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<!DOCTYPE html>
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<TITLE>Linear and Curved Amplitude Shapes</TITLE>
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<H1><A ID="SECTION00820000000000000000"></A>
<A ID="sect4.curved"></A>
<BR>
Linear and Curved Amplitude Shapes
</H1>
<P>
Suppose you wish to fade a signal in over a period of ten seconds--that is,
you wish to multiply it by an amplitude-controlling signal <IMG
WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img2.png"
ALT="$y[n]$"> which rises
from 0 to 1 in value over <IMG
WIDTH="31" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img358.png"
ALT="$10R$"> samples, where <IMG
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img36.png"
ALT="$R$"> is the sample rate. The
most obvious choice would be a linear ramp: <!-- MATH
$y[n] = n/(10R)$
-->
<IMG
WIDTH="109" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img359.png"
ALT="$y[n] = n/(10R)$">. But this will
not turn out to yield a smooth increase in perceived loudness. Over the first
second <IMG
WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img2.png"
ALT="$y[n]$"> rises from <IMG
WIDTH="31" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img102.png"
ALT="$-\infty$"> dB to -20 dB, over the next four by another
14 dB, and over the remaining five, only by the remaining 6 dB. Over most of
the ten second period the rise in amplitude will be barely perceptible.
<P>
Another possibility would be to ramp <IMG
WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img2.png"
ALT="$y[n]$"> exponentially, so that it rises at
a constant rate in dB. You would have to fix the initial amplitude to be
inaudible, say 0 dB (if we fix unity at 100 dB). Now we have the opposite
problem: for the first five seconds the amplitude control will rise from 0 dB
(inaudible) to 50 dB (pianissimo); this part of the fade-in should only have
taken up the first second or so.
<P>
A more natural progression would perhaps have been to regard the fade-in
as a timed succession of dynamics, 0-ppp-pp-p-mp-mf-f-ff-fff,
with each step taking roughly one second.
<P>
A fade-in ideally should obey some scale in between logarithmic and linear. A
somewhat arbitrary choice, but useful in practice, is the quartic curve:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
y[n] = {{ \left ( {{n} \over {N}} \right ) } ^ 4} ,
\end{displaymath}
-->
<IMG
WIDTH="97" HEIGHT="39" BORDER="0"
SRC="img360.png"
ALT="\begin{displaymath}
y[n] = {{ \left ( {{n} \over {N}} \right ) } ^ 4} ,
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where <IMG
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$N$"> is the number of samples to fade in over (in the example above, it's
<IMG
WIDTH="31" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img358.png"
ALT="$10R$">). So, after the first second of the ten we would have risen to -80 dB,
after five seconds to -24 dB, and after nine, about -4 dB.
<P>
Figure <A HREF="#fig04.03">4.3</A> shows three amplitude transfer functions:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
{f_1} (x) = x \hspace{0.2in} \mathrm{(linear),}
\end{displaymath}
-->
<IMG
WIDTH="144" HEIGHT="28" BORDER="0"
SRC="img361.png"
ALT="\begin{displaymath}
{f_1} (x) = x \hspace{0.2in} \mathrm{(linear),}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\ \ \ \ \ \ \ %
{f_2} (x) = 10^{2(x-1)} \hspace{0.2in}\mathrm{(dB\ to\ linear),}
\end{displaymath}
-->
<IMG
WIDTH="236" HEIGHT="28" BORDER="0"
SRC="img362.png"
ALT="\begin{displaymath}
\ \ \ \ \ \ \ %
{f_2} (x) = 10^{2(x-1)} \hspace{0.2in}\mathrm{(dB\ to\ linear),}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
{f_3} (x) = {x^4} \hspace{0.2in}\mathrm{(quartic).}
\end{displaymath}
-->
<IMG
WIDTH="161" HEIGHT="28" BORDER="0"
SRC="img363.png"
ALT="\begin{displaymath}
{f_3} (x) = {x^4} \hspace{0.2in}\mathrm{(quartic).}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
The second function converts from dB to linear, arranged so that the input
range, from 0 to 1, corresponds to 40 dB. (This input range of 40 dB
corresponds to a reasonable dynamic range, allowing
5 dB for each of 8 steps in dynamic.) The quartic curve imitates the
exponential (dB) curve fairly well for higher amplitudes, but drops off more
rapidly
for small amplitudes, reaching true zero at right (whereas the exponential curve
only goes down to <IMG
WIDTH="43" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img364.png"
ALT="$1/100$">).
<P>
<DIV ALIGN="CENTER"><A ID="fig04.03"></A><A ID="4616"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 4.3:</STRONG>
Three amplitude transfer functions. The horizontal axis is in linear,
logarithmic, or fourth-root units depending on the curve.</CAPTION>
<TR><TD><IMG
WIDTH="628" HEIGHT="388" BORDER="0"
SRC="img365.png"
ALT="\begin{figure}\psfig{file=figs/fig04.03.ps}\end{figure}"></TD></TR>
</TABLE>
</DIV>
<P>
We can think of the three curves as showing
transfer functions, from an abstract control (ranging from 0 to 1) to a
linear amplitude. After we choose a suitable transfer function <IMG
WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img112.png"
ALT="$f$">, we
can compute a corresponding amplitude control signal; if we wish to ramp
over <IMG
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$N$"> samples from silence to unity gain, the control signal would be:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
y[n] = f(n/N) .
\end{displaymath}
-->
<IMG
WIDTH="104" HEIGHT="28" BORDER="0"
SRC="img366.png"
ALT="\begin{displaymath}
y[n] = f(n/N) .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
A block diagram for this is shown in Figure <A HREF="#fig04.04">4.4</A>. Here we are
introducing a new type of block to represent the application of a
<A ID="4620"></A>
<I>transfer function</I>. For now we won't worry about its implementation;
depending on the function desired, this might be best done arithmetically or
using table lookup.
<P>
<DIV ALIGN="CENTER"><A ID="fig04.04"></A><A ID="4624"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 4.4:</STRONG>
Using a transfer function to alter the shape of amplitude curves.</CAPTION>
<TR><TD><IMG
WIDTH="296" HEIGHT="227" BORDER="0"
SRC="img367.png"
ALT="\begin{figure}\psfig{file=figs/fig04.04.ps}\end{figure}"></TD></TR>
</TABLE>
</DIV>
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<B> Next:</B> <A ID="tex2html1446"
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<B> Up:</B> <A ID="tex2html1440"
HREF="node58.html">Automation and voice management</A>
<B> Previous:</B> <A ID="tex2html1434"
HREF="node59.html">Envelope Generators</A>
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<ADDRESS>
Miller Puckette
2006-12-30
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