University of Parma

Angelo Farina Angelo Farina

Industrial Engineering Dept.

Industrial Engineering Dept.

University of Parma

University of Parma - - ITALY ITALY HTTP://

HTTP:// www.angelofarina.it www.angelofarina.it

University of Parma

ROOM IMPULSE ROOM IMPULSE

RESPONSES AS TEMPORAL RESPONSES AS TEMPORAL

AND SPATIAL FILTERS

AND SPATIAL FILTERS

ƒ Traditional time-domain measurements with omnidirectional transducers

ƒ Advanced impulse response measurement methods

ƒ Directional transducers, the first attempts of spatial analysis

ƒ Orthonormal decomposition of the spatial properties in spherical harmonics: the Ambisonics method

ƒ The reciprocity principle: directive microphones and directive sources

ƒ Generalization of higher-order spherical harmonics representation of both source and receiver directivity

ƒ Joining time and space: from Einstein’s view to a

comprehensive data structure representing the acoustical transfer function of a room

ƒ Practical usages of measured (or numerically simulated) temporal-spatial impulse response

Topics

Basic

Basic sound sound propagation propagation scheme scheme

Omnidirectional receiver Direct Sound

Reflected Sound

Point Source

Direct Sound

Reverberant tail

Traditional

Traditional measurement measurement methods methods

ƒƒ PulsivePulsive sources: sources: ballonsballons, , blankblank pistolpistol

Modern

Modern electroacoustical electroacoustical methods methods

ƒ ƒ The sound is The sound is generated generated by by means means of of an an omnidirectional omnidirectional loudspeaker

loudspeaker

ƒ ƒ The The signal signal is is computer- computer -generated generated

ƒ ƒ The The same same computer is computer is also also employed employed for for recording recording the the room room ’response ’ response by by means means of of one one or more omnidirectional or more omnidirectional microphones

microphones

ƒ ƒ Also Also directive directive microphones microphones can be can be used: used : binaural binaural, , figure figure- -of of -eight - eight

ƒ ƒ Different Different types types of of test signals test signals have have been been developed developed , providing , providing good good immunity immunity to to background noise background noise and easy deconvolution and easy deconvolution of of the impulse the impulse response: response :

► MLS (Maximum Lenght Sequence, pseudo-random white noise)

► TDS (Time Delay Spectrometry, which basically is simply a linear sine sweep, also known in Japan as “stretched pulse”)

► ESS (Exponential Sine Sweep)

ƒ ƒ Each Each of of these these test signals test signals can be can be employed employed with with different different deconvolution

deconvolution techniques techniques , resulting , resulting in a number in a number of of “different “ different” ” measurement

measurement methods methods

ƒ ƒ Due to Due to theoretical theoretical and practical and practical considerations, the considerations , the preference preference is is nowadays

nowadays generally generally oriented oriented for for the usage the usage of of ESS with ESS with not- not -circular circular deconvolution

deconvolution

Measurement process

ƒ The desidered result is the linear impulse response of the acoustic propagation h(t). It can be recovered by knowing the test signal x(t) and the measured system output y(t).

ƒ It is necessary to exclude the effect of the not-linear part K and of the background noise n(t).

Not-linear, time variant

system K[x(t)]

Noise n(t)

input x(t)

+

w(t)⊗h(t) distorted signal

w(t)

Edirol FA-101 Firewire sound

card:

10 in / 10 out 24 bit, 192 kHz ASIO and WDM

Hardware: PC and audio interface

Hardware:

Hardware: loudspeaker loudspeaker & & microphone microphone

Dodechaedron loudspeaker Omnidirectional

microphone

Aurora Plugins

Generate MLS Generate MLS

Deconvolve DeconvolveMLSMLS

Generate Sweep Generate Sweep

Deconvolve

DeconvolveSweepSweep

Convolution Convolution

Kirkeby

KirkebyInverse FilterInverse Filter

Speech

Speech TransmTransm. Index. Index

Software

MLS MLS method method

ƒ ƒ X(t) is X(t) is a periodic a periodic binary binary signal

signal obtained obtained with with a a suitable

suitable shift- shift -register register, , configured

configured for for maximum

maximum lenght lenght of of the the period

period. .

