Artificial Reverberation
This chapter summarizes basic results in systems for artificial reverberation. Such systems make extensive use of tapped delay lines, comb filters, and allpass filters (described in the previous chapter).The Reverberation Problem
Consider the requirements for acoustically simulating a concert hall or other listening space. Suppose we only need the response at one or more discrete listening points in space (``ears'') due to one or more discrete point sources of acoustic energy. First, as discussed in §2.2, the direct signal propagating from a sound source to a listener's ear can be simulated using a single delay line in series with an attenuation scaling or lowpass filter. Second, each sound ray arriving at the listening point via one or more reflections can be simulated using a delayline and some scale factor (or filter). Two rays create a feedforward comb filter, like the one in Fig.2.9 on page . More generally, a tapped delay line (Fig.2.19) can simulate many reflections. Each tap brings out one echo at the appropriate delay and gain, and each tap can be independently filtered to simulate air absorption and lossy reflections. In principle, tapped delay lines can accurately simulate any reverberant environment, because reverberation really does consist of many paths of acoustic propagation from each source to each listening point. As we will see, the only limitations of a tapped delay line are (1) it is expensive computationally relative to other techniques, (2) it handles only one ``point to point'' transfer function, i.e., from one pointsource to one ear,^{4.1} and (3) it should be changed when the source, listener, or anything in the room moves.Exact Reverb via TransferFunction Modeling
Figure 3.1 depicts the general reverberation scenario for three sources and one listener (two ears). In general, the filters should also include filtering by the pinnae of the ears, so that each echo can be perceived as coming from the correct angle of arrival in 3D space; in other words, at least some reverberant reflections should be spatialized so that they appear to come from their natural directions in 3D space [248]. Again, the filters change if anything changes in the listening space, including source or listener position. The artificial reverberation problem is then to implement some approximation of the system in Fig.3.1.Complexity of Exact Reverberation
For music, a typical reverberation time^{4.2}is on the order of one second. Suppose we choose exactly one second for the reverberation time. At an audio sampling rate of 50 kHz, each filter in Fig.3.1 requires 50,000 multiplies and additions per sample, or 2.5 billion multiplyadds per second. Handling three sources and two listening points (ears), we reach 30 billion operations per second for the reverberator. This computational load would require at least 10 Pentium CPUs clocked at 3 Gigahertz, assuming they were doing nothing else, and assuming both a multiply and addition can be initiated each clock cycle, with no waitstates caused by the three required memory accesses (input, output, and filter coefficient). While these numbers can be improved using FFT convolution instead of direct convolution (at the price of introducing a throughput delay which can be a problem for realtime systems), it remains the case that exact implementation of all relevant pointtopoint transfer functions in a reverberant space is very expensive computationally. While a tapped delay line FIR filter can provide an accurate model for any pointtopoint transfer function in a reverberant environment, it is rarely used for this purpose in practice because of the extremely high computational expense. While there are specialized commercial products that implement reverberation via direct convolution of the input signal with the impulse response, the great majority of artificial reverberation systems use other methods to synthesize the late reverb more economically.Possibility of a Physical Reverb Model
One disadvantage of the pointtopoint transfer function model depicted in Fig.3.1 is that some or all of the filters must change when anything moves. If instead we had a computational model of the whole acoustic space, sources and listeners could be moved as desired without affecting the underlying room simulation. Furthermore, we could use ``virtual dummy heads'' as listeners, complete with pinnae filters, so that all of the 3D directional aspects of reverberation could be captured in two extracted signals for the ears. Thus, there are compelling reasons to consider a full 3D model of a desired acoustic listening space. Let us briefly estimate the computational requirements of a ``brute force'' acoustic simulation of a room. It is generally accepted that audio signals require a 20 kHz bandwidth. Since sound travels at about a foot per millisecond (see §B.7.14 for a more precise value), a 20 kHz sinusoid has a wavelength on the order of 1/20 feet, or about half an inch. Since, by elementary sampling theory, we must sample faster than twice the highest frequency present in the signal, we need ``grid points'' in our simulation separated by a quarter inch or less. At this grid density, simulating an ordinary 12'x12'x8' room in a home requires more than 100 million grid points. Using finitedifference (Appendix D) or waveguidemesh techniques (§C.14,Appendix E) [518,396], the average grid point can be implemented as a multiplyfree computation; however, since it has waves coming and going in six spatial directions, it requires on the order of 10 additions per sample. Thus, running such a room simulator at an audio sampling rate of 50 kHz requires on the order of 50 billion additions per second, which is comparable to the threesource, twoear simulation of Fig.3.1. However, scaling up to a 100'x50'x20' concert hall requires more than 5 quadrillion operations per second. We may conclude, therefore, that a finegrained physical model of a complete concert hall over the audio band is prohibitively expensive. The remainder of this chapter will be concerned with ways of reducing computational complexity without sacrificing too much perceptual quality.Perceptual Aspects of Reverberation
