Reverberation Algorithms

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Reverberation

algorithms

Augusto Sarti



Summaryn

The Reverb Problemn Reverb Perceptionn Acoustic impulse response:

¨ Formation mechanisms

¨ Parameters

n Early Reflections

n Late Reverbn Numerical reverberation algorithms

¨ Schroeder Reverbs

¨ Feedback Delay Network (FDN) Reverberators

¨ Waveguide Reverberators

n Geometrical reverberation algorithms



Impulse response

n The sounds we perceive heavily depend on thesurrounding environment

n Environment-related sound changes are of

convolutive origin (filtering)

¨ Well-modeled by a space-varying impulse response



Direct

signal Early

reflections Reverberations

S

R

S

R

S

R

Time

A m

p l i t u d e

Impulse response



Reverberation tf Function

n Three sources, onelistener (two ears)

n Filters should include

pinnae filtering

n Filters change if anything

in the room changes

(exact model)



Global descriptorsn Energy decay curve (EDC)

¨ Introduced by Schroeder to define reverberation time

¨ It measures the total signal energy remaining in thereverberator’s impulse response at time t

¨ It decays more smoothly than the impulse response, therefore itworks better than the amplitude’s envelope for defining thereverberation time

¨ In reverberant environments a large amount of the total energy is

contained in the last portion of the impulse responsen Reverberation time

}60)0()(:{60 dB EDC t EDC t T −==



Global descriptors



EDR of a violin body



n In the room’s transfer function we can single out resonant modes

n The spacing between two resonant modes is given by

n which is valid above the threshold frequency

Global descriptors



n Number of echoes in the impulse response before time t

n Derivative of N t :

n Clarity index: ratio btw early reflections energy and latereverberation energy

Global descriptors



Implementation



Moorer reverberator

n accounts for late reverberations by placing anIIR filter after the FIR filter (tapped delay line)



Binaural impulse response

n Our sound perception is affected by our own body

¨ Head Related Transfer Function (HRTF)

Acoustic paths can be

grouped together to

reduce cost



Comb filter



Allpass filter



Steady-state

tones (sinusoids)

really do see the

same gain at

every frequency

in an allpass,

while a comb

filter has widely

varying gains



Comb filters and reverberation time

n The decay between successive samples in comband allpass filters is described by the gain

coefficient gi

n In order for the comb filter’s decay to correspond

to a given reverberation time, we must have



Combination of comb filters

n Single comb filters do notprovide sufficient echo density

n In order to improve the echo

density, we need to combinemultiple comb filters

¨ Cascading comb filters

corresponds to multiplying their

transfer functions

¨ Frequency peaks not shared by all

comb filters are cancelled bymultiplication



Combination of comb filters

n Better to place comb

filters in parallel

¨ Example



Parallel comb filters

n The poles of comb filters are given by

n The poles have the same magnitudes

n The modal density (No. of modes per Hz) is




n Modal density turns out to be the same at allfrequencies, unlike real rooms

n Above a threshold frequency, the modal density

is constant

n The modal density of the comb filters is then setto the modal density above the threshold

frequency




n The echo density of the comb filters isapproximatively given by

n Relating echo density and modal density

provides:



Combination of allpass filters

n Unlike comb filters, allpass filters must becascaded

¨ Multiplying freq. responses corresponds to adding

phase responses



Schroeder’s reverberator (1)



Schroeder’s reverberator

n Delays of the comb and allpass filters are chosen so thatthe ratio of the largest and smallest delay is 1.5 (typically30 and 45 ms)

n The gains gi of the comb filters are chosen to provide adesired reverberation time T

r according to

n Allpass filters delays are set to 5 and 1.7 ms



Feedback Delay Networks…

…

at a glancen Unitary matrix: definition

¨ A matrix is unitary if :

¨ We can also write that a matrix is unitary if

|||||||| uuM =⋅

1|||||||| == MMMMT T



FDN

• Stability of the feedback loop is guaranteed if A = gM where M is an unitarymatrix and |g|<1

• Outputs will be mutually incoherent: we can use the FDN to render the diffusesoundfield with a 4 loudspeaker system

• The early reverbeartions can be simulated by appropriately injecting the inputsignal into the delay lines

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Jot’s reverberator

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The input-output relation of Jot’s reverberator is given by

with and




n System transfer function:

n Zeros:

n Poles:




n Moorer noted that convolving exponentiallydecaying white noise with source signals

produces a very natural sounding

n As a consequence, by introducing absorptive

losses into a lossless prototype, we shouldobtain a natural sounding reverberator

n This is accomplished by associating a gain with

each delay:




n The logarithm of the gain is proportional to the length of the delay:

n The above modification has the effect of replacing z with

z/γ in the transfer function

n The lossless prototype response h[n] will be multiplied byan exponential envelope γn



Modeling the

Environment



Modeling the environment

n Simulate reverberations due to

environment



Motivations

Acoustical environment provides ...n Sense of presence

n Comprehension of space

n Localization of auditory cues

n Selectivity of audio signals (“cocktail party

effect”)



