Tracking of Multiple Moving Sources Using
Recursive EM Algorithm
Pei-Jung Chung
Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109-2122, USA Email:peijung [email protected]
Johann F. B ¨ohme
Signal Theory Group, Department of Electrical Engineering and Information Science, Ruhr-University of Bochum, 44780 Bochum, Germany
Email:[email protected]
Alfred O. Hero
Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109-2122, USA Email:[email protected]
Received 31 December 2003; Revised 17 June 2004
We deal with recursive direction-of-arrival (DOA) estimation of multiple moving sources. Based on the recursive EM algorithm, we develop two recursive procedures to estimate the time-varying DOA parameter for narrowband signals. The first procedure requires no prior knowledge about the source movement. The second procedure assumes that the motion of moving sources is described by a linear polynomial model. The proposed recursion updates the polynomial coefficients when a new data arrives. The suggested approaches have two major advantages: simple implementation and easy extension to wideband signals. Numerical experiments show that both procedures provide excellent results in a slowly changing environment. When the DOA parameter changes fast or two source directions cross with each other, the procedure designed for a linear polynomial model has a better performance than the general procedure. Compared to the beamforming technique based on the same parameterization, our approach is computationally favorable and has a wider range of applications.
Keywords and phrases:sensor array processing, recursive estimation, moving sources, tracking.
1. INTRODUCTION
The problem of estimating the direction of arrival (DOA) of 1
2 3
plane waves impinging on a sensor array is of fundamental
4
importance in many applications such as radar, sonar, geo-physics, and wireless communication. The maximum like-lihood (ML) method is known to have excellent statistical performance and is robust against coherent signals and small sample numbers [1]. However, the high computational cost associated with ML method makes it less attractive in prac-tice.
To improve the computational efficiency of the ML ap-proach, numerical methods such as the expectation and maximization (EM) algorithm [2] were suggested in [3,4,5]. Recursive procedures based on the recursive EM (REM) al-gorithm for estimating constant DOA parameters were dis-cussed in [6, 7]. Similar procedures for tracking multiple moving sources were studied in [8,9]. In [9], the authors focused on narrowband sources and assumed known signal
waveforms.
The REM algorithm is a stochastic approximation pro-cedure for finding ML estimates (MLEs). It was first sug-gested by Titterington [10] and extended to the multidimen-sional case in [6]. As it was pointed out by Titterington, REM can be seen as a sequential approximation of the EM algorithm. The gain matrix of REM is the inversion of the augmented data information matrix. Through proper design of the augmentation scheme, the augmented data and the corresponding information matrixes usually have a simple structure [2]. In this case, the REM algorithm is very easy to 5 implement. For constant parameters , estimates generated by REM are strongly consistent and asymptotically normally distributed. For time-varying parameters , the tracking abil-ity of a stochastic approximation procedure depends mainly on the dynamics of the true parameter, gain matrix, and step size [11].
for estimating time-varying DOA. The first procedure does not require any prior knowledge on the motion model. The only assumption is that the unknown parameter changes slowly with time. The second procedure assumes that the time-varying DOA parameter θ(t) is described by a linear polynomial of time. This model is important since a smooth functionθ(t) can be approximated by a local linear polyno-mial in a short-time interval [12]. The procedure reported in [8] employs a decreasing step size to estimate the polynomial coefficients. However, since the DOA parameterθ(t) and the log-likelihood function change with time, a decreasing step size may not capture the nonstationary feature of the under-lying system over a long period. To overcome this problem, we suggest a constant step size to be used in the algorithm. It is noteworthy that both procedures are aimed at maximiz-ing the expected concentrated likelihood function [13]. In-troducing a linear polynomial model implies increasing the dimension of the parameter space. With the additional de-gree of freedom, the procedure designed for a linear polyno-mial model should perform better than the general one.
In contrast to methods based on subspace tracking [14] or two-dimensional beamforming [12], our approach can be easily generalized to wideband cases including underwater acoustic signals. Unlike the Kalman-type algorithms [15], the recursive procedures considered here have a much simpler implementation.
This paper is outlined as follows. We describe the sig-nal model and the REM algorithm briefly in Sections 2
and 3.Section 4 presents two recursive procedures for
lo-calizing moving sources. Simulation results are discussed in
Section 5. We give concluding remarks inSection 6.