Artificial reverberation is an unusually interesting signal processing problem because, as discussed in the previous sections, the ``obvious'' methods based on physical modeling or inputoutput modeling are too expensive computationally for most applications. This leads to the question of what are the perceptually important aspects of reverberation, and how can these be provided by efficient computational structures.Perception of Echo Density and Mode Density
The reverberation problem can be greatly simplified without sacrificing perceptual quality. For example, it can be shown^{4.3}that for typical rooms, the echo density increases as , where is time. Therefore, beyond some time, the echo density is so great that it can be modeled as some uniformly sampled stochastic process without loss of perceptual fidelity. In particular, there is no need to explicitly compute multiple echoes per sample of sound. For smoothly decaying late reverb (the desired kind), an appropriate random process sampled at the audio sampling rate will sound equivalent perceptually. Similarly, it can be shown^{4.4}that the number of resonant modes in any given frequency band increases as frequency squared, so that above some frequency, the modes are so dense that they are perceptually equivalent to a random frequency response generated according to some statistics. In particular, there is no need to explicitly implement resonances so densely packed that the ear cannot hear them all. In summary, we see that, based on limits of perception, the impulse response of a reverberant room can be divided into two segments. The first segment, called the early reflections, consists of the relatively sparse first echoes in the impulse response. The remainder, called the late reverberation, is so densely populated with echoes that it is best to characterize the response statistically in some way. Section 3.3 discusses methods for simulating early reflections in the reverberation impulse response. Similarly, the frequency response of a reverberant room can be divided into two segments. The lowfrequency interval consists of a relatively sparse distribution of resonant modes, while at higher frequencies the modes are packed so densely that they are best characterized statistically as a random frequency response with certain (regular) statistical properties. Section 3.4 describes methods for synthesizing hiqh quality late reverberation.Perceptual Metrics for Ideal Reverberation
Some desirable controls for an artificial reverberator include [218] desired reverberation time at each frequency
 signal power gain at each frequency
 ``clarity'' = ratio of impulseresponse energy in early reflections to that in the late reverb
 interaural correlation coefficient at left and right ears
Energy Decay Curve
For measuring and defining reverberation time , Schroeder introduced the socalled energy decay curve (EDC) which is the tail integral of the squared impulse response at time :Energy Decay Relief
The energy decay relief (EDR) is a timefrequency distribution which generalizes the EDC to multiple frequency bands [215]:Early Reflections
The ``early reflections'' portion of the impulse response of a reverberant environment is often taken to be the first 100ms or so [314]. However, for greater accuracy, it should be extended to the time at which the reverberation reaches its asymptotic statistical behavior.^{4.5} Since the early reflections are relatively sparse and span a relatively short time, they are often implemented using tapped delay lines (TDL).^{4.6} If the computation is affordable, it is best to spatialize the early reflections [248] so that they come from the right directions in 3D space. It is known [61] that the early reflections have a strong influence on spatial impression, i.e., the listener's perception of the listeningspace shape. Figure 3.3 shows a general schematic of a reverberator with separate implementations of early and late reverberation. The taps on the TDL may include lowpass filtering for simulation of air absorption. While the late reverb logically begins when the early reflections end, as implemented in Fig.3.3, it may be more costeffective in practice to feed the ``late reverb'' unit from an earlier tap (or set of taps) from the TDL, thus overlapping them somewhat. This can help when the late reverberator needs time to build up to full density. It is often the case that early reflections can be worked into the latereverberation simulation. For example, usually there are long delay lines in which the input signal can be summed at various points, thereby implementing a transposed tapped delay line (see §2.5.2). Good concert halls are observed to have to have stereorecorded impulse responses that quickly ``lateralize'', with a smooth decay and overall duration of approximately 1.9 seconds [48].Late Reverberation Approximations
Desired Qualities in Late Reverberation
From a perceptual standpoint, the main qualities desired of a good latereverberation impulse response are a smooth (but not too smooth) decay, and
 a smooth (but not too regular) frequency response.