N stages

XOR k stages

x’(n)

1 2

L = ^N −

MLS MLS deconvolution deconvolution

ƒ ƒ The The re re - - recorded recorded signal signal y(i) y(i) is is cross cross - - correlated correlated with with the the excitation

excitation signal signal thanks thanks to to a fast Hadamard a fast Hadamard transform. The transform . The result

result is is the the required required impulse impulse response response h(i), if h(i), if the system the system was was linear linear and time and time- -invariant invariant

y 1 M

L h 1

~ ⋅ + ⋅

=

z

Where M is the Hadamard matrix, obtained by permutation of the original MLS sequence m(i)

( )

[ ^{i j 2 mod L} ] ¹

m )

j , i (

M ~ = + − −

MLS MLS example example

Portable PC with 4- channels sound board Original Room

SoundField Microphone

B- format 4- channels signal

(WXYZ)

Measurement of B- format Impulse Responses

MLS excitation signal

Portable PC with additional sound card Room

Microphone

Output signal y Measurement of room

Impulse Response

MLS test signal x Loudspeaker

MLS MLS example example

Exponential Sine Sweep method

ƒ x(t) is a sine signal, which frequency is varied

exponentially with time, starting at f

and ending at f

.

⎥ ⎥

⎥ ⎥

⎦

⎤

⎢ ⎢

⎢ ⎢

⎣

⎡

⎟⎟

⎟ ⎟

⎠

⎞

⎜⎜

⎜ ⎜

⎝

⎛

−

⋅

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

⋅

⋅ π

= ⋅ ^{⎟⎟ ⎠}

⎜⎜ ⎞

⎝

⋅ ⎛

1 e

f ln f

T f

sin 2 )

t (

x ¹

2

f ln f T

t

1

2

1

Test Signal – x(t)

Measured signal - y(t)

ƒƒ The notThe not-linear-linear behaviourbehaviourofofthe loudspeakerthe loudspeaker causescauses manymanyharmonicsharmonics toto appear

appear

Inverse Filter – z(t)

The The deconvolutiondeconvolution ofof the IR isthe IR is obtainedobtained convolvingconvolving the the measuredmeasured signalsignal y(t)

y(t) withwith the inverse filterthe inverse filter z(t) [equalizedz(t) [equalized, , time-time-reversedreversed x(t)]x(t)]

Deconvolution of Log Sine Sweep

The “time reversal mirror” technique is employed: the system’s impulse response is obtained by convolving the measured signal y(t) with the time-reversal of the test signal x(-t). As the log sine sweep does not have a “white” spectrum, proper equalization is required

Test Signal x(t) Inverse Filter z(t)

Result of the deconvolution

The last impulse response is the linear one, the preceding are the harmonics distortion products of various orders

2° 1°

5° 3°

IR IR Selection Selection

ƒƒ AfterAfter the sequencethe sequence ofofimpulseimpulse responsesresponseshashas beenbeen obtained, obtained, itit isis possiblepossible toto selectselectand and insulateinsulatejust just oneoneofof them:them:

ESS ESS example example

Portable PC with 4- channels sound board Original Room

SoundField Microphone

B- format 4- channels signal

(WXYZ)

Measurement of B- format Impulse Responses

MLS excitation signal

Portable PC with additional sound card Room

Microphone

Output signal y Measurement of room

Impulse Response

Sweep test signal x Loudspeaker

ESS example

ESS example

Maximum Length Sequence vs. Sweep

Post processing of impulse responses

ƒƒ A special plugin has been developed for the computation of STI according A special plugin has been developed for the computation of STI according to IEC

to IEC--EN 60268EN 60268--16:200316:2003

Post processing of impulse responses

ƒƒ A special plugin has been developed for performing analysis of acoustical A special plugin has been developed for performing analysis of acoustical parameters according to ISO

parameters according to ISO--33823382

The new AQT plugin for Audition

ƒƒ The new module is still under development and will allow for very fast The new module is still under development and will allow for very fast computation of the AQT (Dynamic Frequency Response) curve from w

computation of the AQT (Dynamic Frequency Response) curve from within ithin Adobe Audition

Adobe Audition

Spatial analysis by directive microphones Spatial analysis by directive microphones

ƒƒ The initial approach was to use directive microphones for gathering some The initial approach was to use directive microphones for gathering some information about the spatial properties of the sound field

information about the spatial properties of the sound field ““as perceived by as perceived by the listener

the listener””