2. PROBLEM FORMULATION
Consider an array of N sensors receiving M far-field waves from unknown time-varying directions θ(t) =
[θ1(t)· · ·θM(t)]. The array outputx(t)∈CN×1at time
in-stanttis expressed as
x(t)=Hθ(t)s(t) +u(t), t=1, 2,. . ., (1) where the steering matrix
Hθ(t)=dθ1(t)
· · ·dθM(t)∈CN×M (2)
consists of M steering vectors d(θm(t)) ∈ CN×1 (m = 1,. . .,M). To avoid ambiguity, we assume thatM < N. The signal waveform s(t) = [s1(t)· · ·sM(t)]T ∈ CM×1 is con-sidered unknown and deterministic. (·)T denotes the trans-pose of a vector. Furthermore, the noise processu(t)∈CN×1 is independent identically complex and normally distributed with zero mean and covariance matrixνI, whereνrepresents the unknown noise spectral parameter andI is the identity matrix.
In the following, we assume that the number of sources
Mis known. Standard procedures based on minimum de-6
7 scription length (MDL) criteria [16] or multiple hypoth-esis testing [7] can be used to determineM. The problem of
interest is to estimate the time-varying DOA parameterθ(t) recursively from the observationx(t). We assume that a good initial estimateθ0is available at the beginning of the recur-sion.
3. RECURSIVE PARAMETER ESTIMATION USING INCOMPLETE DATA
The REM algorithm suggested by Titterington is a stochastic approximation procedure for finding MLEs. As pointed out in [10], there is a strong relationship between this procedure and the EM algorithm [2]. Using Taylor expansion, Tittering-ton showed that, approximately, REM maximizes EM’s aug-mented log likelihood sequentially. The unknown parame-ter is considered as constant in [10]. In the fixed parameter case, a properly chosen decreasing step size is necessary to ensure strong consistency and asymptotic normality of the algorithm [10,17].
Suppose x(1),x(2),. . . are independent observations, each with underlying probability density function (pdf)
f(x;ϑ), whereϑ denotes an unknown constant parameter. The augmented data associated with the EM y(1),y(2),. . . 8
is characterized by the pdf f(y;ϑ). According to [2], the augmented data y(t) is so specified thatM(y(t))=x(t) is a many-to-one mapping. Letϑtdenote the estimate aftert ob-servations. The following procedure is aimed at finding the true parameterϑwhich may coincide with the MLE in the asymptotic sense [18]:
ϑt+1=ϑt+tIEM
ϑt−1γx(t),ϑt, (3)
wheretis a decreasing step size and 9
IEM
ϑt= E−∇ϑ∇Tϑlogf(y;ϑ)|x(t),ϑ|ϑ=ϑt, (4)
γx(t),ϑt=∇ϑlogfx(t);ϑ|ϑ=ϑt (5)
represent the augmented information matrix and gradient vector, respectively. ∇ϑ is a column gradient operator with respect toϑ. We assume that both (4) and (5) exist. Under mild conditions, the estimates generated by (3) are strongly
consistent, asymptotic, and normally distributed. In view 10 of the well-known singularities and multiple maxima that are on likelihood surfaces, one could not of course expect con-sistency irrespective of the starting point [10].
The augmented data y usually has a simpler structure than the observed data x. Therefore, the augmented data information matrix IEM(ϑt) is easier to compute and in-vert than the observed data information matrix I(ϑt) =
E−∇ϑ∇Tϑlogf(x;ϑ)|x(t),ϑ|ϑ=ϑt. Although REM does not
have the optimal convergence rate in the Cram´er-Rao sense as the following procedure [10]:
ϑt+1=ϑt+tIϑt−1γx(t),ϑt, (6) it is much easier to implement than (6). UsingIEM(ϑt)−1as the gain matrix is a trade-offbetween the convergence rate and computational cost.
When the parameter of interest varies with time, a de-creasing step size such as t = t−α, 1/2 < α ≤ 1, cannot capture the nonstationary feature of the underlying system. A classical way to overcome this difficulty is to replacetwith
a constant step size. In general, a large step size reduces the bias and increases the variance of the estimates [11]. A small step size has opposite effects. Since the time-varying param-eterϑ(t) may follow a complicated dynamics, an exact inves-tigation of the convergence behavior of the algorithm
ϑt+1=ϑt+IEM
ϑt−1γx(t),ϑt (7) is only possible when certain assumptions are made on the parameter model. More discussion about convergence prop-erties of a stochastic approximation procedure in a nonsta-tionary environment can be found in [11].
4. LOCALIZATION OF MOVING SOURCES
The REM algorithm with constant step size (7) is applied to estimate the time-varying DOA parameterθ(t). We start with a general case in whichθ(t) changes slowly with time and then consider a linear polynomial model.
4.1. General case
From the signal model inSection 2, we know that the array observationx(t) is complex and normally distributed with the log likelihood function
logfx(t);ϑ = − Nlogπ+Nlogν +1 ν x(t)−Hθ(t)s(t)Hx(t)−Hθ(t)s(t), (8) whereϑ = [θ(t)Ts(t)Tν]T and (·)H denotes the Hermitian transpose.