Schroeder Allpass Sections
Manfred Schroeder's original papers on the use of allpass filters for artificial reverberation [417,412,152,153] started a lively thread of research which continues to the present. For many years thereafter, digital reverberation algorithms were designed along the lines suggested by Schroeder using delay lines, comb filters, and allpass filterselements described in Chapter 2. There was even specialpurpose hardware developed to implement these structures efficiently in real time [393]. Today, these elements continue to serve as the basis for commercial devices for artificial reverberation and related effects [104]. They are also still typically used in software for artificial reverberation [86]. We will see some examples starting in §3.5 below. Schroeder's suggested use of allpass filters was especially brilliant because there is nothing in nature to suggest them. Instead, he recognized the conceptual and practical utility of separating the coloration of reverberation from its duration and density aspects. While Schroeder's 1961 paper is entitled ``Colorless Artificial Reverberation,'' there is no such thing as colorless (exactly allpass) reverberation in the real world. However, it makes sense as an idealization of natural reverb. Colorless reverberation is an idealization only possible in the ``virtual world''. In Schroeder's original work, and in much work which followed, allpass filters are arranged in series, as shown in Fig.3.4.Nested Allpass Filters
Another common method for increasing the density of an allpass impulse response is to nest two or more allpass filters, as described in §2.8.2 and shown in Fig.2.32 on page . In general, a nested allpass filter is created when one or more of its delay elements is replaced by another allpass filter. As we saw in §2.8.2, firstorder nested allpass filters are equivalent to lattice filters. This equivalence implies that any order transfer function (any poles and zeros) may be obtained from a linear combination of the delay elements of nested firstorder allpass filters, since this is a known property of the lattice filter [297]. In general, any delayelement or delayline inside a stable allpassfilter can be replaced with any stable allpassfilter, and the result will be a stable allpass.Schroeder Reverberators
The subject of artificial reverberation was initiated in the early 1960s by Manfred Schroeder [417,412]. Early Schroeder reverberators consisted of the following elements [412]: A series connection of several allpass filters
 A parallel bank of feedback comb filters
 A mixing matrix
denotes a Schroeder allpass section with delay length samples and coefficients (see Fig.2.30 and associated discussion),
denotes a feedback comb filter with delay length and coefficient (diagrammed in Fig.2.24), and MM in Fig.3.5 denotes the mixing matrix
MM
which can be efficiently implemented using four adders and two
negations:


``There are about 15 large response peaks in every 100 cps [Hz] interval for a room with 1 sec reverberation time. Thus, one might hope that if an artificial reverberator has a comparable number of response peaks it might sound just as good as a real room. We have been able to confirm this expectation by subjective evaluations of the responses of reverberators consisting of several comb filters ... connected in parallel. For a delay of 0.04 sec, the number of response peaks per 100 cps [Hz] is 4. Thus, between 3 and 4 comb filters in parallel ... with incommensurate delays, are required to approximate the number of peaks in the frequency reponse of a room having a reverberation time of T[60] = 1 sec. Also, the open loop gain of the comb filters should not exceed about 0.85 or 1.4 dB to keep the response fluctuations from being excessive.''Thus, one may choose the combfilter delayline lengths more or less arbitrarily, and then use enough of them in parallel (with mutually prime delayline lengths) to achieve a perceptually adequate fluctuation density in the frequencyresponse magnitude. In [412], four such delays are chosen between 30 and 45 ms, and the corresponding feedback coefficients are set to give the desired overall decay time. The delay lengths shown in Fig.3.7 were optimized by ear by John Chowning (and perhaps others at CCRMA) for an audio sampling rate of kHz. Finally, for multichannel listening, Schroeder suggested [412] a mixing matrix at the reverberator output. The goal of the mixing matrix is to bring out any number of uncorrelated audio channels of reverberation (for any number of output speakers) [153, p. 111112].