ƒƒ Two apparently different approaches emerged: binaural dummy heads and Two apparently different approaches emerged: binaural dummy heads and pressure

pressure--velocity microphones:velocity microphones:

Binaural Binaural microphone (left) microphone (left)

and and

variable

variable--directivity directivity microphone (right) microphone (right)

“ “ objective objective ” ” spatial parameters spatial parameters

ƒƒ It was attempted to “It was attempted to “quantifyquantify”” the “the “spatialityspatiality”” of a room by means of of a room by means of

“objective“objective”” parameters, based on 2-parameters, based on 2-channels impulse responses measured channels impulse responses measured with directive microphones

with directive microphones

ƒƒ The most famous The most famous “spatial“spatial”” parameter is IACC (Inter Aural Cross parameter is IACC (Inter Aural Cross Correlation), based on binaural IR measurements

Correlation), based on binaural IR measurements

LeftLeft

Right Right

80 ms 80 ms

( )

( ) ( )

( ) ( )

pp_LL(τ(τ))

pp_RR((ττ))

[ ] ( )

[

]

Max IACC

d t p

d p

d t p

p

t _E

ms 80

0 2R ms

80

0 2L ms 80

0

R L

+

−

∈ ρ

= τ

⋅ + τ

⋅ τ

⋅ τ

τ

⋅ + τ

⋅ τ

= ρ

∫

∫

∫

“ “ objective objective ” ” spatial parameters spatial parameters

ƒƒ Other “Other “spatialspatial”” parameters are the Lateral Energy ratios: LE, LF, LFCparameters are the Lateral Energy ratios: LE, LF, LFC

ƒƒ These are defined from a 2-These are defined from a 2-channels impulse response, the first channel is a channels impulse response, the first channel is a standard omni microphone, the second channel is a

standard omni microphone, the second channel is a ““figurefigure--ofof--eighteight”” microphone:

microphone:

( ) ( )

∫ ( )

∫

τ

⋅ τ

τ

⋅ τ

⋅ τ

=

ms 0

o2 ms 80

ms 5

o 8

d h

d h

h LFC

Figure Figure

of 8 of 8 OmniOmni

( )

∫ ( )

∫

τ

⋅ τ

τ

⋅ τ

=

ms 0

o2 ms 80

ms 5

82

d h

d h

LF

hh_oo(τ(τ))

hh₈₈((ττ))

( )

∫ ( )

∫

τ

⋅ τ

τ

⋅ τ

=

ms 0

o2 ms 80

ms 25

82

d h

d h

LE

Robustness of spatial parameters Robustness of spatial parameters

ƒƒ Both IACC and LF depend strongly on the orientation of the microphonesBoth IACC and LF depend strongly on the orientation of the microphones

ƒƒ Binaural and pressure-Binaural and pressure-velocity measurements were performed in 2 velocity measurements were performed in 2 theatres employing a rotating table for turning the microphones theatres employing a rotating table for turning the microphones

(1-LF) Auditorium Parma – Sorgente a sx

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(1-LF) Auditorium Roma (Sala 1200) – Sorgente a sx

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Theatre 1-LF IACC Parma 0.725 0.266 Roma 0.676 0.344

IACC Auditorium Parma - Sorgente a sx

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IACC Auditorium Roma (Sala 1200) - Sorgente a sx

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Are binaural measurents reproducible?

Are binaural measurents reproducible?

ƒƒ Experiment performed in anechoic room Experiment performed in anechoic room --same loudspeaker, same source same loudspeaker, same source and receiver positions, 5 binaural dummy heads

and receiver positions, 5 binaural dummy heads

Are binaural measurents reproducible?

Are binaural measurents reproducible?

ƒƒ 90°90° incidence incidence --at low frequency IACC is almost 1, at high frequency the at low frequency IACC is almost 1, at high frequency the difference between the heads becomes evident

difference between the heads becomes evident

IACCe - 90° incidence

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

31.5 63 125 250 500 1000 2000 4000 8000 16000

Frequency (Hz)

IACCe

B&K4100 Cortex Head Neumann

Are binaural measurents reproducible?

Are binaural measurents reproducible?

ƒƒ Diffuse field Diffuse field --the difference between the heads is now dramaticthe difference between the heads is now dramatic

IACCe - random incidence

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

31.5 63 125 250 500 1000 2000 4000 8000 16000

Frequency (Hz)

IACCe

B&K4100 Cortex Head Neumann

Are LF measurents reproducible?

