2022-04-12 22:02:59 -03:00
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2022-04-12 21:54:18 -03:00
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<!--Converted with LaTeX2HTML 2002-2-1 (1.71)
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original version by: Nikos Drakos, CBLU, University of Leeds
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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<TITLE>Linear and Curved Amplitude Shapes</TITLE>
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<H1><A NAME="SECTION00820000000000000000"></A>
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<A NAME="sect4.curved"></A>
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<BR>
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Linear and Curved Amplitude Shapes
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</H1>
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<P>
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Suppose you wish to fade a signal in over a period of ten seconds--that is,
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you wish to multiply it by an amplitude-controlling signal <IMG
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WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img2.png"
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ALT="$y[n]$"> which rises
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from 0 to 1 in value over <IMG
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WIDTH="31" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img358.png"
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ALT="$10R$"> samples, where <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img36.png"
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ALT="$R$"> is the sample rate. The
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most obvious choice would be a linear ramp: <!-- MATH
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$y[n] = n/(10R)$
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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="img359.png"
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ALT="$y[n] = n/(10R)$">. But this will
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not turn out to yield a smooth increase in perceived loudness. Over the first
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second <IMG
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WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img2.png"
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ALT="$y[n]$"> rises from <IMG
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WIDTH="31" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img102.png"
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ALT="$-\infty$"> dB to -20 dB, over the next four by another
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14 dB, and over the remaining five, only by the remaining 6 dB. Over most of
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the ten second period the rise in amplitude will be barely perceptible.
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<P>
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Another possibility would be to ramp <IMG
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WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img2.png"
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ALT="$y[n]$"> exponentially, so that it rises at
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a constant rate in dB. You would have to fix the initial amplitude to be
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inaudible, say 0 dB (if we fix unity at 100 dB). Now we have the opposite
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problem: for the first five seconds the amplitude control will rise from 0 dB
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(inaudible) to 50 dB (pianissimo); this part of the fade-in should only have
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taken up the first second or so.
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<P>
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A more natural progression would perhaps have been to regard the fade-in
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as a timed succession of dynamics, 0-ppp-pp-p-mp-mf-f-ff-fff,
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with each step taking roughly one second.
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<P>
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A fade-in ideally should obey some scale in between logarithmic and linear. A
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somewhat arbitrary choice, but useful in practice, is the quartic curve:
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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] = {{ \left ( {{n} \over {N}} \right ) } ^ 4} ,
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\end{displaymath}
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-->
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<IMG
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WIDTH="97" HEIGHT="39" BORDER="0"
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SRC="img360.png"
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ALT="\begin{displaymath}
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y[n] = {{ \left ( {{n} \over {N}} \right ) } ^ 4} ,
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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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where <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$"> is the number of samples to fade in over (in the example above, it's
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<IMG
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WIDTH="31" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img358.png"
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ALT="$10R$">). So, after the first second of the ten we would have risen to -80 dB,
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after five seconds to -24 dB, and after nine, about -4 dB.
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<P>
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Figure <A HREF="#fig04.03">4.3</A> shows three amplitude transfer functions:
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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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{f_1} (x) = x \hspace{0.2in} \mathrm{(linear),}
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\end{displaymath}
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-->
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<IMG
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WIDTH="144" HEIGHT="28" BORDER="0"
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SRC="img361.png"
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ALT="\begin{displaymath}
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{f_1} (x) = x \hspace{0.2in} \mathrm{(linear),}
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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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<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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\ \ \ \ \ \ \ %
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{f_2} (x) = 10^{2(x-1)} \hspace{0.2in}\mathrm{(dB\ to\ linear),}
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\end{displaymath}
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-->
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<IMG
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WIDTH="236" HEIGHT="28" BORDER="0"
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SRC="img362.png"
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ALT="\begin{displaymath}
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\ \ \ \ \ \ \ %
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{f_2} (x) = 10^{2(x-1)} \hspace{0.2in}\mathrm{(dB\ to\ linear),}
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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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<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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{f_3} (x) = {x^4} \hspace{0.2in}\mathrm{(quartic).}
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\end{displaymath}
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-->
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<IMG
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WIDTH="161" HEIGHT="28" BORDER="0"
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SRC="img363.png"
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ALT="\begin{displaymath}
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{f_3} (x) = {x^4} \hspace{0.2in}\mathrm{(quartic).}
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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 second function converts from dB to linear, arranged so that the input
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range, from 0 to 1, corresponds to 40 dB. (This input range of 40 dB
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corresponds to a reasonable dynamic range, allowing
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5 dB for each of 8 steps in dynamic.) The quartic curve imitates the
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exponential (dB) curve fairly well for higher amplitudes, but drops off more
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rapidly
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for small amplitudes, reaching true zero at right (whereas the exponential curve
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only goes down to <IMG
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WIDTH="43" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img364.png"
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ALT="$1/100$">).
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig04.03"></A><A NAME="4616"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 4.3:</STRONG>
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Three amplitude transfer functions. The horizontal axis is in linear,
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logarithmic, or fourth-root units depending on the curve.</CAPTION>
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<TR><TD><IMG
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WIDTH="628" HEIGHT="388" BORDER="0"
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SRC="img365.png"
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ALT="\begin{figure}\psfig{file=figs/fig04.03.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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We can think of the three curves as showing
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transfer functions, from an abstract control (ranging from 0 to 1) to a
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linear amplitude. After we choose a suitable transfer 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="$f$">, we
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can compute a corresponding amplitude control signal; if we wish to ramp
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over <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$"> samples from silence to unity gain, the control signal would be:
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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] = f(n/N) .
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\end{displaymath}
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-->
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<IMG
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WIDTH="104" HEIGHT="28" BORDER="0"
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SRC="img366.png"
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ALT="\begin{displaymath}
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y[n] = f(n/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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A block diagram for this is shown in Figure <A HREF="#fig04.04">4.4</A>. Here we are
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introducing a new type of block to represent the application of a
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<A NAME="4620"></A>
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<I>transfer function</I>. For now we won't worry about its implementation;
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depending on the function desired, this might be best done arithmetically or
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using table lookup.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig04.04"></A><A NAME="4624"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 4.4:</STRONG>
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Using a transfer function to alter the shape of amplitude curves.</CAPTION>
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<TR><TD><IMG
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WIDTH="296" HEIGHT="227" BORDER="0"
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SRC="img367.png"
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ALT="\begin{figure}\psfig{file=figs/fig04.04.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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HREF="node61.html">Continuous and discontinuous control</A>
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HREF="node58.html">Automation and voice management</A>
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HREF="node59.html">Envelope Generators</A>
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<B> <A NAME="tex2html1442"
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HREF="node4.html">Contents</A></B>
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Miller Puckette
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2006-12-30
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