Multiple Source Localization using Independent Component Analysis
Hiroshi Sawada∗ , Ryo Mukai, Shoko Araki, and Shoji Makino
NTT Communication Science Laboratories, Kyoto 619-0237, Japan
This paper presents a method for estimating location information about multiple sources.
The proposed method uses independent component analysis (ICA) as a main statistical tool.
The nearfield model as well as the farfield model can be assumed in this method. As anapplication of the method, we show experimental results for the direction-of-arrival (DOA)
estimation of three sources that were positioned 3-dimensionally.

Introduction
Source localization is a basic and important technique in array signal processing not
only for wireless communication systems, but also for audio/speech processing systems[1, 2]. Many source localization methods have been proposed. A well-known method isthe MUSIC (MUltiple SIgnal Classification) algorithm [3]. It identifies the noise subspacewith second order statistics and searches for location parameters that orthogonalize thesteering vector and the noise subspace. The proposed ICA-based method, on the otherhand, employs higher order statistics and directly identifies basis vectors, each of whichcontains the location information about a source. Consequently, the resolution of source
localization for a multiple source case is superior with the ICA-based method [4], and it iseasy to adapt it to the nearfield model as well as the farfield model.
The ICA-based source localization technique is a by-product of research on frequencydomain blind source separation (BSS), where the permutation ambiguity of the ICA solution in each frequency bin should be aligned in order to reconstruct time-domain signals
properly [5]. One of the approaches for aligning this permutation ambiguity involves clustering estimated source locations [6–9]. Earlier works [6, 7] calculated directivity patterns
from the separation matrix W obtained by ICA, and then searched for null directions,which correspond to the directions of sources [10]. However, it is simpler and more effective to estimate the directions directly from the basis vectors ai , which are given by thepseudoinverse of W. The method presented in this paper is based on this idea.

Independent Component Analysis
We consider a situation where N sources are convolutively mixed and observed atM sensors. Convolutive mixtures in the time domain can be approximated as multiple
instantaneous mixtures in the frequency domain:x(f, τ ) =N

i=1 hi (f )si (f, τ ),(1)(f, τ )]T is a sensor vector at frequency f and frame

where x(f, τ ) = [x1 (f, τ ), . . . , xM
index τ , and hi (f ) = [h1i (f ), . . . , hM i (f )]T is the vector of the frequency responses fromsource si (f, τ ) to all M sensors. We obtain only x(f, τ ) by applying the short-time Fouriertransform (STFT) to time-domain sensor observations.
The first step of the proposed method is to solve complex-valued ICAy(f, τ ) = W(f )x(f, τ ),(2)(f, τ )]T is a vector of separated signals and W(f ) is anwhere y(f, τ ) = [y1 (f, τ ), . . . , yN
N ×M separation matrix. There are various algorithms available for ICA as summarized in
0-7803-8883-6/05/$20.00 ©2005 IEEE

Figure 1: Nearfield (left) and farfield (right) modele.g. [11]. Then, we calculate the Moore-Penrose pseudoinverse W+ (which is equivalent
to the inverse W−1 if N = M ) of W
[a1 , · · · , aN ] = W+ , ai = [a1i , . . . , aM i ]T .

(3)

It is not difficult to make W invertible by using an appropriate ICA procedure, such as
whitening followed by unitary transformation (e.g. FastICA [11]). By multiplying bothsides of (2) by W+ , the sensor vector x(f, τ ) is represented by a linear combination ofbasis vectors a1 , . . . , aN :x(f, τ ) =N

i=1 ai (f )yi (f, τ ).

(4)

By comparing (1) and (4), we observe the following fact. If the ICA algorithm works
well and the outputs y1 , . . . , yN are the estimation of the sources s1 , . . . , sN , then the basis
vectors a1 , . . . , aN are also estimations of the mixing vectors h1 , . . . , hN up to permutationand scaling ambiguity: ai = αi hΠ(i) . Here, scalar αi represents scaling ambiguity andΠ : {1, . . . , N } → {1, . . . , N } represents permutation ambiguity.

Source Localization with Nearfield Model
Let pj and qi be 3-dimensional vectors representing the locations of sensor j andsource i, respectively. To estimate source location qi from basis vectors ai and sensor positions pj , we approximate the frequency response hji (f ) with a nearfield model (Fig. 1):
hji (f ) =1−1e2πf c (||qi −pj ||−||qi||) ,||qi − pj ||(5)where c is the propagation velocity of the signals. We assume that the phase depends onthe difference between the distances ||qi − pj || − ||qi || from the source to the sensor andto the origin o = [0, 0, 0]T . This makes the phase zero at the origin. In accordance with themodel (5), the ratio between two elements aji , aj  i of the same basis vector ai provides thekey equation for source localization:||qi − pj  || 2πf c−1 (||qi −pj ||−||qi −pj ||)αi hjiajie==,aj  i
αi hj  i||qi − pj ||(6)where the scaling ambiguity αi is canceled out by calculating the ratio. The permutation
ambiguity still remains. However, if we estimate the location qi for all i = 1, . . . , N , theset of all estimated locations does not depend on the permutation.
The set of vectors qi in the argument of (6)||qi − pj || − ||qi − pj  || =arg(aji /aj  i ).
2πf c−1(7)hyperboloid