Are LF measurents reproducible?

ƒƒ Experiment performed in the Auditorium of Parma -Experiment performed in the Auditorium of Parma - same loudspeaker, same loudspeaker, same source and receiver positions, 5 pressure

same source and receiver positions, 5 pressure--velocity microphonesvelocity microphones

Are LF measurents reproducible?

Are LF measurents reproducible?

ƒƒ At 7.5 m distance, the results already exhibit significant scatterAt 7.5 m distance, the results already exhibit significant scatter

Comparison LF - measure 1 - 7.5m distance

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

31.5 63 125 250 500 1000 2000 4000 8000 16000

Frequency (Hz)

LF

Schoeps Neumann Soundfield B&K

Are LF measurents reproducible?

Are LF measurents reproducible?

ƒƒ At 25 m distance, the scatter is even larger....At 25 m distance, the scatter is even larger....

Comparison LF - measure 2 - 25m distance

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

31.5 63 125 250 500 1000 2000 4000 8000 16000

Frequency (Hz)

LF

Schoeps Neumann Soundfield B&K

3D extension of the pressure

3D extension of the pressure- -velocity measurements velocity measurements

ƒƒ The Soundfield microphone allows for simultaneous measurements of the The Soundfield microphone allows for simultaneous measurements of the omnidirectional pressure and of the three cartesian components o

omnidirectional pressure and of the three cartesian components of particle f particle velocity (figure

velocity (figure--ofof--8 patterns)8 patterns)

3D Impulse Response (Gerzon, 1975) 3D Impulse Response (Gerzon, 1975)

Portable PC with 4- channels sound board Original Room

Sound Source

SoundField Microphone

B-format 4-channels signal (WXYZ) Measurement of B-format

Impulse Responses

MLS or sweep excitation signal

Convolution of dry signals with the B-format Impulse

Responses Sound Source

Mono Mic.

B-format Imp. Resp.

of the original room

B-format 4-channels signal (WXYZ) Convolver

Ambisonics decoder

Speaker array in the reproduction room

The Waves project (2003) The Waves project (2003)

ƒƒ The original idea of Michael Gerzon was finally put in practice in 2003, The original idea of Michael Gerzon was finally put in practice in 2003, thanks to the Israeli

thanks to the Israeli--based company WAVESbased company WAVES

ƒƒ More than 50 theatres all around the world were measured, capturing 3D More than 50 theatres all around the world were measured, capturing 3D IRs (4

IRs (4-channels B-channels B--format with a Soundfield microphone)format with a Soundfield microphone)

ƒƒ The measurments did also include binaural impulse responses, andThe measurments did also include binaural impulse responses, and a a circular

circular--array of microphone positionsarray of microphone positions

ƒƒ More details on WWW.ACOUSTICS.NETMore details on WWW.ACOUSTICS.NET

Directivity of transducers Directivity of transducers

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LookLine D200 dodechaedron LookLine D200 dodechaedron

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Directivity of transducers Directivity of transducers

Soundfield ST

Soundfield ST--250 microphone250 microphone

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What about source directivity ? What about source directivity ?

ƒƒ Current 3D IR sampling is still based on the usage of an “Current 3D IR sampling is still based on the usage of an “omnidirectionalomnidirectional”” source

source

ƒƒ The knowledge of the 3D IR measured in this way provide no information The knowledge of the 3D IR measured in this way provide no information about the soundfield generated inside the room from a directive

about the soundfield generated inside the room from a directive source source (i.e., a musical instrument, a singer, etc.)

(i.e., a musical instrument, a singer, etc.)

ƒƒ Dave Malham suggested to represent also the source directivity with a set Dave Malham suggested to represent also the source directivity with a set of spherical harmonics, called O

of spherical harmonics, called O--format format -- this is perfectly reciprocal to the this is perfectly reciprocal to the representation of the microphone directivity with the B

representation of the microphone directivity with the B--format signals format signals (Soundfield microphone).

(Soundfield microphone).