Example Schroeder Reverberators
Additional example Schroeder Reverberators, drawn from CCRMA software listings, are shown in Figures 3.6 and 3.7. The notation used in the figures is explained above in Equations (3.23.3).Freeverb
A more recently developed Schroeder reverberator is ``Freeverb''  a public domain C++ program by ``Jezar at Dreampoint'' used extensively in the freesoftware world. It uses four Schroeder allpasses in series and eight parallel SchroederMoorer filteredfeedback combfilters (§2.6.5) for each audio channel, and is said to be especially well tuned.Freeverb Main Loop
The C++ code for the main processing loop of Freeverb is shown in Fig.3.9. Notice that it sums the two stereo input channels to create a mono signal that is fed to the reverberator, which then computes a stereo output signal.
void revmodel::processreplace(float *inputL, float *inputR, float *outputL, float *outputR, long numsamples, int skip) { float outL,outR,input; int i; while(numsamples > 0) { outL = outR = 0; input = (*inputL + *inputR) * gain; // Accumulate comb filters in parallel for(i=0; i<numcombs; i++) { outL += combL[i].process(input); outR += combR[i].process(input); } // Feed through allpasses in series for(i=0; i<numallpasses; i++) { outL = allpassL[i].process(outL); outR = allpassR[i].process(outR); } // Calculate output REPLACING anything already there *outputL = outL*wet1 + outR*wet2 + *inputL*dry; *outputR = outR*wet1 + outL*wet2 + *inputR*dry; // Increment sample pointers, allowing for interleave // (if any) inputL += skip; // For stereo buffers, skip = 2 inputR += skip; outputL += skip; outputR += skip; } } 
LowpassFeedback Comb Filter
Inspection of comb.h in the Freeverb source shows that Freeverb's ``comb'' filter is more specifically a lowpassfeedbackcomb filter (LBCF^{4.11}§2.6.5). It is constructed using a delay line whose output is lowpassfiltered and summed with the delayline's input. The particular lowpass used in Freeverb is a unitygain onepole lowpass having the transfer functionFreeverb Allpass Approximation
In Eq.(3.2) we defined the allpass notation byConclusions
Since Freeverb is a SchroederMoorer reverberator, and such reverbs have been around since the 1970s, its relatively recent success underscores the value of careful parameter tuning (typically by ear, but automatic optimizations are possible). The idea of synthesizing rightchannel processing from leftchannel processing to obtain ``stereo spreading'' by enlarging all the delay lines by a fixed amount appears to have been introduced by Freeverb.FDN Reverberation
Feedback Delay Networks (FDN) were introduced earlier in §2.7. An example is shown in Fig.2.29 on page . After a brief historical summary, this section will cover some practical considerations for the use of FDNs as reverberators.History of FDNs for Artificial Reverberation
Feedback delay networks were first suggested for artificial reverberation by Gerzon [156], who proposed an ``orthogonal matrix feedback reverberation unit''. He noted that individual feedback comb filters yielded poor quality, but that several such filters could sound good when crosscoupled. An ``orthogonal matrix feedback'' around a parallel bank of delay lines was suggested as a means of obtaining maximally rich crosscoupling. He was especially concerned with good stereo spreading of the reverberation at a time when most artificial reverberators sought merely to decorrelate the reverberation in each output channel. Later, and apparently independently, Stautner and Puckette [473] suggested a specific fourchannel FDN reverberator and gave general stability conditions for the FDN. They proposed the feedback matrixChoice of Lossless Feedback Matrix
As mentioned in §3.4, an ``ideal'' late reverberation impulse response should resemble exponentially decaying noise [314]. It is therefore useful when designing a reverberator to start with an infinite reverberation time (the ``lossless case'') and work on making the reverberator a good ``noise generator''. Such a starting point is ofen referred to as a lossless prototype [153,430]. Once smooth noise is heard in the impulse response of the lossless prototype, one can then work on obtaining the desired reverberation time in each frequency band (as will be discussed in §3.7.4 below). In reverberators based on feedback delay networks (FDNs), the smoothness of the ``perceptually white noise'' generated by the impulse response of the lossless prototype is strongly affected by the choice of FDN feedback matrix as well as the (ideally mutually prime) delayline lengths in the FDN (discussed further in §3.7.3 below). Following are some of the better known feedbackmatrix choices.Hadamard Matrix
A secondorder Hadamard matrix may be defined byHouseholder Feedback Matrix
