Array di microfoni

A. Farina, A. Capra

SPEAKERS ARRAYS FOR A PHYSICAL MODELLING SPEAKERS ARRAYS FOR A PHYSICAL MODELLING PIANOPIANO

3D Techniques:• Wave Field Synthesis• Beam Forming• CrossTalk Cancellation

GOAL: the 3D sound of a real “grancoda” piano

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

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.

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.

Piano recording in anechoic roomPiano recording in anechoic room

Impulse Responses of the sound board

Neumann stereo mics

Neumann Dummy Head

Eigenmike®

31 Bruel&Kjaer microphones

MICROPHONE ARRAYS: TYPES AND PROCESSINGMICROPHONE ARRAYS: TYPES AND PROCESSING

Linear Array Planar Array Spherical Array

Processing Algorithm

SIGNAL

PROCESSOR

x1(t)

x2(t)

xM(t)

M inputs

y1(t)

y2(t)

yV(t)

. . . . . . V outputs

. . .

x1(t)

x2(t)

xM(t)

h1,1(t)

h2,1(t)

hM,1(t)

y1(t)

General ApproachGeneral Approach

• Whatever theory or method is chosen, we always start with M microphones, providing M signals xm, and we derive from them V signals yv

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

processor

M inputsV outputs

M

mvmmv thtxty

1, )()()(

Traditional approachesTraditional approaches

• The processing filters hm,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

Traditional Spherical Harmonics Traditional Spherical Harmonics approachapproach

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.

Novel approachNovel approach

• No theory is assumed: the set of hm,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.)

The microphone array impulse responses cm,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

cki

Novel approachNovel approach

D,Md,M2,M1,M

D,md,m2,m1,m

D,2d,22,21,2

D,1d,12,11,1

c...c...cc

..................

c...c...cc

..................

c...c...cc

c...c...cc

C

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 DirectivityTarget Directivity

nnQ )cos()cos(5.05.0,

,dd Qp

Novel approachNovel approach

m = 1…M microphones

d = 1…Ddirections

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)

hM(t)

pd(t)

Target function

c1,d(t)

AM,v

δ(t)

A1,vδ(t)

A2,v

δ(t)

M

mdmdm Ddphc

1, ..1

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

c2,d(t)

cM,d(t)

Novel approachNovel approachWe 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:

2/..0

..1

1,,, Nk

DdPHC

M

mdkmkdm

DxMk

DxVMxVk C

PH

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

Least-squares solutionLeast-squares solutionWe 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

MxMkDxMkMxDk

kjDxVMxDk

MxVkICC

eQCH

*

*

Spectral shape of the regularization Spectral shape of the regularization parameter parameter kk

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

H

L

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 hReal-time synthesis of the filters h

MxMkDxMkMxDk

kjMxDk

MxDkICC

eCR

*

*

DxVMxDkMxVk QRH

Time-domain windowing

[Rk]MxD

[Qk]DxV

[Hk]MxV

[hn]MxV N-pointreal-IFFT

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

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

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)

Linear array - polar plots

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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)

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

Linear array - test results (small loudspeaker)

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Linear array - test results (rectangular wood panel)

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