According to (7), all elements in ϑshould be updated simultaneously. Since we are mainly interested in the DOA parameterθ(t) and since including{s(t),ν}in the recursion will complicate the gain matrix IEM(ϑt)−1, then the proce-dure (7) is only applied toθ(t). The estimate for signal wave-form and noise level, denoted byst =[st1 st2 · · · stM]T and
νt, respectively, is updated by computing their MLEs once the
current DOA estimate is available. For simplicity, we use θ instead ofθ(t) in the following discussion.
Taking the first derivative on the right-hand side of (8) with respect toθm, we obtain themth element of the gradient vectorγ(x(t),ϑt) [17]:
γx(t),ϑtm= ν2tRex(t)−HθtstHdθtmstm, (9)
whered(θm)=∂d(θm)/∂θm.
The augmented datay(t) is obtained by decomposing the array output into its signal and noise parts. Formally it is ex-pressed as
y(t)=y1(t)T· · ·ym(t)T· · ·yM(t)T
T
. (10) The augmented data associated with themth signal
ym(t)=dθmsm(t) +um(t) (11)
is complex and normally distributed with meand(θm)sm(t) and covariance matrix νmI with the constraintMm=1νm =
ν. A convenient choice isνm =ν/M. The corresponding log likelihood is given by logfy(t);ϑ = −M m=1 Nlogπ+Nlog ν M +M ν ym(t)−dθmsm(t)H ×ym(t)−dθmsm(t). (12)
Since the signals are decoupled through the augmentation scheme (10),IEM(ϑt) is anM×Mdiagonal matrix when we only consider the DOA parameterθ. By definition (4), the
mth diagonal element ofIEM(ϑt) is the conditional expecta-tion of the second derivative of the augmented log likelihood
IEM ϑtmm= E−∂θ∂22 mlogf y(t);ϑx(t),ϑt, (13) which is given by IEM(ϑt) mm=ν2tRe −dθtmstmHx(t)−Hθtst +Mdθmtstm2 , (14) whered(θm)=∂2d(θ m)/∂θ2m.
Once the estimateθt+1is available, the signal and noise parameters are obtained by computing their MLEs at current θt+1andx(t) as follows: st+1=Hθt+1# x(t), νt+1= 1 Ntr Pθt+1⊥Cx(t) , (15)
whereH(θt+1)#is the generalized left inverse of the matrix H(θt+1),P(θt+1)⊥ =I−P(θt+1) is the orthogonal comple-ment of the projection matrixP(θt+1) =H(θt+1)H(θt+1)#, andCx(t)=x(t)x(t)H.
Given a constant step size, the number of sourcesM, and the current estimateθt, the (t+ 1)st recursion of the al-gorithm proceeds as shown inAlgorithm 1.
4.2. Linear polynomial model
We consider moving sources described by the linear polyno-mial model
(1) Calculate the gradient vectorγ(x(t),θt) by (9) and the matrixIEM(θt) by (14).
(2) Update DOA parameters by
θt+1=θt+[IEM(θt)]−1γ(x(t),θt).
(3) Update the signal and noise parameters st,νtby (15).
Algorithm1: Recursive EM algorithm I (REM I) (arbitrary mo-tion).
where θ0 = [θ01,. . .,θ0M]T andθ1 = [θ11,. . .,θ1M]T. The
linear polynomial (16) can be seen as a truncated Taylor ex-pansion which gives a good description for the source mo-tion in a small observamo-tion interval [12].
The REM algorithm is applied to estimateθ0andθ1. For notational simplicity, we define the extended DOA parame-ter asΘ =[ΘT1· · ·ΘT
m· · ·ΘTM]T whereΘm =[θ0m,θ1m]T.
Similarly to the procedure presented inSection 4.1, REM is 11
only applied to update the DOA parameterΘ rather than ϑ=[ΘTs(t)Tν]T.
Based on this approach, the 2mth and (2m+ 1)st element of the gradient vectorγ(x(t),ϑt) are given by
∂ ∂θ0m logfx(t);ϑ|ϑ=ϑt =ν2tRex(t)−HΘtstHdΘtmstm, ∂ ∂θ1m logfx(t);ϑ|ϑ=ϑt =2νttRex(t)−HΘtstHdΘtmstm, (17) respectively, whered(Θtm)=∂d(θm)/∂θm|θm=Θt m0+tΘtm1. Note
thatθ is calculated at the current estimateΘt according to the linear model (16).