ƒƒ Consequently, a complete and reciprocal spatial transfer function can be Consequently, a complete and reciprocal spatial transfer function can be defined, employing a 4

defined, employing a 4--channels Ochannels O--format source and a 4format source and a 4--channels Bchannels B-- format receiver:

format receiver:

Portable PC with 4in-4out channels sound board O-format 4-channels source

B-format 4-channels microphone (Soundfield)

1st order MIMO impulse response 1st order MIMO impulse response

ƒ ƒ If only spherical harmonics of order 0 and 1 are taken into account, If only spherical harmonics of order 0 and 1 are taken into acco unt, a complete spatial transfer function measurement requires

a complete spatial transfer function measurement requires 16 impulse responses:

16 impulse responses:

{ } ^y [ ] ^h { } ^x

or x

x x x

h h

h h

h h

h h

h h

h h

h h

h h

y y y y

z y x w

zz zy

zx zw

yz yy

yx yw

xz xy

xx xw

wz wy

wx ww

z y x w

⊗

=

⎪ ⎪

⎭

⎪ ⎪

⎬

⎫

⎪ ⎪

⎩

⎪ ⎪

⎨

⎧

⊗

⎥ ⎥

⎥ ⎥

⎥

⎦

⎤

⎢ ⎢

⎢ ⎢

⎢

⎣

⎡

=

⎪ ⎪

⎭

⎪ ⎪

⎬

⎫

⎪ ⎪

⎩

⎪ ⎪

⎨

⎧

ƒ ƒ Once these 16 IRs have been measured, it is possible to compute Once these 16 IRs have been measured, it is possible to compute the response of the room with a source and a receiver having

the response of the room with a source and a receiver having arbitrary directivity patterns, given by the O

arbitrary directivity patterns, given by the O -format source functions - format source functions {r {r

, r , r

, r , r

, r , r

}, and the B- }, and the B -format receiver functions {r format receiver functions {r

, r , r

, r , r

, r , r

}: }:

{ } { } { } ^{r y r} [ ] ^h { } ^s

t _sr = ⊗ = ⊗ ⊗

ƒ ƒ In which also each of {s} and {r} are sets of 4 impulse responses, In which also each of {s} and {r} are sets of 4 impulse response s, representing the frequency

representing the frequency- -dependent directivities of the source dependent directivities of the source and of the receiver

and of the receiver

Limits of the 1

Limits of the 1 ^{st st} - - order method order method

ƒƒ Albeit mathematically elegant and easy to implement with currentlyAlbeit mathematically elegant and easy to implement with currently--existing hardware, existing hardware, the 1

the 1^stst-order method presented here cannot represent faithfully the comp-order method presented here cannot represent faithfully the complex directivity lex directivity pattern of an human voice or of an human ear:

pattern of an human voice or of an human ear:

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345

Real

1st order approx.

W X Y

1.421 0.289 0.000

Limits of the 1

Limits of the 1 ^{st st} - - order method order method

ƒƒ The polar pattern of a binaural dummy head is even more complex,The polar pattern of a binaural dummy head is even more complex,as shown as shown here (1 kHz, right ear):

here (1 kHz, right ear):

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285 290

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315320325330335340345350355

Measured 1st order approx.

W X Y

0.395 -0.015 0.191

How to get better spatial resolution?

How to get better spatial resolution?

ƒƒ The answer is simple: analyze the spatial distribution of both source and receiver by The answer is simple: analyze the spatial distribution of both source and receiver by means of higher

means of higher--order spherical harmonics expansionorder spherical harmonics expansion

ƒƒ Spherical harmonics analysis is the equivalent, in space domain,Spherical harmonics analysis is the equivalent, in space domain,of the Fourier of the Fourier analysis in time domain

analysis in time domain

ƒ

ƒ As a complex time-As a complex time-domain waveform can be though as the sum of a number of domain waveform can be though as the sum of a number of sinusoidal and cosinusoidal functions, so a complex spatial dist

sinusoidal and cosinusoidal functions, so a complex spatial distribution around a ribution around a given notional point can be expressed as the sum of a number of

given notional point can be expressed as the sum of a number of spherical harmonic spherical harmonic functions

functions

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

0 0.5 1 1.5 2 2.5 3

dc Cos(5) Sin(5) Cos(10) Sin(10)

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0 0.5 1 1.5 2 2.5 3

Sum

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Higher

Higher - - order spherical harmonics expansion order spherical harmonics expansion

3 3 ° ° - - order microphone (Trinnov order microphone (Trinnov - - France) France)