One choice of lossless feedback matrix for FDNs, especially nice in the case, is a specific Householder reflection proposed by Jot [217]:where can be interpreted as the specific vector about which the input vector is reflected in dimensional space (followed by a sign inversion). More generally, the identity matrix can be replaced by any permutation matrix [153, p. 126]. It is interesting to note that when is a power of 2, no multiplies are required [430]. For other , only one multiply is required (by ). Another interesting property of the Householder reflection given by Eq.(3.4) (and its permuted forms) is that an matrixtimesvector operation may be carried out with only additions (by first forming times the input vector, applying the scale factor , and subtracting the result from the input vector). This is the same computation as physical wave scattering at a junction of identical waveguides (§C.8). An example implementation of a Householder FDN for is shown in Fig.3.11. As observed by Jot [153, p. 216], this computation is equivalent to parallel feedback comb filters with one new feedback path from the output to the input through a gain of . A nice feature of the Householder feedback matrix is that for , all entries in the matrix are nonzero. This means every delay line feeds back to every other delay line, thereby helping to maximize echo density as soon as possible. Furthermore, for , all matrix entries have the same magnitude:
Householder Reflections
For completeness, this section derives the Householder reflection matrix from geometric considerations [451]. Let denote the projection matrix which orthogonally projects vectors onto , i.e.,Most General Lossless Feedback Matrices
As shown in §C.15.3, an FDN feedback matrix is lossless if and only if its eigenvalues have modulus 1 and its eigenvectors are linearly independent. A unitary matrix is any (complex) matrix that is inverted by its own (conjugate) transpose:Triangular Feedback Matrices
An interesting class of feedback matrices, also explored by Jot [216], is that of triangular matrices. A basic fact from linear algebra is that triangular matrices (either lower or upper triangular) have all of their eigenvalues along the diagonal.^{4.13} For example, the matrixChoice of Delay Lengths
Following Schroeder's original insight, the delay line lengths in an FDN ( in Fig.3.10) are typically chosen to be mutually prime. That is, their prime factorizations contain no common factors. This rule maximizes the number of samples that the lossless reverberator prototype must be run before the impulse response repeats. The delay lengths should be chosen to ensure a sufficiently high mode density in all frequency bands. An insufficient mode density can be heard as ``ringing tones'' or an uneven amplitude modulation in the late reverberation impulse response.Mean Free Path
A rough guide to the average delayline length is the ``mean free path'' in the desired reverberant environment. The mean free path is defined as the average distance a ray of sound travels before it encounters an obstacle and reflects. An approximate value for the mean free path, due to Sabine, an early pioneer of statistical room acoustics, isMode Density Requirement
A guide for the sum of the delayline lengths is the desired mode density. The sum of delayline lengths in a lossless FDN is simply the order of the system :Prime Power DelayLine Lengths
When the delayline lengths need to be varied in real time, or interactively in a GUI, it is convenient to choose each delayline length as an integer power of a distinct prime number [457]:
round
where is the desired length in samples. That is, can be
simply obtained by rounding
to the
nearest integer (max 1). The primepower delayline length
approximation is then of course
Achieving Desired Reverberation Times
A lossless prototype reverberator, as in Fig.3.10 when , has all of its poles on the unit circle in the plane, and its reverberation time is infinity. To set the reverberation time to a desired value, we need to move the poles slightly inside the unit circle. Furthermore, due to air absorption (§2.3,§B.7.15), we want the highfrequency poles to be more damped than the lowfrequency poles [314]. As discussed in §2.3, this type of transformation can be obtained using the substitutionwhere denotes the filtering per sample in the propagation medium (a lowpass filter with gain not exceeding 1 at all frequencies).^{4.14}Thus, to set the FDN reverberation time to at frequency , we want propagation through samples to result in attenuation by dB, i.e.,
Solving for , the propagation attenuation persample, gives
The last form comes from ln, where denotes the time constant of decay (time to decay by ) [451], i.e.,
Series expanding and assuming samples ( seconds) provides the practically useful approximation
Conformal Map Interpretation of Damping Substitution
The relation [Eq.(3.7)] can be written down directly from [Eq.(3.5)] by interpreting Eq.(3.5) as an approximate conformal map [326] which takes each pole , say, from the unit circle to the point . Thus, the new pole radius is approximately , where the approximation is valid when is approximately constant between the new pole location and the unit circle. To see this, consider the partial fraction expansion [449] of a proper thorder lossless transfer function mapped to :Damping Filters for Reverberation Delay Lines
