Array di microfoni

(1)

Array di microfoni

A. Farina, A. Capra

(2)

SPEAKERS ARRAYS FOR A PHYSICAL MODELLING PIANO SPEAKERS ARRAYS FOR A PHYSICAL MODELLING PIANO

3D Techniques:

• Wave Field Synthesis

• Beam Forming

• CrossTalk Cancellation GOAL: the 3D sound of a real

“grancoda” piano

(3)

Spatialization Effects (1/4)

1. Stereo:

Is the “reference” technique.

2 . Recursive Ambiophonic Crosstalk Elimination (RACE):

- Cancellation of the signal coming from the R speaker and arriving to the Left ear using, in the L speaker, a copy of that signal with a proper delay and a proper attenuation

- The cancellation signal arrives to the Right ear: it needs to be cancelled with another copy coming from the R

speaker…

- …

… and so on

(4)

3. Beam Forming:

2 plane waves from +30° and -30°.

30°

Spatialization Effects (2/4)

Speaker arrays generate sound fields

simulating sound sources placed to a

finite or infinite distance. This effect is

created by means of different gains

and delays.

(5)

4. Convolution with real IRs :

- Points of percussion: F#1, G#3, C6, F#7.

Spatialization Effects (3/4)

5. Gains and delays :

- 4 point sources placed at a fixed distance from the listener.

- Calculation of the delay

from source to speaker.

(6)

Piano recording in anechoic room Piano recording in anechoic room

Impulse Responses of the sound board

(7)

Neumann stereo mics

Neumann Dummy Head Eigenmike®

31 Bruel&Kjaer microphones

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MICROPHONE ARRAYS: TYPES AND PROCESSING MICROPHONE ARRAYS: TYPES AND PROCESSING

Linear Array Planar Array Spherical Array

Processing Algorithm

(9)

General Approach General Approach

• Whatever theory or method is chosen, we always start with M microphones, providing M signals x

_m

, and we derive from them V signals y

v

• And, in any case, each of these V outputs can be expressed by:

processor M inputs

V outputs

 



 ^M

m

v m m

v t x t h t

y

1 , ( )

) ( )

(

(10)

Traditional approaches Traditional approaches

• The processing filters h

_m,v

are usually

computed following one of several, complex mathematical theories, based on the solution of the wave equation (often under certaing simplifications), and assuming that the

microphones are ideal and identical

• In some implementations, the signal of each

microphone is processed through a digital

filter for compensating its deviation, at the

expense of heavier computational load

(11)

Traditional Spherical Harmonics approach Traditional Spherical Harmonics approach

Spherical Harmonics (H.O.Ambisonics)

Virtual microphones

A fixed number of “intermediate” virtual microphones is computed, then the

dynamically-positioned virtual microphones are obtained by linear combination of these intermediate signals.

(12)

Novel approach Novel approach

• No theory is assumed: the set of h

_m,v

filters are derived directly from a set of impulse response measurements, designed according to a least-squares principle.

• In practice, a matrix of impulse responses is measured, and the matrix has to be numerically inverted (usually employing some regularization technique).

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

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

diffractions, reflections, etc.)

(13)

The microphone array impulse responses c

_m,d

, are measured for a number of D incoming directions.

We get a matrix C of measured impulse responses for a large number P of directions

m=1…M mikes d=1…D sources

c_ki

Novel approach Novel approach

 





 







D , M d

, M 2

, M 1

, M

D , m d

, m 2

, m 1

, m

D , 2 d

, 2 2

, 2 1

, 2

D , 1 d

, 1 2

, 1 1

, 1

c ...

c c

...

c ...

c c

...

c ...

c c

c ...

c c

C

(14)

The virtual microphone which we want to synthesize must be specified in the same D directions where the impulse responses had been measured. Let’s choose a high-order cardioid of order n as our target virtual

microphone.

This is just a direction-dependent gain.