Geometric acoustic modeling

n Spatialize sound by computing reverberationpaths from source to receiver



Similarities to Graphics

n Both model wave propagatation



Differences from Graphics I

n Sound has longer wavelengths than light¨ Diffractions are significant

¨ Specular reflections dominate diffuse reflections

¨ Occlusions by small objects have little effect



Differences from Graphics II

n Sound waves are coherent¨ Modeling phase is important



n Sound travels more slowly than light¨ Reverberations are perceived over time

Differences from Graphics III



Overview of approaches

n Finite element methodsn Boundary element methods

n Image source methods

n Ray tracing

n Beam tracing



Finite element methods

n Solve wave equation over grid-alignedmesh



Boundary element methods

n Solve wave equation over discretizedsurfaces



Boundary Element Trade-offs

n Advantages¨ Works well for low frequencies

¨ Simple formulation



n Disadvantages¨ Complex function stored with each element

¨ Form factors must model diffractions &

specularities

¨ Elements must be much smaller thanwavelength

Boundary Element Trade-offs



Image source methods

n Consider direct paths from “virtualsources”



Image source trade-offs

n Advantages¨ Simple for rectangular rooms



n Disadvantages¨ O(nr ) visibility checks in arbitrary

environments

¨ Specular reflections only

Image source trade-offs



Path tracing

n Trace paths between source andreceiver



Path Tracing Trade-offs

n Advantages¨ Models all types of surfaces and scattering

¨ Simple to implement

Incoming ray

Sampledreverberation

s



Path Tracing Disadvantages

n Disadvantages¨ Subject to sampling errors (aliasing)

¨ Depends on receiver position



Beam Tracing

n Trace beams (bundles of rays) fromsource



Beam Tracing Trade-offs

n Advantages¨ Takes advantage of spatial coherence

¨ Predetermines visible virtual sources



Beam Tracing Disadvantages

n Disadvantages¨ Difficult for curved surfaces or refractions

¨ Requires efficient polygon sorting and

intersection

BSPsCell adjacency graphs



Complex 3D Environments

n Precompute beam tree for stationarysource



Interactive Performance

n Lookup beams containing moving receiver



Summary

n FEM/BEM¨ best for low frequencies

n Image source methods

¨ best for rectangular rooms (very common)

n Path tracing

¨ best for high-order reflections (very common)

n Beam tracing

¨ best for precomputation



Current research in

interactive audio

spatialization



81

Back to the problem

n Path/ray tracingaccording to the laws of geometric optics

n Applications to

¨ Simulation of acoustic

reverberations incomplex environments

¨ Prediction of EMpropagation for wirelesssystems (multipathfading)



82

n Construction of the beam tree through space

subdivision

n Construction of paths through beam tree lookup

Beam tracing



83

Using space subdivision



84

What is missing?

n Traditional beam tracing assumes that thesource be fixed

¨ Every time the source moves, the BT needs tobe rebuilt from scratch (lengthy process basedon space subdivision)

n Is it possible to avoid space subdivision?

n Is it possible to settle all visibility issues inadvance (irrespective of the source

location)?n Is it ultimately possible to build the BT

through a simple lookup process?



85

Reformulating the problem

n Define environment’s visibility

independently from the source’s

location

n Compute the environment’s visibilityn Build the beam tree using

¨ Visibility info

¨ Source’s location

n Build the paths using

¨ Beam tree

¨ Receiver’s location



86

Environment’s characterizationn Sources and Receivers

¨ Assumed to be point-like

n Reflectors

¨ Oriented surface of a reflecting wall

n A reflecting wall defines two reflectors

n Assumed as flat

n Identified by an indexn Byproducts:

¨ Beams

n Compact bundle of rays originated by the same source

n Identified by a source (real or virtual) and the illuminatedportion of a reflector

¨ Active reflectors

n That portion of a reflector illuminated by a beam

n Identified by a beam and a reflector



87

Visibility

n Visibility function¨ Function that associates the index of the visible

reflector to a viewpoint and a viewing direction

¨ Piece-wise constant function that takes on values in theparameter space that characterizes viewpoint andviewing direction

n Visibility function from a reflector ¨ Visibility function where viewpoints are constrained on

the pts of the reflector

n Environment’s visibility description

¨ Set of the visibility functions associated to all the

environment’s reflectorsn M reflecting walls => 2M visibility functions



88

Defining the parameter spacen

Parameter space: viewpoint and viewing direction¨ If the point lies on a reflector

n 4D parameter space in the 3D case

n 2D parameter space in the 2D case

¨ Reflector’s normalization

n affine transformation (rigid motion + scaling) of the

geometric space that remaps the reflector onto thesegment that goes from (0,-1) to (0,1), with reflectingsurface facing x≥0

¨ This way viewpoint and viewing direction can bedescribed by the eq. y = a x + b, where -1≤b≤1describes the point on the reflector and a the viewing

directionn Parameter space: (a,b)



