Study and Design of First-Order Differential Arrays - Study and Design of Differential Microphone Arrays - page 56

′

The only way to be able to match (3.57) with a beamformer h ( ω ) is to have

a steering vector of the form:

T

e ωτ 0 cos( θ − ψ 1 ) e ωτ 0 cos( θ − ψ 2 ) ···e ωτ 0 cos( θ − ψ M )

d ( ω,θ )= ,

(3.58)

where ψ m is an angle depending on the m th microphone position. This implies
that, in order to be able to steer anywhere in a plane, we need at least three
noncolinear microphones.
One simple way to have some (limited) discrete steering without any com-
plicated signal processing, is to add one microphone and form an array having
the shape of an equilateral triangle. With this structure, we can form three
distinguished pairs of microphones and we see that we can realize all first-
order directional patterns at six different directions: 0 ◦ , 60 ◦ , 120 ◦ , 180 ◦ , 240 ◦ ,
and 300 ◦ .

References

1. G. W. Elko, “Superdirectional microphone arrays,” in Acoustic Signal Processing for
Telecommunication , S. L. Gay and J. Benesty, Eds. Boston, MA: Kluwer Academic
Publishers, 2000, Chapter 10, pp. 181-237.
2. J. Merimaa, “Applications of 3-D microphone array,” in Proc. AES 112th Convention ,
2002, pp. 1-11.
3. E. De Sena, H. Hacihabiboglu, and Z. Cvetkovic, “On the design and implementa-
tion of higher-order differential microphones,” IEEE Trans. Audio, Speech, Language
Process. , vol. 20, pp. 162-174, Jan. 2012.
4. G. H. Golub and C. F. Van Loan, Matrix Computations . Third Edition. Baltimore,
Maryland: The Johns Hopkins University Press, 1996.
5. H. Cox, R. M. Zeskind, and M. M. Owen, “Robust adaptive beamforming,” IEEE
Trans. Acoust., Speech, Signal Process. , vol. ASSP-35, pp. 1365-1376, Oct. 1987.
6. G. W. Elko and A.-T. Nguyen Pong, “A steerable and variable first-order differential
microphone array,” in Proc. IEEE ICASSP , 1997, pp. 223-226.
7. R. M. M. Derkx and K. Janse, “Theoretical analysis of a first-order azimuth-steerable
superdirective microphone array,” IEEE Trans. Audio, Speech, Language Process. , vol.
17, pp. 150-162, Jan. 2011.