$({a^2} +
|
|
{b^2})/2$"> simply by filtering out all the other components; ideally the result
|
|
will be a constant (DC) signal. As long as we filter out all the partials, and
|
|
also all the difference tones, we end up with a stable output that correctly
|
|
estimates the average power.
|
|
|
|
<P>
|
|
|
|
<DIV ALIGN="CENTER"><A ID="fig08.27"></A><A ID="10551"></A>
|
|
<TABLE>
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.27:</STRONG>
|
|
Envelope following from the spectral point of view: (a) an
|
|
incoming signal with two components; (b) the result of squaring it.</CAPTION>
|
|
<TR><TD><IMG
|
|
WIDTH="388" HEIGHT="328" BORDER="0"
|
|
SRC="img997.png"
|
|
ALT="\begin{figure}\psfig{file=figs/fig08.27.ps}\end{figure}"></TD></TR>
|
|
</TABLE>
|
|
</DIV>
|
|
|
|
<P>
|
|
Envelope followers may also be used on noisy signals, which may be thought of
|
|
as signals with dense spectra. In this situation there will be difference
|
|
frequencies arbitrarily close to zero, and filtering them out
|
|
entirely will be impossible; we will always get fluctuations in the output, but
|
|
they will decrease proportionally as the filter's passband is narrowed.
|
|
|
|
<P>
|
|
Although a narrower passband will always give a cleaner output, whether for
|
|
discrete or continuous spectra, the filter's settling time will lengthen
|
|
proportionally as the passband is narrowed. There is thus a tradeoff between
|
|
getting a quick response and a smooth result.
|
|
|
|
<P>
|
|
<HR>
|
|
<!--Navigation Panel-->
|
|
<A ID="tex2html2857"
|
|
HREF="node154.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="next.png"></A>
|
|
<A ID="tex2html2851"
|
|
HREF="node151.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="up.png"></A>
|
|
<A ID="tex2html2845"
|
|
HREF="node152.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="prev.png"></A>
|
|
<A ID="tex2html2853"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="contents.png"></A>
|
|
<A ID="tex2html2855"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A ID="tex2html2858"
|
|
HREF="node154.html">Single Sideband Modulation</A>
|
|
<B> Up:</B> <A ID="tex2html2852"
|
|
HREF="node151.html">Applications</A>
|
|
<B> Previous:</B> <A ID="tex2html2846"
|
|
HREF="node152.html">Subtractive synthesis</A>
|
|
<B> <A ID="tex2html2854"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A ID="tex2html2856"
|
|
HREF="node201.html">Index</A></B>
|
|
<!--End of Navigation Panel-->
|
|
<ADDRESS>
|
|
Miller Puckette
|
|
2006-12-30
|
|
</ADDRESS>
|
|
</BODY>
|
|
</HTML>
|