Maximum Directivity Beamformer for Spherical ... - Semantic Scholar
Maximum Directivity Beamformer for Spherical-Aperture Microphones Morag Agmon [email protected]
Boaz Rafaely [email protected]
Joseph Tabrikian [email protected]
Electrical & Computer Engineering Ben-Gurion University of the Negev, Beer-Sheva, Israel Oct. 20th, 2009
Abstract
Maximum Directivity Beamformer
Performance of microphone arrays at the high-frequency range is typically limited by aliasing, which is a result of the spatial sampling process. A potential approach to avoid spatial aliasing is by using continuous sensors, in which spatial sampling is not required. The proposed beamforming technique is used to compute the optimal real-valued aperture weighting function for a spherical-aperture microphone. Real-valued aperture weighting functions are required to ensure the realizability of the sensor.
Finds the optimal coefficients dn that maximize the directivity of a spherical-aperture microphone with a given order of N. Directivity [Rafaely, 2005b]: dT bbH d∗ Q= T ∗ (9) d Cd 2 2 2 1 2 C = 4π diag |b0| , 3 |b1| , ..., (2N + 1) |bN | d – weight coefficients vector b – look-direction steering vector
Spherical microphone arrays Drawbacks: ◮ Spatial aliasing, limited high-frequency performance ◮ Highly directional microphone arrays require many microphones and an expensive system for connecting the microphones
Formulation: dopt = arg max Q
|y0|2 = 1
s.t.
d
Simplifying the problem formulation: dopt = arg min dT Cd∗
dT bbH d∗ = 1
s.t.
d
(10)
(11)
Solution for this problem is known... but complex: Figure: mh acoustics em32 Eigenmiker microphone array
Spherical aperture microphone
C−1b dcomplex = T −1 b C b A real-valued solution does not exist, and needs to be fully derived.
(12)
Using Lagrange multipliers method:
Advantages: ◮ Enables 3D processing ◮ Beamforming, create high-directivity beam patterns ◮ Does not suffer from spatial aliasing ◮ Could replace systems with a large number of sensors, reduced realization cost
J , dT Cd + λ dT bbH d − 1
(13)
Taking the gradient of J with respect to d and setting to zero: n o dT C + λdT Re bbH = 0
(14)
Imposing the constraint and simplifying:
Drawbacks: ◮ Weighting cannot be applied with simple electronics ◮ Real-valued weighting is required to ensure realizability
−1
C c dreal = T −1 c C c
A practical method for embedding weight function - varying electrode density [Franc¸ois et al., 2003]:
,
c = Re be
j − 2 ∠(bT C−1b)
(15)
Simulation Examples (a)
EMFi material:
(b)
40
25
20
Directivity
20
Directivity Index [dB]
Magnitude [dB]
30
Real kr=5 Complex kr=5 Real kr=10 Complex kr=10
10
0
15
10
Sensitivity 5
−10
Plane wave response Spherical Fourier Transform [Driscoll & Healy, 1994]: Z 2π Z π m∗ f (θ, φ) Yn (θ, φ) sin θdθdφ fnm = 0
−20 0
5
10 kr
5
10 kr
15
20
Figure: (a) narrowband performance and sensitivity (b) broadband performance
(1)
∞ X n X
40
fnm Ynm (θ, φ)
(2) 30
n=0 m=−n
0 n X
∗ pnm (k , r ) wnm
Magnitude [dB]
Directivity
The spherical-aperture microphone output [Rafaely, 2005a]: Z 2π Z π y (k , r ) = p (k , r , θ, φ) w ∗ (θ, φ) sin θdθdφ =
0 0
20
0
f (θ, φ) =
0 N X
15
(3)
20
10
Sensitivity 0
(4)
−10
n=0 m=−n
∗ – Spherical Fourier transform of p and w respectively. pnm and wnm
Assumptions [Rafaely, 2005a]: 1. Single plane wave 2. Unit amplitude 3. Known direction of arrival 4. Symmetrical beam pattern around the look direction
−20 0
⇒
m∗ pnm (k , r ) = bn (kr ) Yn (Ω0) ∗ = dnYnm (Ωl ) wnm
(5) (6)
15
20
N X
n=0 N X
Conclusions Initial feasibility of spherical-aperture microphones was shown. ◮ No spatial aliasing. ◮ Easier realization than equivalent spherical microphone array. ◮ Simple beamforming techniques. ◮ Aperture weighting by varying electrode density. ◮
References Driscoll, J. R. & Healy, Jr., D. M. (1994). Adv. Appl. Math. 15, 202–250.
Aperture weighting function, and microphone output:
yl (k , r , Θ) =
10 kr
Figure: Maximum directivity beamformer with bounded sensitivity
bn (kr ) – mode gains for a rigid sphere Ω0 – plane wave direction of arrival Ωl – microphone look direction. dn – acts as the only parameter in the weighting function, must be real.
w (Θ) =
5
dn 2n+1 4π Pn (cos Θ)
(7)
Franc¸ois, A., Man, P. D. & Preumont, A. (2003). In 6th National Congress on Theoretical and Applied Mechanics, Ghent, Belgium.
dnbn (kr ) 2n+1 4π Pn (cos Θ)
(8)
Rafaely, B. (2005a). IEEE Trans. on Speech and Audio Processing 13, 135–143.
n=0
Pn – Legendre polynomial of order n Θ – spatial angle between the Ω0 and Ωl
Rafaely, B. (2005b). IEEE Signal Processing Letters 12, 713–716.