In an FDN, such as the one shown in Fig.3.10, delays appear in long delayline chains . Therefore, the filter needed at the output (or input) of the th delay line is (replace by in Fig.3.10).^{4.15} However, because is so close to in magnitude, and because it varies so weakly across the frequency axis, we can design a much lowerorder filter that provides the desired attenuation versus frequency to within psychoacoustic resolution. In fact, perfectly nice reverberators can be designed in which is merely first order for each [314,217].DelayLine Damping Filter Design
Let denote the desired reverberation time at radian frequency , and let denote the transfer function of the lowpass filter to be placed in series with the th delay line which is samples long. The problem we consider now is how to design these filters to yield the desired reverberation time. We will specify an ideal amplitude response for based on the desired reverberation time at each frequency, and then use conventional filterdesign methods to obtain a loworder approximation to this ideal specification. In accordance with Eq.(3.6), the lowpass filter in series with a length delay line should approximateThis is the same formula derived by Jot [217] using a somewhat different approach. Now that we have specified the ideal delayline filter in terms of its amplitude response in dB, any number of filterdesign methods can be used to find a loworder which provides a good approximation to satisfying Eq.(3.9). Examples include the functions invfreqz and stmcb in Matlab. Since the variation in reverberation time is typically very smooth with respect to , the filters can be very low order.
FirstOrder DelayFilter Design
The firstorder case is very simple while enabling separate control of lowfrequency and highfrequency reverberation times. For simplicity, let's specify and , denoting the desired decaytime at dc () and half the sampling rate ( ). Then we have determined the coefficients of a onepole filter:Orthogonalized FirstOrder DelayFilter Design
In [217], firstorder delayline filters of the formdenotes the ratio of reverberation time at half the sampling rate divided by the reverberation time at dc.^{4.16}
Multiband DelayFilter Design
In §3.7.5, we derived firstorder FDN delayline filters which can independently set the reverberation time at dc and at half the sampling rate. However, perceptual studies indicate that reverberation time should be independently adjustable in at least three frequency bands [217]. To provide this degree of control (and more), one can implement a multiband delayline filter using a generalpurpose filter bank [370,500]. The output, say, of each delay line is split into bands, where is recommended, and then, from Eq.(3.6), the gain in the th band for a length delayline can be set toSpectral Coloration Equalizer
In the previous section, a ``graphical equalizer'' was used to set the reverberator decay time independently in each spectral band slice. While this gives much control over decay time, there is no control over the initial spectral gain in each band. Therefore, another good place for a graphical equalizer is at the reverberator input or output. Such an equalizer allows control of the initial spectral coloration of the reverberator. In the example of Fig.3.10, a spectral coloration equalizer is most efficiently applied to the input signal , before entering the FDN (but after splitting off the direct signal to be scaled by and added to the output), or the output of in Fig.3.10.Tonal Correction Filter
Let denote the component of the impulse response arising from the th pole of the system. Then the energy associated with that pole isFDNs as Digital Waveguide Networks
As discussed in §C.15, the FDN using a Householderreflection feedback matrix is equivalent to a network of digital waveguides intersecting at a single scattering junction [463,464,385]. The wave impedance in the th waveguide is simply , the th element of the axisofreflection vector . The choice corresponds to all of the waveguides having the same impedance (the ``isotrophic junction'' case).FDN Reverberators in Faust
The Faust example reverb_designer.dsp brings up a FDN reverberator in which the signal out of each delay line is split into five bands so that can be controlled independently in each band. The 16 delayline lengths are distributed exponentially between a minimum and maximum length set by two min/maxlength sliders, but rounded to the nearest integerpower of a distinct prime, as introduced above in §3.7.3). The FDN reverberator is implemented in Faust's effect.lib. The bandsplitting is carried out by the filterbank function in Faust's filter.lib. The Faust function filterbank(order,freqs) implements a filter bank having the needed properties using Butterworth lowpass/highpass bandsplitting arranged in a dyadic tree (normally a good choice for audio filter banks). That is, the whole spectrum is split at the highest crossover frequency, the lowpass region is then split into two bands at the next crossover frequency down, and so on, splitting the lowpass band at each stage in the dyadic tree [455,500]. The number of poles in each Butterworth lowpass/highpass filter is specified by order, and freqs