MP05-2
sice02-0445
Theory and Experiment of Dual Sound Sources Localization with Five Proximate Microphones Masaki Kurihara, Nobutaka Ono and Shigeru Ando Graduate School of Information Science and Technology, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan
[email protected] Abstract: This paper describes a new method for the sound source localization, where dual sound sources simultaneously exist. While conventional array processing methods require the frequency decomposition as pre-processing, our method estimates the directions of sound sources with higher temporal resolution, directly from the original signal and its spatio-temporal derivatives. We show an algorithm for dual sound source localization and a fabrication of the microphone array system. Keywords: sound source localization, differential equation, spatio-temproal derivative
1. Introduction Most of conventional sound source localization methods1) are based on the antena array signal processing. In them, input signals are assumed to have narrow band-width. So, on applying them to wide-band signals like many acoustic signals, frequency decomposition requires as a pre-processing. This is one of reasons to decrease temporal resolution. In this paper, extending the spatio-temporal gradient method2) , we propose a new algorithm for the sound source localization, where dual sound sources simultaneously exist. Since observed signals are spatio-temporal gradients of the sound field, we can treat a wide-band signal directly and localize the sound sources with high temporal resolution. In this paper, we show 1) the sound field generated by dual sound sources satisfies second order differential equations, 2) determining coefficients of these equations gives us directions of dual sound sources.
When multiple plane waves are arrived, f (x, y, z, t) is represented as ! Ã N (n) (n) (n) X xτx + yτy + zτz . (5) g t+ f (x, y, z, t) = c n=1
In this case, the following lemma is satisfied.
Lemma 2 Eq(5) satisfies the following differential equation. Ã ! N (n) Y τw Dw − (6) Dt f = 0 c n=1 where w is any valiable of x, y, z.
2.2 Localization algorithm sound sources case
2. Theory 2.1 General form of differential equation satisfied with multiple plane waves Let f (x, y, z, t) be a sound pressure at an observation point (x y z). First, we assume that a plane wave arrives. Then, f (x, y, z, t) is written as µ ¶ xτx + yτy + zτz f (x, y, z, t) = g t + (1) c where (τx τy τz ) is a normalized vector which represents the direction of arrival. In this case, f (x, y, z, t) satisfies the following lemma. Lemma 1 Eq.(1) satisfies the following differential equations. ³ τx ´ Dx − Dt f = 0, (2) c
SICE 2002 Aug. 5–7, 2002, Osaka
where Dw to w.
³ τy ´ Dy − Dt f = 0, (3) c ³ ´ τz Dz − Dt f = 0, (4) c represents a partial derivative with respect
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for
dual
Assume that dual sound sources exist. As the direction vector τ = (τx τy τz ) is normalized, its degree of freedom is 2. Therefore, we estimate only τx , τy to localize the sound sources. According to Lemma 2, the sound field generated by dual sound sources satisfies the following four differential equations. à !à ! (1) (2) τx τx Dx − Dx − Dt Dt f = 0 (7) c c à !à ! (1) (2) τx τy Dx − Dt Dt f = 0 (8) Dy − c c à !à ! (1) (2) τy τx Dy − Dx − Dt Dt f = 0 (9) c c !à ! à (1) (2) τy τy Dy − Dy − Dt Dt f = 0 (10) c c
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3. Fabrication
So, we obtain 1) one differential equation about fxx , fxt , ftt , 2) two differential equations about fxy , fxt , fyt , ftt , 3) one differential equation about fyy , fyt , ftt . 1) indicates that by minimizing Z |fxx (t) + αx fxt (t) + βx ftt (t)|2 dt, J=
(11)
Γ
Figure 1 shows fabricated localization system. Five electret microphones are arranged in a cross form. The distance between neighboring microphones are 5cm. The linear combination of their outputs give us the 1st and the 2nd order spatio derivatives with x, y, approximately. The recieved signals are amplified by electric circuits, read out by an A/D board, and processed by a Desktop PC.
the coefficients of eq.(7), αx and βx , are estimated as the follows. αx
=
βx
=
Stt,xt Stt,xx − Stt,tt Sxt,xx , 2 Sxt,xt Stt,tt − Stt,xt Stt,xt Sxt,xx − Sxt,xt Stt,xx , 2 Sxt,xt Stt,tt − Stt,xt
where Sp,q =
Z
fp (t)fq (t)dt,
(12) (13)
(14)
Γ
and Γ is an observing interval (about 5[ms]). Then, we obtain p −αx ± αx2 − 4βx (i) τx = c · , (15) 2 where i = 1, 2. In the same way, 3) gives us the es(i) timation for τy . So, we call eq.(7) and eq.(10) the decision equations. Here, the sound source directions have two possibilities as the following. (1)
i) (τx
(1)
ii) (τx
(1)
(2)
τy ), (τx (2)
(2)
τy ), (τx
Figure 1: Dual sound sources localization sensor system with five proximate microphones
4. Conclusion
(2)
τy ) (1)
τy )
This ambiguity is solved by evaluating eq.(8) and eq.(9). Therefore, we call them the judgement equations.
In this paper, we porposed a novel sensor system with five proximate microphone, which localizes dual sound sources, simultaneously. We show experimental results in the presentation.
References
2.3 Discussion for minimum setup In order to obtain spatio differential signals we use linear combinations of signals received by multi microphones. Two decision equations include f, fx , fy , fxx , fyy . In addition to them, the judgement equations include fxy . However, two judgement equations are independent and eliminating fxy gives us another judgement equation as follows. Ã (2) (1) (1) (2) τy − τy τx − τx Dxt + Dyt c c ! (1) (2) (2) (1) τx τy − τx τy (16) + Dtt f = 0 c2 Therefore, by our method, the minimum number of microphones to determine directions of dual sound sources is five and five proximate microphones arrayed in a cross form are minimum setup.
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[1] H.Krim and M.Viberg, “Two Decades of Array Signal Processing Research, “ IEEE Signal Processing Magazine, vol.13, no.4, pp.67—94, July, 1996 [2] S. Ando, H. Shinoda, K. Ogawa, and S. Mitsuyama. “A Three-Dimensional Sound Localization Sensor System Based on the Spatio-Temporal Gradient Method,” Trans. Soc. Instrument and Control Engineers (SICE), vol.29, no.5, pp.520-528 1993 (in Japanese)