ƒ ƒ Arnoud Laborie developed a 24- Arnoud Laborie developed a 24 -capsule compact microphone capsule compact microphone array

array - - by means of advanced digital filtering, spherical ahrmonic by means of advanced digital filtering, spherical ahrmonic signals up to 3

signals up to 3° ° order are obtained (16 channels) order are obtained (16 channels)

4 4 ° ° - - order microphone (France Telecom) order microphone (France Telecom)

ƒ ƒ Jerome Daniel and Sebastien Moreau built samples of 32- Jerome Daniel and Sebastien Moreau built samples of 32 -capsules capsules spherical arrays

spherical arrays - - these allow for extractions of microphone signals these allow for extractions of microphone signals up to 4

up to 4 ° ° order (25 channels) order (25 channels)

Multichannel software for high

Multichannel software for high- - order order

ƒƒ Plogue Bidule can be used as multichannel host software, runningPlogue Bidule can be used as multichannel host software, runninga number of VST a number of VST plugins developed by France Telecom

plugins developed by France Telecom --these include spherical harmonics extraction from these include spherical harmonics extraction from the spherical microphone arrays, rotation and manipulation of th

the spherical microphone arrays, rotation and manipulation of the multichannel Be multichannel B--format format signals, and final rendering either on head

signals, and final rendering either on head--.tracked headphones or on a static array of .tracked headphones or on a static array of loudspeakers (high

loudspeakers (high--order Ambisonics)order Ambisonics)

Verification of high

Verification of high - - order patterns order patterns

ƒƒ Sebastien Moreau and Olivier Warusfel verified the directivity pSebastien Moreau and Olivier Warusfel verified the directivity patterns of the atterns of the 44°-°-order microphone array in the anechoic room of IRCAM (Paris)order microphone array in the anechoic room of IRCAM (Paris)

5 kHz 5 kHz

10 kHz 10 kHz

Frequency extension of the patterns

Frequency extension of the patterns

High High - - order sound source order sound source

ƒƒ University of California Berkeley's Center for New Music and Audio University of California Berkeley's Center for New Music and Audio Technologies (CNMAT) developed a new 120

Technologies (CNMAT) developed a new 120--loudspeakers, digitally loudspeakers, digitally controlled sound source, capable of synthesizing sound emission controlled sound source, capable of synthesizing sound emission according to spherical harmonics patterns up to 5

according to spherical harmonics patterns up to 5°° order.order.

Technical details of high

Technical details of high - - order source order source

ƒƒ Class-Class-D embedded amplifiersD embedded amplifiers

ƒƒ Embedded ethernet interface Embedded ethernet interface and DSP processing

and DSP processing

ƒƒ Long-Long-excursion special excursion special Meyer Sound drivers Meyer Sound drivers

Accuracy of spatial synthesis Accuracy of spatial synthesis

ƒƒ The spatial reconstruction error of a 120-The spatial reconstruction error of a 120-loudspeakers array is frequency loudspeakers array is frequency dependant, as shown here:

dependant, as shown here:

ƒƒ The error is acceptably low over an extended frequency range up to 5The error is acceptably low over an extended frequency range up to 5°°--orderorder

Advanced digital filtering techniques Advanced digital filtering techniques

ƒ ƒ A set of digital filters can be employed for sinthesizing the A set of digital filters can be employed for sinthesizing the

required spatial pattern (spherical harmonis), either when deali required spatial pattern (spherical harmonis), either when dealing ng with a microphone array or when dealing with a loudspeaker array with a microphone array or when dealing with a loudspeaker array

ƒ ƒ Whatever theory or method is chosen, we always start with N Whatever theory or method is chosen, we always start with N input signals x

input signals x

, and we derive from them M output signals y , and we derive from them M output signals y

ƒ ƒ And, in any case, each of these M outputs can be expressed by: And, in any case, each of these M outputs can be expressed by:

i N

1 i

ij

j h x

y ∑

=

⊗

=

processor N inputs

M outputs

Example with a microphone array Example with a microphone array

ƒƒ The sound field is sampled in N points by means of a microphone arrayThe sound field is sampled in N points by means of a microphone array

Is the time-domain sampled waveform of a wave with well defined spatial characteristics, for example:

