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