contains a list of desired crossover frequencies separating the bands. A certain amount of dispersion is also introduced, since the filter bank is causal and delayequalized (so that the bands may be summed without phase cancellation artifacts at the band edges). Also note that the lower bands are effectively produced by higher order filters than the upper bands. When the reverberation time is longer than the dispersion delay, the dispersion should not be audible as such, although it can affect the ``sound'' of the reverberation. In general, however, artificial reverberators normally benefit from additional allpass dispersion. Figure 3.12 shows the block diagram of a FDN reverberator made from Faust's reverb_designer.dsp by changing 16 to 4. Figure 3.13 shows the Faust block diagram of the associated Hamard matrix multiplication. As it shows, multiplication by a Hadamard matrix can be implemented (ignoring the normalizing scale factor) as a series of block sums and differences (often called butterflies or shufflers) in which the block size decreases by a factor of 2 each stage. Figures for the remaining components of the reverberator may be perused via the shell command faust2firefox reverb_designer.dsp followed by clicking on the blocks in the browser.ZitaRev1
A FOSS^{4.17} reverberator that combines elements of Schroeder (§3.5) and FDN reverberators (§3.7) is zitarev1,^{4.18}written in C++ for Linux systems by Fons Adriaensen. A Faust version of the zitarev1 stereomode functionality is zita_rev1 in Faust's effect.lib. A highlevel block diagram appears in Fig.3.14.ZitaRev1 DelayLine Filters
In zitarev1, the damping filter for each delay line consists of a lowshelf filter [449],^{4.19}in series with a unique firstorder lowpass filter that sets the highfrequency to be half that of the middleband at a particular frequency (specified as ``HF Damping'' in the GUI). Since the filter is constrained to be a lowpass, for , i.e., the decay time gets shorter at higher frequencies. Viewing the resulting damping filter as a threeband filter bank (§3.7.5), let and denote the desired band gains at dc and ``middle frequencies'', respectively.^{4.20} Then the low shelf may be set for a desired dcgain of , and its input (or output) signal multiplied by to obtain the resulting filterFurther Extensions
Schroeder's original structures for artificial reverberation were comb filters and allpass filters made from two comb filters. Since then, they have been upgraded to include specific early reflections and persample airabsorption filtering (Moorer, Schroeder), precisely specified frequency dependent reverberation time (Jot), and a nearly independent factorization of ``coloration'' and ``duration'' aspects (Jot). The evolution from comb filters to feedback delay networks (Gerzon, Stautner, Puckette, Jot) can be seen as a means for obtaining greater richness of feedback, so that the diffuseness of the impulse response is greater than what is possible with parallel and/or series comb filters. In fact, an FDN can be seen as a richly crosscoupled bank of feedback comb filters whenever the diagonal of the feedback matrix is nonzero. The question then becomes what aspects of artificial reverberation have not yet been fully addressed?Spatialization of Reverberant Reflections
While we did not go into the subject here, the early reflections should be spatialized by including a headrelated transfer function (HRTF) on each tap of the earlyreflection delay line [248].^{4.21} Some kind of spatialization may be needed also for the late reverberation. A true diffuse field (§3.2.1) consists of a sum of plane waves traveling in all directions in 3D space. Since we do not know how to achieve this effect using current systems for reverberation, the typical goal is to simply extract uncorrelated outputs from the reverberation network and feed them to the various output channels, as discussed in §3.5. However, this is not ideal, since the resulting sound field consists of wavefronts arriving from each of the speakers, and it is possible for the reverberation to sound like it is emanating from discrete speaker locations. It may be that spatialization of some kind can better fool the ear into believing the late reverberation is coming from all directions.Distribution of Mode Frequencies
Another way in which current reverberation systems are ``artificial'' is the unnaturally uniform distribution of resonant modes with respect to frequency. Because Schroeder, FDN, and waveguide reverbs are all essentially a collection of delay lines with feedback around them, the modes tend to be distributed as the superposition of the resonant modes of feedback comb filters. Since a feedback comb filter has a nearly harmonic set of modes (see §2.6.2), aggregates of comb filters tend to provide a uniform modal density in the frequency domain. In real reverberant spaces, the mode density increases as frequency squared, so it should be verified that the uniform modes used in a reverberator are perceptually equivalent to the increasingly dense modes in nature. Another aspect of perception to consider is that frequencydomain