REVERBERATION: COMBOLUTION & ALGORITHMS

J. Dario R. Barbosa 311275087 (SID)

Digital Audio Systems, DESC9115, Semester 1 2012 Graduate Program in Audio and Acoustics Faculty of Architecture, Design and Planning, The University of Sydney

ABSTRACT One of the most interesting and fundamental phenomenon in acoustics is Reverberation. Reverberation is how sounds propagate from the source to the listener. Focusing in enclosure spaces, when a signal is emitted by a Figure 2. Simple comb filter. sound source a torrent of echoes from different surfaces in the room strikes the listener’s ear, producing an impression of 1.2. ARTIFICIAL REVERBERATORS space. During this trip the radiated sound is modified by the environment, interacting with objects, surfaces, physical Since the 1950’s engineers and acousticians have tried to parameters and geometric characteristics of the room. Such develop reverberation devices capable to simulate sound reflections give to the listener an impression of space, position, propagation in enclosures. The two most popular ways to create orientation, directional characteristics of the sound source, size artificial reverberation is by convolution and by algorithms. and shape of the room [1, 2, 3, 4]. ‘Magnetic tape reverberator’ introduced in 1961 by John Artificial reverberation can be created by convolution and Kellner is another way; please refer to reference [6] for further by algorithms. information.

1. INTRODUCTION Reverberation by convolution shown in Figure 3, is the mathematical process to create a third signal from two other signals, one signal is the sound and the other signal is the 1.1. ACOUSTIC BACKGROUND impulse response of the reverberation enclosure we want to simulate. Convolution is a FIR based system. Reverberation can be divided in the frequency spectrum in three sections: direct sound, early reflections and echo density also named late reverberation; Figure 1.

Figure 3. Convolution

Using algorithms is another method to create artificial reverberation. These algorithms are IIR based system. Figure 1. Direct sound, early reflections and echo density in the frequency spectrum.

The reverberation time is defined by W. C. Sabine as the 2. CONVOLUTION time interval in which reverberation level drops by 60dB [5]. Direct sound is the sound that travels from the source to the Reverberation can be classified as linear invariant system; listener. Early reflections are reflections that arrive between 50 therefore it can be described by its input response and can be and 80 ms after the direct signal. The time between direct sound implemented by means of a Finite Impulse Response (FIR) and early reflections is named Pre-Delay or Initial Time Gap. filter [7].

Considering sound pressure, the particle velocity of air is in There are two ways to create an impulse response signal. phase with pressure for long distances and in quadrature for One is by exiting the room with a pure tone signal which short distances. If the source is at long distance we consider the changes continuously from low to high frequencies creating a condition of propagation of plane waves. In short distances the full frequency spectrum and capturing its response by means of air velocity and pressure are in quadrature and therefore energy microphones refer to Figure 4, and the second one is by doesn’t radiate. Therefore, we will focus on plane waves. designing on a computer the impulse response of the room.

The harmonic series of plane waves can be reproduced by a comb filter refer to Figure 2, by means of a feedback delay line.

Figure 5. All pass filter

The z transform of the all pass filter is given by

− = Figure 4. Exiting a room with a pure signal to capture the 1 − impulse response of the room. All pass filters has a flat magnitude response, since poles and zeros have conjugate reciprocal locations, giving a flat The number of coefficients in the FIR filter determines its frequency response. order and those coefficients are the samples of the impulse response of the room. The reverberation time T60 for one comb filter can be calculated by Implementing this type of reverberation demands a lot of computational operations and therefore it needs more time to 3 produce the signal response. And time means money which is = the main reason of why it is very expensive. log 1⁄ To define a desired reverberation time for a network of

parallel comb filters every gain has to be set according to the following equation. 3. ALGORITHMS

3.1. M. R. Schroeder = 10 In 1961 M. R. Schroeder introduce a reverberation IIR The stability of the current and the following reverberators based system consisting of four recursive comb filters in is guaranteed if g < 1. parallel and in series with two all pass filters. This system allows a dense impulse response and a flat frequency response. The all pass filters increase the echo density and do not 3.2. MOORER’S REVERBERATOR introduce coloration (timbre) unless the delay time is greater than the integration time of the ear, i.e. about 50m [8] Moore enhance Schroeder’s reverberation algorithm with several improvements [4]. To simulate early reflections Moorer The z transform of the comb filter is inserted a FIR delay line. Another improvement was to insert a one pole low pass filter into the feedback loop of each filter to simulate the absorption of the air; shown in Figure 6. All pass filter in time response can give a metallic character to the = 1 − sound, or add some roughness and granularity. Where m is the delay in samples and g is the feedback gain. The frequency response of a comb filter shows n periodic peaks that correspond to the poles frequencies while the distance between two peaks can be calculated with: . The ∆ = ⁄ echo density Dt and modal density Df for one comb filter is Figure 6. Low pass filter inserted on the feedback of a comb give by filter

