Array di microfoni
A. Farina, A. Capra
SPEAKERS ARRAYS FOR A PHYSICAL MODELLING PIANO
GOAL: the 3D sound of a real
“grancoda” piano
3D Techniques:
• Wave Field Synthesis
• Beam Forming
• CrossTalk Cancellation
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
Spatialization Effects (2/4)
3. Beam Forming:
2 plane waves from +30° and -30°.
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.
30
°
Spatialization Effects (3/4)
4. Convolution with real IRs:
- Points of percussion: F#1, G#3, C6, F#7.
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 room
Impulse Responses of the sound board
Neumann stereo mics
Eigenmike®
31 Bruel&Kjaer microphones
Neumann Dummy Head
MICROPHONE ARRAYS: TYPES AND PROCESSING
Linear Array
Planar Array
x1(t)
M inputs
x2(t)
..
.
xM(t)
x1(t)
x2(t)
..
.
xM(t)
Spherical Array
y1(t)
SIGNAL
PROCESSOR
h1,1(t)
h2,1(t)
hM,1(t)
Processing Algorithm
y2(t)
..
.
yV(t)
y1(t)
V outputs
General Approach
processor
V outputs
M inputs
• 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:
M
y v (t )   xm (t )  hm ,v (t )
m 1
Traditional 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 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.
Novel 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.)
Novel approach
The microphone array impulse responses cm,d , are
measured for a number of D incoming directions.
d=1…D sources
m=1…M mikes
cki
 c1,1 c1,2
c
c 2, 2
 2,1
...
 ...
C
 c m,1 c m,2
 ...
...

c M,1 c M,2
...
c1,d
...
...
c 2, d
...
...
...
...
... c m,d
...
...
...
...
... c M,d ...
c1, D 
c 2, D 

... 

c m, D 
... 

c M, D 
We get a matrix C of measured impulse responses for a large
number P of directions
Target Directivity
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.
Qn ,   0.5  0.5  cos( )  cos( )
n
This is just a direction-dependent gain.
The theoretical impulse response
coming from each of the D directions is:
pd  Qd  ,   
Novel approach
Applying the filter matrix H to the measured impulse responses C, the system should
behave as a virtual microphone with wanted directivity
δ(t)
d = 1…D
directions
δ(t)
c1,d(t)
c2,d(t)
h1(t)
pd(t)
A2,v
A1,v
h2(t)
AM,v
m = 1…M
microphones
cM,d(t)
hM(t)
Target function
δ(t)
M
c
m ,d
 hm

pd
d  1.. D
m 1
But in practice the result of the filtering will never be exactly equal to the prescribed
functions pd…..
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:
M
C
m 1
m ,d ,k
 H m,k  Pd
 d  1..D

k  0.. N / 2
We now try to invert this linear equation system at every frequency k, and for every
virtual microphone v:
H k MxV

P DxV

Ck DxM
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 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:
P
The inversion of this matrix system is now performed adding a regularization
parameter b, in such a way to minimize the total error (Nelson/Kirkeby approach):
H k MxV

C   Q DxV  e

*
Ck MxD  Ck DxM  bk  I MxM
*
k MxD
 jk
It revealed to be advantageous to employ a frequency-dependent regularization
parameter bk.
Spectral shape of the regularization
parameter bk
• At very low and very high frequencies it is
advisable to increase the value of b.
eH
eL
Real-time synthesis of the filters h
It is possible to compute just once the following term:
*

Ck MxD  e jk
Rk MxD  *
Ck MxD  Ck DxM  bk  I MxM
Then, whenever a new set filters is required, this is generated simply applying to R the
gains Q of the target microphone:
Hk MxV  Rk MxD  QDxV
FIR filters realtime synthesis algorithm:
[Rk]MxD
N-point
real-IFFT
[Qk]DxV
Time-domain
windowing
[hn]MxV
[Hk]MxV
Thanks to Hermitian symmetry properties, a real-FFT algorithm can be employed
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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-20
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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)
468 Hz
469 Hz
1359 Hz
1312 Hz
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