perception of resonances actually decreases with frequency. To summarize, in nature the modes get denser with frequency, while in perception they are less resolved, and in current reverberation systems they stay more or less uniform with frequency; perhaps a uniform distribution is a good compromise between nature and perception? At low frequencies, however, resonant modes are accurately perceived in reverberation as boosts, resonances, and cuts. They are analogous to early reflections in the time domain, and we could call them the ``early resonances.'' It is interesting that no system for artificial reverberation except waveguide mesh reverberation (of which the author is aware) explicitly attempts precise shaping of the lowfrequency amplitude response of a desired reverberant space, at least not directly. The lowfrequency response is shaped indirectly by the choice of early reflections, and the use of parallel combfilter banks in Schroeder reverberators serves also to shape the lowfrequency response significantly. However, it would be possible to add filters for shaping more carefully the lowfrequency response. Perhaps a reason for this omission is that hall designers work very hard to eliminate any explicit resonances or antiresonances in the response of a room. If uneven resonance at low frequencies is always considered a defect, then designing for a maximally uniform mode distribution, as has been discussed for the highfrequency modes, would be ideal also at low frequencies. Quite the opposite situation exists when designing ``smallbox reverberators'' to simulate musical instrument resonators [428,203]; there, the lowfrequency modes impart a characteristic timbre on the lowfrequency resonance of the instrument (see Fig.3.2).Digital Waveguide Reverberators
It was mentioned in §3.7.8 above that FDNs can be formulated as special cases of Digital Waveguide Networks (DWN) (see Appendix C for a fuller development of DWNs). Specifically, an FDN is obtained from a DWN consisting of a single scattering junction (§C.15). It follows that the DWN paradigm provides a more generalized framework in which to pursue further improvements of reverberation architecture. For example, when multiple FDNs are embedded within a single DWN, it becomes possible to richly crosscouple them in an energycontrolled manner in order to create richer recursive structures than either alone. General DWNs were proposed for artificial reverberation in [430,433].The Digital Waveguide Mesh for Reverberation
A special case of digital waveguide networks known as the digital waveguide mesh has also been proposed for use in artificial reverberation systems [396,518]. As discussed in §2.4, a digital waveguide (bidirectional delay line) can be considered a computational acoustic model for traveling waves in opposite directions. A mesh of such waveguides in 2D or 3D can simulate waves traveling in any direction in the space. As an analogy, consider a tennis racket in which a rectilinear mesh of strings forms a pseudomembrane. A major advantage of the waveguide mesh for reverberation applications is that wavefronts are explicitly simulated in all directions, as in real reverberant spaces. Therefore, a true diffuse field can be developed in the late reverberation. Also, the echo density grows with time and the mode density grows with frequency in a natural manner for the 2D and 3D mesh. Finally, the lowfrequency modes of the reverberant space can be simulated very precisely (for better or worse). The computational cost of a waveguide mesh is made tractable relative to more conventional finitedifference simulations by (1) the use of multiplyfree scattering junctions and (2) very coarse meshes. Use of a coarse mesh means that the ``physical modeling'' aspects of the mesh are only valid at low frequencies. As practical matter, this works out well because the ear cannot hear mode tuning errors at high frequencies. There is no error in the mode dampings in a lossless reverberator prototype, because the waveguide mesh is lossless by construction. Therefore, the only errors relative to an ideal simulation of a lossless membrane or space are (1) mode tuning error, and (2) finite band width (cut off at half the sampling rate). The tuning error can be understood as due to dispersion of the traveling waves in certain directions [518,399]. Much progress has been made on the problem of correcting this dispersion error in various mesh geometries (rectilinear, triangular, tetrahedral, etc.) [521,398,399]. See §C.14 for an introduction to the digital waveguide mesh and a few of its properties.Time Varying Reverberators
In real rooms, thermal convention currents cause the propagation path delays to vary over time [58]. Therefore, for greater physical accuracy, the delay lines within a digital reverberator should vary over time. From a more practical perspective, time variation helps to break up and obscure unwanted repetition in the late reverberation impulse response [430,104].Next Section:
DelayLine and Signal Interpolation
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Acoustic Modeling with Digital Delay