The theoretical impulse response

coming from each of the D directions is:

Target Directivity Target Directivity

   

ⁿ

Q

n

 ,   0 . 5  0 . 5  cos(  )  cos(  )

 ^ ^   ^



_d

,

d

Q

p

(15)

Novel approach Novel approach

m = 1…M microphones

d = 1…D directions

Applying the filter matrix H to the measured impulse responses C, the system should behave as a virtual microphone with wanted directivity

h₁(t) h₂(t)

h_M(t)

p

_d

(t)

Target function

c

_1,d

(t)

A_M,v

δ(t)

A_1,v δ(t)

A_2,v δ(t)









M



m

d m

d

m

h p d D

c

1

,

1 ..

But in practice the result of the filtering will never be exactly equal to the prescribed functions p_d…..

c

_2,d

(t)

c

_M,d

(t)

(16)

Novel approach Novel approach

We go now to frequency domain, where convolution becomes simple multiplication at every frequency k, by taking an N-point FFT of all those impulse responses:

We now try to invert this linear equation system at every frequency k, and for every virtual microphone v:

 





 

 



0 .. / 2

..

1

, ,

,

k N

D P d

H C

M m

d k

m k

d m

   

 

_k ^DxV_DxM

k MxV

C H  P

This over-determined system doesn't admit an exact solution, but it is possible to find an approximated solution with the Least Squares method

(17)

Least-squares solution Least-squares solution

We compare the results of the numerical inversion with the theoretical response of our target microphones for all the D directions, properly delayed, and sum the squared

deviations for defining a total error:

The inversion of this matrix system is now performed adding a regularization parameter , in such a way to minimize the total error (Nelson/Kirkeby approach):

It revealed to be advantageous to employ a frequency-dependent regularization parameter _k.

P

     

 

_k _MxD

 

_k _DxM _k

 

_MxM

k j MxD DxV

k k MxV

I C

C

e Q

H C









 

^





*

(18)

Spectral shape of the regularization Spectral shape of the regularization

parameter parameter  

_k_k

• At very low and very high frequencies it is advisable to increase the value of  .

_H

_L

(19)

It is possible to compute just once the following term:

Then, whenever a new set filters is required, this is generated simply applying to R the gains Q of the target microphone:

FIR filters realtime synthesis algorithm:

Thanks to Hermitian symmetry properties, a real-FFT algorithm can be employed

Real-time synthesis of the filters h Real-time synthesis of the filters h

   

 

_k _MxD

 

_k _DxM _k

 

_MxM

k j k MxD

k MxD

I C

C

e R C







  ^





*

  ^H

_k _MxV

^   ^R

_k _MxD

^   ^Q

_DxV

Time-domain windowing

[R_k]_MxD

[Q_k]_DxV

[H_k]_MxV

[h_n]_MxV

N-point real-IFFT

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Critical aspects

• LOW frequencies: wavelength longer than array width - no phase difference between mikes - local approach provide low spatial

resolution (single, large lobe) - global approach simply fails (the linear system becomes singular)

• MID frequencies: wavelength comparable with array width -with local approach secondary lobes arise in spherical or plane wave detection (negligible if the total bandwidth is sufficiently wide) - the global approach works fine, suppressing the side lobes, and providing a narrow spot.

• HIGH frequencies: wavelength is shorter than twice the average mike spacing (Nyquist limit) - spatial undersampling - spatial

aliasing effects – random disposition of microphones can help

the local approach to still provide some meaningful result - the

global approach fails again

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Linear array

• 16 omnidirectional mikes mounted on a 1.2m aluminium beam, with exponential spacing

• 16 channels acquisition system:

2 Behringer A/D converters + RME Hammerfall digital sound card

• Sound recording with Adobe Audition

• Filter calculation, off-line processing and visualization with Aurora plugins

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Linear array - calibration

• The array was mounted on a rotating table, outdoor

• A Mackie HR24 loudspeaker was used

• A set of 72 impulse responses was measured employing Aurora plugins under Adobe Audition (log sweep method) - the sound card controls the rotating table.