• a spherical wave centered in a precise emission point P

• a plane wave with a certain direction

• a spherical harmonic referred to a receiver point P

Pre-calculated filters

i N

1 i

ij

j

h x

y ∑

=

⊗

=

x

(t) x

(t) x

(t) x

(t)

y

(t)

Traditional design of digital filters Traditional design of digital filters

ƒ ƒ The processing filters h The processing filters h

are usually computed are usually computed following one of several, complex mathematical following one of several, complex mathematical

theories, based on the solution of the wave theories, based on the solution of the wave

equation (often under certaing simplifications), equation (often under certaing simplifications),

and assuming that the microphones are ideal and and assuming that the microphones are ideal and

identical identical

ƒ ƒ In some implementations, the signal of each In some implementations, the signal of each

microphone is processed through a digital filter microphone is processed through a digital filter

for compensating its deviation, at the expense of for compensating its deviation, at the expense of

heavier computational load

heavier computational load

Novel approach Novel approach

ƒ ƒ No theory is assumed: the set of h No theory is assumed: the set of h

filters are filters are derived directly from a set of impulse response derived directly from a set of impulse response

measurements, designed according to a least measurements, designed according to a least - -

squares principle.

squares principle.

ƒ ƒ In practice, a matrix of filtering coefficients, is In practice, a matrix of filtering coefficients, is formed, and the matrix has to be numerically formed, and the matrix has to be numerically

inverted (usually employing some regularization inverted (usually employing some regularization

technique).

technique).

ƒ ƒ This way, the outputs of the microphone array are This way, the outputs of the microphone array are maximally close to the ideal responses prescribed maximally close to the ideal responses prescribed

ƒ ƒ This method also inherently corrects for transducer This method also inherently corrects for transducer deviations and acoustical artifacts (shielding,

deviations and acoustical artifacts (shielding, diffractions, reflections, etc.)

diffractions, reflections, etc.)

Example: synthesizing 0

Example: synthesizing 0 - - order shape order shape

The microphone array impulse responses c

, are measured for a number of P incoming directions.

…we get the filters to be applied to the microphonic signals from processing the matrix of measured impulse responses

⎥ ⎥

⎥ ⎥

⎦

⎤

⎢ ⎢

⎢ ⎢

⎣

⎡

=

N , P N

, k N

, 2 N

, 1

i , P i

, k i

, 2 i

, 1

2 , P 2

, k 2

, 2 2

, 1

1 , P 1

, k 1

, 2 1

, 1

c c

c c

c c

c c

c c

c c

c c

c c

i=1…N mikes C

k=1…P sources

c

Example: synthesizing 0

Example: synthesizing 0 - - order shape order shape

δ

=

⊗

δ

=

⊗

δ

=

⊗

δ

=

⊗

∑

∑

∑

∑

=

=

=

=

N 1 i

i , P 0

, i N

1 i

i , k 0

, i N

1 i

i , 2 0

, i N

1 i

i , 1 0

, i

c h

...

c h

...

c h

c h

Lets call v

the right-hand vector of known results (they will be different for higher-order shapes).

Once this matrix of N inverse filters are computed (for example, employing the Nelson/Kirkeby method), the output of the

microphone array, synthesizing the prescribed 0th-order shape, will again be simply:

∑ =

⊗

= ^N

1 i

0 , i i

0 x h

y

We design the filters in such a way that the response of the system is the prescribed theoretical function v

for the k-th source (an unit- amplitude Dirac’s Delta function in the case of the example, as the 0th-order function is omnidirectional).

So we set up a linear equation system of P equations, imposing

that:

System

System ’ ’ s least s least - - squares inversion squares inversion

ƒƒ For computing the matrix of N filtering coefficients hFor computing the matrix of N filtering coefficients h_i0i0, a least-, a least-squares squares method is employed.

method is employed.

ƒƒ A “A “total squared errortotal squared error”” εε_tottot is defined as:is defined as:

( )

∑ ∑ = = ⎥ ⎥

⎦

⎤

⎢ ⎢

⎣

⎡ ⊗ −

=

ε ^P

1 k

2 k N

1 i

ki 0

i

tot h c v

ƒ ƒ A set of N linear equations is formed by minimising A set of N linear equations is formed by minimising ε ε

, , imposing that:

imposing that:

) N ...

1 i

( h _{i 0} 0

tot = =

∂ ε

∂