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<H1><A ID="SECTION001160000000000000000">
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Artificial reverberation</A>
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</H1>
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<P>
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Artificial reverberation is widely used to improve the sound of recordings, but
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has a wide range of other musical applications [<A
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HREF="node202.html#r-dodge85">DJ85</A>, pp.289-340].
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Reverberation in real, natural spaces arises from a complicated pattern of
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sound reflections off the walls and other objects that define the space. It is
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a great oversimplification to imitate this process using recirculating,
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discrete delay networks. Nonetheless, modeling reverberation using
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recirculating delay lines can, with much work, be made to yield good results.
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<P>
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The central idea is to idealize any room (or other reverberant space) as a
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collection of parallel delay lines that models the memory of the air inside the
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room. At each point on the walls of the room, many straight-line paths
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terminate, each carrying sound to that point; the sound then reflects into many
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other paths, each one originating at that point, and leading eventually to some
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other point on a wall.
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<P>
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Although the wall (and the air we passed through to get to the wall) absorbs
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some of the sound, some portion of the incident power is reflected and makes
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it to another wall. If most of the energy recirculates, the room reverberates
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for a long time; if all of it does, the reverberation lasts forever. If at
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any frequency the walls reflect more energy overall than they receive, the
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sound will feed back unstably; this never happens in real rooms (conservation
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of energy prevents it), but it can happen in an artificial reverberator if it
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is not designed correctly.
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<P>
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To make an artificial reverberator using a delay network, we must
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fill two competing demands simultaneously. First, the delay lines must
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be long enough to prevent
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<A ID="8091"></A><I>coloration</I> in the output as a result of comb filtering.
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(Even if we move beyond the simple comb filter
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of Section <A HREF="node109.html#sect7.recirculatingcomb">7.4</A>, the frequency response will tend to
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have peaks and
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valleys whose spacing varies inversely with total delay time.) On the other
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hand, we should not hear individual echoes; the
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<A ID="8094"></A><I>echo density</I> should ideally be at least one thousand per second.
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<P>
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In pursuit of these aims, we assemble some number of delay lines and
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connect their outputs back to their inputs. The feedback path--the connection
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from the outputs back to the inputs of the delays--should have an aggregate
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gain that varies gently as a function of frequency, and never exceeds one for
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any frequency. A good starting point is to give the feedback path a flat
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frequency response and a gain slightly less than one; this is done using
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rotation matrices.
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<P>
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Ideally this is all we should need to do, but in reality we will not always
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want to use the thousands of delay lines it would take to model the paths
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between every possible pair of points on the walls. In practice we
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usually use between four and sixteen delay lines to model the room. This
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simplification sometimes reduces the echo density below what we would wish,
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so we might use more delay lines at the input of the recirculating network
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to increase the density.
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<P>
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Figure <A HREF="#fig07.15">7.15</A> shows a simple reverberator design that uses this
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principle. The incoming sound, shown as two separate signals in this example,
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is first thickened by progressively delaying one of the two signals and then
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intermixing them using a rotation matrix. At each stage the number of
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echoes of the original signal is doubled; typically we would use between 6 and 8
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stages to make between 64 and 256 echos, all with a total delay of between 30
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and 80 milliseconds. The figure shows three such stages.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig07.15"></A><A ID="8099"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 7.15:</STRONG>
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Reverberator design using power-preserving transformations and
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recirculating delays.</CAPTION>
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<TR><TD><IMG
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WIDTH="248" HEIGHT="453" BORDER="0"
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SRC="img773.png"
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ALT="\begin{figure}\psfig{file=figs/fig07.15.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Next comes the recirculating part of the reverberator. After the initial
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thickening, the input signal is fed into a bank of parallel delay
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lines, and their outputs are again mixed using a rotation matrix. The
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mixed outputs are attenuated by a gain <IMG
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WIDTH="40" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img774.png"
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ALT="$g \le 1$">, and fed back into the
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delay lines to make a recirculating network.
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<P>
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The value <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img29.png"
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ALT="$g$"> controls
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the reverberation time. If the average length of the recirculating delay lines
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is <IMG
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WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img28.png"
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ALT="$d$">, then any incoming sound is attenuated by a factor of <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img29.png"
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ALT="$g$"> after a
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time delay of <IMG
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WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img28.png"
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ALT="$d$">. After time <IMG
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WIDTH="9" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img82.png"
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ALT="$t$"> the signal has recirculated <IMG
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WIDTH="25" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img775.png"
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ALT="$t/d$"> times,
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losing <!-- MATH
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$20 {\log_{10}} (g)$
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-->
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<IMG
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WIDTH="74" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img776.png"
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ALT="$20 {\log_{10}} (g)$"> decibels each time around, so the total gain, in
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decibels, 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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20 {t\over d} {\log_{10}} (g)
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\end{displaymath}
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-->
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<IMG
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WIDTH="80" HEIGHT="38" BORDER="0"
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SRC="img777.png"
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ALT="\begin{displaymath}
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20 {t\over d} {\log_{10}} (g)
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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 usual measure of reverberation time (RT) is the time at which the gain drops
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by sixty decibels:
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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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20 {\mathrm{RT}\over d} {\log_{10}}( g ) = -60
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\end{displaymath}
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-->
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<IMG
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WIDTH="144" HEIGHT="39" BORDER="0"
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SRC="img778.png"
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ALT="\begin{displaymath}
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20 {\mathrm{RT}\over d} {\log_{10}}( g ) = -60
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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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\mathrm{RT} = {{-3d} \over {{\log_{10}}(g) }}
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\end{displaymath}
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-->
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<IMG
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WIDTH="100" HEIGHT="43" BORDER="0"
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SRC="img779.png"
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ALT="\begin{displaymath}
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\mathrm{RT} = {{-3d} \over {{\log_{10}}(g) }}
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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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If <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img29.png"
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ALT="$g$"> is one, this formula gives <IMG
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WIDTH="19" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img305.png"
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ALT="$\infty$">, since the logarithm of one is zero.
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<P>
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The framework shown above is the basis for many modern reverberator designs.
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Many extensions of this underlying design have been proposed. The most
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important next step would be to introduce filters in the recirculation path so
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that high frequencies can be made to decay more rapidly than low ones; this is
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readily accomplished with a very simple low-pass filter, but we will not work
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this out here, having not yet developed the needed filter theory.
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<P>
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In general, to use this framework to design a reverberator involves making many
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complicated choices of delay times, gains, and filter coefficients. Mountains
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of literature have been published on this topic; Barry Blesser has published a
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good overview [<A
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HREF="node202.html#r-blesser01">Ble01</A>]. Much more is known about reverberator design
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and tuning that has not been published; precise designs are often kept secret
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for commercial reasons. In general, the design process involves painstaking
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and lengthy tuning by trial, error, and critical listening.
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<P>
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<BR><HR>
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<!--Table of Child-Links-->
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<A ID="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<UL>
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<LI><A ID="tex2html2215"
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HREF="node112.html">Controlling reverberators</A>
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</UL>
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<ADDRESS>
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Miller Puckette
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2006-12-30
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