• The inverse filters were designed with the local approach (separate inversion of the 16 on-axis responses, employing Aurora’s “Kirkeby4” plugin)

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Linear array - polar plots

8000 Hz

-30 -25 -20 -15 -10 -5 0 0

5 10 1520 2530

35 40

45 50

55 60

65 70

75 80 85 90 95 100 105 110 115 120 125 130 135 140 150145 160155 170165 175 180 190185 200195 210205 215 220 225 230 235 240 245 250 255 260 265 270 275 280 285

290 295

300 305

310

315320325330335340345350355

250 Hz

-30 -25 -20 -15 -10 -5 0 0

5 10 1520 2530

3540 45

50 55

60 65

70 75

80 85 90 95 100 105 110 115 120 125 130 135 145140 155150 165160 175170 180 190185 200195 210205 220215 225 230 235 240 245 250 255 260 265 270 275 280 285

290 295

300 305

310

315320325330335340345350355

500 Hz

-30 -25 -20 -15 -10 -5 0 0

5 10 1520 25

3035 40

45 50

55 60

65 70

75 80 85 90 95 100 105 110 115 120 125 130 135 140 150145 160155 170165 175 180 190185 200195 205 215210 220 225 230 235 240 245 250 255 260 265 270 275 280 285

290 295

300 305

310 315

320

325330335340 345350 355

1000 Hz

-30 -25 -20 -15 -10 -5 0 0

5 10 1520 25

3035 40

45 50

55 60

65 70

75 80 85 90 95 100 105 110 115 120 125 130 135 140 150145 160155 170165 175 180 190185 200195 205 215210 220 225 230 235 240 245 250 255 260 265 270 275 280 285

290 295

300 305

310 315

320

325330335340 345350 355

2000 Hz

-30 -25 -20 -15 -10 -5 0 0

5 10 1520 25

3035 40

45 50

55 60

65 70

75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 150 160155 170165 175 180 190185 200195 205 210 215 220 225 230 235 240 245 250 255 260 265 270 275 280 285

290 295

300 305

310 315

320

325330335340 345350 355

4000 Hz

-30 -25 -20 -15 -10 -5 0 0

5 10 1520 2530

35 40

45 50

55 60

65 70

75 80 85 90 95 100 105 110 115 120 125 130 135 140 150145 160155 170165 175 180 190185 200195 210205 215 220 225 230 235 240 245 250 255 260 265 270 275 280 285 290

295 300

305 310

315320325330335340345350355

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Linear array - practical usage

• The array was mounted on an X-Y scanning apparatus

• a Polytec laser vibrometer is mounted along the array

• The system is used for mapping the velocity and sound pressure along a thin board of “resonance” wood (Abete della val di Fiemme, the wood

employed for building high-quality musical instruments)

• A National Instruments board controls the step motors through a Labview

interface

• The system is currently in usage at IVALSA (CNR laboratory on wood, San Michele all’Adige, Trento, Italy)

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Linear array - practical usage

• The wood panel is excited by a small piezoelectric transducer

• When scanning a wood panel, two types of results are obtained:

• A spatially-averaged spectrum of either radiated pressure, vibration velocity, or of their product (which provides an estimate of the radiated sound power)

• A colour map of the radiated pressure or of the vibration velocity at each

resonance frequency of the board

(26)

Linear array - test results (small loudspeaker)

- 5 0 - 4 0 - 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 4 0 5 0

2 5 0 H z

- 5 0 - 4 0 - 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 4 0 5 0

5 0 0 H z

- 5 0 - 4 0 - 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 4 0 5 0

1 0 0 0 H z

- 5 0 - 4 0 - 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 4 0 5 0

2 0 0 0 H z

- 5 0 - 4 0 - 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 4 0 5 0

4 0 0 0 H z

- 5 0 - 4 0 - 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 4 0 5 0

8 0 0 0 H z

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Linear array - test results (rectangular wood panel)

0 1 0 2 0 3 0 4 0 5 0

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0

4 6 8 H z

0 1 0 2 0 3 0 4 0 5 0

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0

4 6 9 H z

0 1 0 2 0 3 0 4 0 5 0

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0

1 3 5 9 H z

0 1 0 2 0 3 0 4 0 5 0

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0

1 3 1 2 H z

SPL (dB) velocity (m/s) SPL (dB) velocity (m/s)