- Transfer-Function Models
- A Physical Modeling Approach
- Perceptual Aspects of Reverberation
- Early Reflections
- Late Reverberation
- Feedback Delay Networks (FDN)

- The simulation of room reverberation ideally involves two transfer functions per sound source per listener (one for each ear). The tranfer functions or filter representations will change if anything in the room changes.
**Figure 9:**Transfer function approach to reverberation simulation for three sources and one listener. - For the three source, one listener setup depicted in Fig. 9, the output signals would be computed via six convolutions:

(1) *h*_{ij}[*n*] is an FIR filter representation of the impulse response from source*j*to ear*i*and*M*_{ij}is the length of the filter. - For impulse responses of one second (appropriate when the
*t*_{60}= 1 second) and a sample rate*f*_{s}= 50 kHz, each filter would require 50,000 multiplies and additions per sample or 2.5 billion multiply-adds per second. For three sources and two listening points (ears), this corresponds to 30 billion operations per second. - In addition to being very computationally demanding, this approach requires new filter representations whenever the room setup changes. In general, it is difficult to implement a flexible reverberation control scheme using convolution-based approaches.

- A ``distributed'' physical modeling approach to a reverberant space would allow for dynamic modifications of listener and source positions during an acoustic simulation.
- However, a brute force acoustic simulation of a room response using three-dimensional physical modeling techniques would require nearly 150 million ``mesh'' grid points to simulate a room of only 4 x 4 x 3 meters at a sample rate of 50 kHz.
- In addition, current three-dimensional modeling techniques are plagued by dispersion errors that would limit the quality of this approach. It is possible to use warping techniques to minimize the dispersion errors but this would significantly increase the already prohibitive computational burden.

- Above some frequency, the mode density (which increases as
*f*^{2}) of a reverberation response becomes so high that it can be approximated by a random frequency distribution. - Beyond some time, the echo density (which increases as
*t*^{2}) of a reverberation response becomes so high that it can be approximated by a random time distribution. - Based on perceptual limits, the impulse response of a reverberant space can be divided into two segments:
- The beginning of the impulse response consists of distinct, relatively sparse, early reflections.
- The remainder of the impulse response, called the late reverberation, consists of densely-packed echoes which become impossible to distinguish in time.

- The frequency response of a reverberant space can likewise be divided into two segments:
- The low-frequency region consists of a relatively sparse distribution of resonant modes.
- Higher-frequency modes are packed so densely that they are best characterized by a random frequency distribution with certain statistical properties.

- Parametric controls for an artificial reverberator should include:
*t*_{60}in at least three frequency bands- ``clarity'' (ratio of impulse response energy in early reflections to that in the late reverb section)
- inter-aural correlation coefficient at the left and right ears

- Early reflections, within the first 100 milliseconds or so, are typically implemented using tapped delay lines (suggested by Schroeder (1970) and implemented by Moorer (1979)).
**Figure 10:**Early reverberation implemented with a tapped delay line, followed by a late reverberation processing block. - Early reflections should be calculated for a given geometry and spatialized.
- The delay-line tap outputs should be scaled in proportion to propagation distance.
- Most room surfaces are not perfectly flat, resulting in diffuse scattering. Thus, attempts to exactly reproduce the response of a given room via techniques such as ray tracing are generally unsuccessful.

- A good late reverberation should have a smooth decay and a smooth frequency response.
- Some fluctuation in the short-term energy is needed to achieve a natural sound (Blesser, 2001; Dattorro, 1997).
- Moorer's ideal late reverb: exponentially decaying white noise. But it would be better to say exponentially decaying ``colored'' noise, since the high-frequency energy should decay faster than the low-frequency energy.
- Schroeder (1962) suggested the use of parallel comb filters and cascaded allpass filters to synthesize reverberation.
- Allpass filters produce frequency-dependent time shifts, which help diffuse the sound. For this reason, Schroeder allpass sections are sometimes referred to as impulse expanders or impulse diffusers.
- The gain values are typically set around
*g*= 0.7. The delay-line lengths*M*_{i}should be mutually prime and span successive orders of magnitude.**Figure 12:**Impulse response of three cascaded Schroeder allpass sections (*g*= 0.7 and*M*_{i}= [113, 337, 1051]). - The impulse response, calculated with the Matlab script allpass.m, of three cascaded Schroeder allpass sections is shown in Fig. 12.
- The feedback comb filters provide coloration and the delay-line lengths are set to mutually prime values.
- The STK classes
`PRCRev`,`JCRev`, and`NRev`implement Schroeder reverberators of various complexities. In particular:`PRCRev`implements two series allpass units and two parallel comb filters.`JCRev`implements three series allpass units, four parallel comb filters, and two decorrelation delay lines in parallel at the output.`NRev`implements six parallel comb filters, three series allpass units, a lowpass filter, another allpass filter in series, followed by two allpass filters in parallel at the output.

- Figure 14 illustrates an example FDN reverberator using three delay lines proposed by Jot (1992).
- An FDN can be seen as a vector feedback comb filter, with
*N*feedback ``channels'' (*N*=3 in Fig. 14). - The ``mixing matrix'' provides diffusion by ``scattering'' energy amongst the
*N*channels. Assuming decay control is handled by the*g*_{i}coefficients, this matrix should be ``lossless''. - To achieve frequency-dependent decay control, the
*g*_{i}coefficients can be replaced by low-order digital filters. - The ``tonal correction'' filter
*E*(*z*) is a low-order filter which serves to equalize modal energy amongst the three bands. - The delay-line lengths are generally chosen to be mutually prime. System ``tuning'' remains a manual, trial and error process.

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