spherehyperboloid

Figure 2: Source localization by the intersection of two hyperboloids and a spheredefines a surface where the difference between the distances from pj and pj  is constant.
The surface is one sheet of a two-sheeted hyperboloid. Alternatively, the modulus of (6)||qi − pj  ||  aji = aj  i ||qi − pj ||(8)defines a sphere where the ratio of the distances from pj and pj  is constant. With these
two equations (7) and (8), we can estimate the possible location qi of source si . Such a
hyperboloid and sphere are defined for a pair of sensors j and j  . If we select another pair,a different hyperboloid and sphere are obtained. In this way, the location qi is estimated as
the intersection of several hyperboloids and spheres. An example is shown in Fig. 2.

DOA Estimation with Farfield Model
Although it is useful to estimate a 3-dimensional location, calculating the intersectionsof hyperboloids and spheres is computationally demanding. There are many cases where itis sufficient to estimate just the DOA of source si . If we assume that the source location qi
is far from sensors pj and pj  , (7) can be approximated as a farfield model (Fig. 1)(pj − pj  )Targ(aji /aj  i )qi=.
||qi ||2πf c−1(9)The set of vectors qi satisfying (9) represents a cone [8], which is the asymptotic surfaceof the corresponding hyperboloid (7). The intersections of several cones specify the DOA
of a source. Let us assume that we select u cones whose corresponding sensor pairs are(j1 , j1 ), . . . , (ju , ju ). Then, the set of equations (9) for the u pairs is represented asriqi=,(10)D||qi ||
2πf c−1
where D = [pj1 − pj1 , . . . , pju − pju ]T, ri = [ arg(aj1 i /aj1 i ), . . . , arg(aju i /aju i ) ]T .
In practical situations, there is no exact solution for (10) because the u conditions do notcoincide exactly. We typically solve it in the least-square sense by using the Moore-Penrose
pseudoinverse [9]:
D+ ri
qi=.
(11)
||qi ||
2πf c−1
If rank(D) ≥ 3, the set of vectors qi that satisfy (11) represents a line in 3-dimensional

10.5

0−0.5
−1−1−10−0.5

00.5

1 1Figure 3: 3-dimensional arrangement of eight microphones and three loudspeakers (left)and DOA estimation results for this case (right))
space, and thus the DOA of a source i is estimated as ||qqii (f(f )|| .
The left photograph in Fig. 3 shows a case where eight microphones and three loudspeakers are arranged 3-dimensionally, and the right plot shows the DOA estimations for)this case. Each point shows a location vector normalized to unit norm qi (f ) ← ||qqii (f(f )|| .
The estimations are obtained for all frequencies f and all output indexes i. As shown inthe plot, they form clusters, each of which corresponds to the location of each source.

References[1] H. Krim and M. Viberg, “Two decades of array signal processing research: the parametricapproach,” IEEE Signal Processing Magazine, vol. 13, no. 4, pp. 67–94, July 1996.
[2] M. Brandstein and D. Ward, Eds., Microphone Arrays, Springer, 2001.
[3] R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE Trans.
Antennas and Propagation, vol. 34, pp. 276–280, Mar. 1986.
[4] H. Sawada, R. Mukai, and S. Makino, “Direction of arrival estimation for multiple source
signals using independent component analysis,” in Proc. International Symposium on Signal
Processing and its Applications, July 2003, pp. 411–414.
[5] H. Sawada, R. Mukai, S. Araki, and S. Makino, “A robust and precise method for solving
the permutation problem of frequency-domain blind source separation,” IEEE Trans. Speech
Audio Processing, vol. 12, pp. 530–538, Sept. 2004.
[6] H. Saruwatari, S. Kurita, K. Takeda, F. Itakura, T. Nishikawa, and K. Shikano, “Blind source
separation combining independent component analysis and beamforming,” EURASIP Journal
on Applied Signal Processing, vol. 2003, no. 11, pp. 1135–1146, 2003.
[7] M. Z. Ikram and D. R. Morgan, “A beamforming approach to permutation alignment formultichannel frequency-domain blind speech separation,” in Proc. ICASSP 2002, May 2002,
pp. 881–884.
[8] R. Mukai, H. Sawada, S. Araki, and S. Makino, “Frequency domain blind source separation
using small and large spacing sensor pairs,” in Proc. ISCAS 2004, May 2004, vol. V, pp. 1–4.
[9] R. Mukai, H. Sawada, S. Araki, and S. Makino, “Frequency domain blind source separation
for many speech signals,” in Proc. ICA 2004 (LNCS 3195), Sept. 2004, pp. 461–469.
[10] S. Araki, S. Makino, Y. Hinamoto, R. Mukai, T. Nishikawa, and H. Saruwatari, “Equivalence
between frequency domain blind source separation and frequency domain adaptive beamforming for convolutive mixtures,” EURASIP Journal on Applied Signal Processing, vol. 2003, no.
11, pp. 1157–1166, 2003.
[11] A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis, John Wiley &Sons, 2001.