Boaz Rafaely [email protected]
Joseph Tabrikian [email protected]
Electrical & Computer Engineering Ben-Gurion University of the Negev, Beer-Sheva, Israel Oct. 20th, 2009
Abstract
Maximum Directivity Beamformer
Performance of microphone arrays at the high-frequency range is typically limited by aliasing, which is a result of the spatial sampling process. A potential approach to avoid spatial aliasing is by using continuous sensors, in which spatial sampling is not required. The proposed beamforming technique is used to compute the optimal real-valued aperture weighting function for a spherical-aperture microphone. Real-valued aperture weighting functions are required to ensure the realizability of the sensor.
Finds the optimal coefficients dn that maximize the directivity of a spherical-aperture microphone with a given order of N. Directivity [Rafaely, 2005b]: dT bbH d∗ Q= T ∗ (9) d Cd 2 2 2 1 2 C = 4π diag |b0| , 3 |b1| , ..., (2N + 1) |bN | d – weight coefficients vector b – look-direction steering vector
Spherical microphone arrays Drawbacks: ◮ Spatial aliasing, limited high-frequency performance ◮ Highly directional microphone arrays require many microphones and an expensive system for connecting the microphones
Formulation: dopt = arg max Q
|y0|2 = 1
s.t.
d
Simplifying the problem formulation: dopt = arg min dT Cd∗
dT bbH d∗ = 1
s.t.
d
(10)
(11)
Solution for this problem is known... but complex: Figure: mh acoustics em32 Eigenmiker microphone array
Spherical aperture microphone
C−1b dcomplex = T −1 b C b A real-valued solution does not exist, and needs to be fully derived.
(12)
Using Lagrange multipliers method:
Advantages: ◮ Enables 3D processing ◮ Beamforming, create high-directivity beam patterns ◮ Does not suffer from spatial aliasing ◮ Could replace systems with a large number of sensors, reduced realization cost
J , dT Cd + λ dT bbH d − 1
(13)
Taking the gradient of J with respect to d and setting to zero: n o dT C + λdT Re bbH = 0
(14)
Imposing the constraint and simplifying:
Drawbacks: ◮ Weighting cannot be applied with simple electronics ◮ Real-valued weighting is required to ensure realizability
−1
C c dreal = T −1 c C c
A practical method for embedding weight function - varying electrode density [Franc¸ois et al., 2003]:
,
c = Re be
j − 2 ∠(bT C−1b)
(15)
Simulation Examples (a)
EMFi material:
(b)
40
25
20
Directivity
20
Directivity Index [dB]
Magnitude [dB]
30
Real kr=5 Complex kr=5 Real kr=10 Complex kr=10
10
0
15
10
Sensitivity 5
−10
Plane wave response Spherical Fourier Transform [Driscoll & Healy, 1994]: Z 2π Z π m∗ f (θ, φ) Yn (θ, φ) sin θdθdφ fnm = 0
−20 0
5
10 kr
5
10 kr
15
20
Figure: (a) narrowband performance and sensitivity (b) broadband performance
(1)
∞ X n X
40
fnm Ynm (θ, φ)
(2) 30
n=0 m=−n
0 n X
∗ pnm (k , r ) wnm
Magnitude [dB]
Directivity
The spherical-aperture microphone output [Rafaely, 2005a]: Z 2π Z π y (k , r ) = p (k , r , θ, φ) w ∗ (θ, φ) sin θdθdφ =
0 0
20
0
f (θ, φ) =
0 N X
15
(3)
20
10
Sensitivity 0
(4)
−10
n=0 m=−n
∗ – Spherical Fourier transform of p and w respectively. pnm and wnm
Assumptions [Rafaely, 2005a]: 1. Single plane wave 2. Unit amplitude 3. Known direction of arrival 4. Symmetrical beam pattern around the look direction
−20 0
⇒
m∗ pnm (k , r ) = bn (kr ) Yn (Ω0) ∗ = dnYnm (Ωl ) wnm
(5) (6)
15
20
N X
n=0 N X
Conclusions Initial feasibility of spherical-aperture microphones was shown. ◮ No spatial aliasing. ◮ Easier realization than equivalent spherical microphone array. ◮ Simple beamforming techniques. ◮ Aperture weighting by varying electrode density. ◮
References Driscoll, J. R. & Healy, Jr., D. M. (1994). Adv. Appl. Math. 15, 202–250.
Aperture weighting function, and microphone output:
yl (k , r , Θ) =
10 kr
Figure: Maximum directivity beamformer with bounded sensitivity
bn (kr ) – mode gains for a rigid sphere Ω0 – plane wave direction of arrival Ωl – microphone look direction. dn – acts as the only parameter in the weighting function, must be real.
w (Θ) =
5
dn 2n+1 4π Pn (cos Θ)
(7)
Franc¸ois, A., Man, P. D. & Preumont, A. (2003). In 6th National Congress on Theoretical and Applied Mechanics, Ghent, Belgium.
dnbn (kr ) 2n+1 4π Pn (cos Θ)
(8)
Rafaely, B. (2005a). IEEE Trans. on Speech and Audio Processing 13, 135–143.
n=0
Pn – Legendre polynomial of order n Θ – spatial angle between the Ω0 and Ωl
Rafaely, B. (2005b). IEEE Signal Processing Letters 12, 713–716.