89

Visibility region

n The visibility region of a given reflector w.r.t. a reference reflector is the region of

the parameter space (a,b) that

corresponds to viewpoints on the

reference reflector from which the givenone results as visible

¨ Due to occlusions, this region can be empty or

made of a set of convex polygons

n The visibility region of reflector i is the

region where the visibility function is equal

to i



90

Visibility region

n A generic reflector can be described by x = e t + f

y = g t + h

0≤t ≤1

Substituting in y=ax+b we obtaing t + h = a (e t + f) + b, 0≤t ≤1

-1≤b≤1

Visibility region:

Intersection btw a bundle of rays (a beam inparameter space) and the strip -1≤b≤1

(f,h)

(e+f,g+h)



91

Examples



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Visibility regionn Potential visibility region: visibility region with no other reflectors

n Potential visibility regions may overlapn Actual visibility region is contained within the potential one

¨ Overlaps must be resolved considering occlusions

n Approach for evaluating visibility function

¨ Compute potential visibility regions

¨ Resolve overlaps and identify actual visibility regions¨ Label actual visibility regions



93

Resolving overlaps

n When two potential visibility regionsoverlap, the corresponding reflectors

exhibit a partial occlusion w.r.t. the

reference reflector

n Who occludes who decides which regioneats which on the overlap

n This can be done by tracing a sample ray

within the overlapping region



94

Parameter space (dual space)



95

Reflectors in the dual space



96

Normalized dual space



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Building a beam tree from visibility

n Evaluating the global visibility of theenvironment corresponds to building

one visibility function per reflector

¨ This corresponds to constructing and

labeling all the actual visibility regions for each reflector

n All this ignores the location of the source

n Given source location and visibility,

how do we build the beam tree?



98

Sources in parameter space

n A source in parameter space is a line (dual of a pt)

n Source and active portion of a reflector define a beam¨ The branching of a beam is defined by the intersection btw

the line and the actual visibility regions



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Beam tracing

n Given a beam reflected by the i-threflector, use visibility to to determine itsbranching in sub-beams (one per visiblereflector)

¨ Determine virtual source location in thewarped space corresponding to i-th reflector

¨ Determine illuminated portion of reflector andthe corresponding “narrowed” reference strip

¨ Scan actual visibility regions over the line

corresponding to the source in parameter space

¨ Update beam tree



100

Beam tracing



101

Computational efficiency



102

Computational efficiency



103



104

Modeling diffraction

n Use geometric theory of diffractionn Diffraction modeled by placing sources

(and the relative beam trees) at diffracting

wedges

n Beam trees computed in advance jointly

with visibility information



A comparison btw imp.

responsesSim.

Meas.



Comparison

n Parameters:

¨ EDC

¨ Early Decay Time (EDT): time that imp. resp. takes to dim down of

10 dB.

¨ Center Time (CT): centroid of squared impulse response

n Imp. Resp. envelope:



EDC

Simulated refl. only

Sim. of refl. + diffr. Complete simulation

recorded



EDT

Simulated refl. only

Sim. of refl. + diffr. Complete simulation\

recorded



Centre TimeSimulated refl. only

Sim. of refl. + diffr. Complete simulation

recorded



110

Auralization



Active Beamshaping



112

Rendering beams Physical approach: WFS

Huygens principle

data-based (needs

wavefield acquisition)

Works on wavefronts

Geometric appraoch:

Beam Tracing

Implements general

solution according to

geometric propagation

principles

Boundary conditions

become components

of the implementation

G l



113

Goal Reconstruction of an arbitrary source (arbitrary

radiation function) in an arbitrary location using an array

of speaker in close range

Si l t f



114

Signal transfer

bhG =

=

Nx1

Mx1NxM

Matrix form

Example (1) central beam



115

Example (1) – central beam

n M = 16;

n f = 700Hz

n Δy = not uniform

n Gaussian mask

Example 2 skewed beam



116

Example 2 – skewed beam

n M = 64;

n f = 700Hz

n Δy = 10 cm

n Maschera Gauss

Rendered beam



Rendered beam

Wid b d t i



118

Wideband extension

¨ Instead of setting constraints at a singlefrequency, we apply them to multiplefrequencies (wideband minimization)

¨ 4 parameters:n F, No. of frequencies where we minimize

n M, No. of speakers

n N, No. of angles

n T, No. of taps of the filter

E l 3 k d b



119

Example 3 – skewed beam

Interface



Interface

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Testing in a dry room



121

Testing in a dry room

15-speaker non-uniform array

Multiple audio cards in daisy-chain configuration

8-16-24 synchronized outputs

Results



Resultsn Expected contrasting needs

¨ Low frequencies require extensive arrays

¨ High frequencies require closely-spaced

speakers

¨ Cost constrains limit the No. of speakers

n With 15-16 speakers we do not go beyond

17-18 db of attenuation btw main lobe and

side lobes with a limited frequency range

(300Hz-6kHz)

122

C l i



Conclusions

n Results are comparable to those achievedwith WFS but we control them in a

geometric fashion. Therefore we can

¨ Reconstruct an arbitrary source in an arbitrary

location¨ Combine multiple beams through

superposition principle, therefore it can be

used as a geometric engine for synthesizing

the response of the environment as well (earlyreverberation for spatial impression)

Reverberation Algorithms

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