The z transform of the absorbent comb filter is given by = ∙ = ∙ Where Ts= 1/fs. In the time domain we get an exponential decaying sequence of impulses with a distance of m samples in = 1 − between. For a natural sounding artificial reverberation and The cutoff frequency depends on the distance between the echo density of 1000 reflections per second and a modal density source and the listener. The virtual distance can be calculated of 0.15 eigenfrequencies per hertz are required [8] from the maximum delay length of the comb filters and the The second module is the all pass filter which is a comb speed of sound. For a desire cutoff frequency the α of the low filter with an additional weighted forward path. pass filter results from

= − ∙ − ∙ − −

Where fc is the cutoff frequency and fs is the sampling rate. The special on this reverberator is that it has simple knobs The last modification was to increase the number of comb referred as diffusion that controls the decay rate, high-frequency filters by 6 to get a higher echo and modal density for longer damping, and input signal bandwidth of the reverberated signal. reverberation times. See Annex A.

3.3. GARDNER’S REVERBERATOR 3.5. JOT’S REVERBERATOR

Jot proposed a method for designing a multiple feedback Gardner’s proposed three different algorithms with a ‘reverberant filter’ simulating the late mono reverberation. He common global structure see Figure 7. All pass reverberator proposed to solve the problem of unpleasant resonances by with low pass filtered feedback path and multiple weighted ensuring that all resonances in any frequency band have the output taps. same decay time. His method is designed to simulate the final part of the reverberation decay. His intention is not to simulate the exact content of the room response but to obtain a natural sound reverberation with a cost effective structure. He states that a further investigation is needed to determine how to control the spatial impression. For further information refer to [11].

Figure 7. Gardner’s global structure 4. AKNOWLEDGMENTS Depending on the reverberation time they were categorized on small, medium, and large rooms. The input signal pass I want to thank to my mother for gave me her personal support through a series of all pass sections AP and recirculated through no matter what, as well as to all my friends, especially to my a low pass filter. With the factor g reverberation time and the Nora. degree of the low pas filter can be affected. The cutoff frequency can be calculated by 5. REFERENCES

⁄ [1] Udo Zolzer, ‘DAFX: Digital Audio Effects’, Chapter 6, = 200 Copyright 2002 John Wiley & Sons, Ltd. Where d is the virtual source receiver distance. [2] ‘Auditory Spatial Impression’, Hilmar Lehnert, Ruhr- Universitiit Bochum, Lehrstuhl fiir allge-meine With the all pass reverberator the harshness, buzziness, and Elektrotechnik und Akustik, D-4630 Bochum,Germany metallic sound of the all pass system is smoothed out, possible [3] Aspects of Reverberation Echo Density, Huang, Patty; as a result of the increased of the echo density caused by the Abel, Jonathan S, CCRMA, Stanford University, Presented outermost feedback path. This outermost feedback is a comb at the 123rd Convention 2007 October 5–8 New York, NY filter. Inserting a low pass filter into this feedback we can [4] J.Moorer, “About this Reverberation Business,” Computer simulate the effect of the air absorption. Music Journal 3(2):13-28, 1979 [5] Artificial Reverberation: Comparing algorithms by using For Gardner was impossible to design a single diffuse monaural analysis tools. Presented at the 121ts convention, reverberator to cover all desire reverberation times. Thus, he 2006 October 5-8 San Francisco, CA, USA. designed by trial and error three different reverberators to cover [6] ‘Musical aspects of synthetic reverberation’, John Kellner small, medium and large rooms. The detailed description can be Presented at the 13the anual meeting, oct.9 – oct.13, 1961 found on [9]. [7] Convolution Processing for Realistic Reverberation. Andrew Reilly and David McGrath, Lake DSP, Sydney,Australia. [8] M.Schroeder, “Natural Sounding Artificial Reverberation,” 3.4. DATTORRO’S REVERBERATOR J.Audio Eng. Soc.47(2):219-223, 1962 July. The input signal flows through a pre-delay followed by a [9] W.Gardner, “The Virtual Acoustic Room,” Master’s low pass which defines the bandwidth. After this, four cascade Thesis, MIT Media Lab, 1992 August. all pass filters lead to a buildup of the echo density. Finally the [10] J.Dattorro. “Effect Design - Part 1: Reverberator and other signal enters a structure called ‘Tank’ where it recirculates Filters,”J.Audio Eng. Soc. 45(19):660-684, 1997 indefinitely. The ‘Tank’ consists of two blocks which are cross- September couple. Every block contains two all pass filters, two delays, a [11] J.M.Jot. “Digital delay networks for designing artificial low pass and a decay coefficient for attenuation. The four decay reverberation,”Proc.94th AES Convention, Preprint coefficients determine the rate of the decay. When the decay No.3030, 1991. coefficients are very close to 1 and the damping filter within the Tank is turn off the sound will remain held in the Tank indefinitely. The purpose of the four all pass filters are to decorrelate the incoming signal before reaches the Tank [10].

Annex A

Dattorro’s Reverberator