Because each source is described by two unknown pa-rameters, the augmented data information matrix becomes block diagonal. Unfortunately, this matrix is singular under current parameterization. To avoid singularity and simplify the recursion, rather than using this block diagonal matrix in the recursion directly, we consider an alternative matrix
˜
IEM(ϑt) which is the diagonal part ofIEM(ϑt). Letd(Θt)=∂2d(θ m)/∂θm2|θ m=Θ t m0+tΘ t m1 . According to the augmentation scheme specified above, the 2mth and 12
(2m+ 1)st diagonal components of ˜IEM(Θt) are given by 2 νtRe −dΘtmstmHx(t)−HΘtstm +MdΘmt stm2, 2t2 νt Re −dΘtmstmHx(t)−HΘtstm +MdΘmt stm2, (18) respectively.
Similarly to the general case, the signal and noise parameters are updated by (15) once the estimateΘt+1 is available. The parameterθt+1in (15) is replaced byΘt+1.
(1) Calculate the gradient vectorγ(x(t),Θt) by (17) and the matrix ˜IEM(Θt) by (18).
(2) Update DOA parameters by
Θt+1=Θt+[˜IEM(Θt)]−1γ(x(t),Θt).
(3) Update the signal and noise parameters st,νtby (15)
withθtreplaced byΘt.
Algorithm2: Recursive EM algorithm II (REM II) (linear polyno-mial model).
Given the step size, the number of sourcesM, and the current estimateΘt, the (t+ 1)st recursion of the algorithm proceeds as shown inAlgorithm 2.
For simplicity, the REM for the general case and the REM for the linear polynomial model are referred to as “REM I” and “REM II,” respectively.
From (9), (14), and (15), the computational complexity of REM I lies approximately between O(MN +MN2) and
O(MN +N3). The dominant termMN2 (orN3) is associ-ated with st+1 given by (15) which is a solution to a least square (LS) problem. Different LS algorithms yield diff er-ent computational loads [19]. Due to the increased num-ber of unknowns, REM II requires twice as many compu-tations as REM I in computing the gradient vector and aug-mented information matrix. Clearly, REM II is computation-ally more efficient than the local-polynomial-approximation (LPA) based beamforming technique [12] whose computa- 13 tional complexity is given byO(NTLP) where T represents the number of snapshots,Ldenotes the number of points in the angular search domain, andPdenotes the number of an-gular velocity search domain.
It was pointed out in [13] that REM for constant DOA es-timation is indeed a recursive procedure for finding the max-imum of the expected concentrated likelihood function
L(θ)= −tr logP(θ)Cx(t)
, (19)
whereCx(t)= Ex(t)x(t)H. The constant step size
consid-ered in REM I captures the time-varying character of the likelihood function. Similarly, REM II is aimed at finding the maximum ofL(θ). Using a different parameterization, such as a linear polynomial model implies increasing the dimen-sion of the parameter space. With the additional degree of freedom, REM II is expected to have a better tracking abil-ity. Later inSection 5we will show that in critical situations where two source directions cross with each other, REM II provides more accurate estimates than REM I.
Choosing a proper step size plays an important role in the algorithms’ tracking ability. The optimal step size depends on the dynamics of the true parameters, for instance, rate of change. Interested readers can find general guidelines in [11] and an adaptive procedure designed for REM with a decreas-ing step size in [20].
4.3. Extension to broadband signals
The algorithms presented previously are derived under the narrowband signal assumption. Extension to the
broad-90 80 70 60 50 40 30 20 10 0 DO A (d egr ee ) 5 10 15 20 25 30 35 40 45 50 Time index True trajectory Estimated trajectory
Figure1: True trajectory and estimated trajectory by REM I for the narrowband case.θ0=[10◦, 60◦, 66◦],θ1=[0.6◦,−1.0◦, 0.4◦]. SNR =20 dB.
band case is straightforward. From the asymptotic theory of Fourier transform [21], we know that each frequency bin is asymptotically independent of the other [22]. The log likeli-hood function associated with the broadband signal is a sum of the log likelihoods of individual frequency bins. Corre-spondingly, the gradient vector and augmented information matrix can be easily obtained by adding up the gradient vec-tors and augmented data information matrices of relevant frequency bins. Similarly to the narrowband case, the signal and noise parameters at each frequency are updated by calcu-lating their MLEs once the current DOA estimate is available. 5. SIMULATION
The proposed algorithms are tested by numerical experi-ments. In the first part, we consider REM algorithms’ appli-cation in narrowband and broadband cases. In the second part, we compare REM II with the LPA-based beamforming technique [12].
5.1. Recursive EM algorithms I and II
The narrowband signals generated by three sources of equal power are received by a uniformly linear array of 15 sen-sors with interelement spacings of half a wavelength. The signal-to-noise ratio (SNR), defined as 10 log(sm(t)2/ν),m= 1, 2, 3, is kept at 10, 20 dB. The motion of the moving 14
sources is described by the linear polynomial model (16). 15
Three-different-parameter sets {θ0,θ1}are assumed in the 16 experiments. Each experiment performs 200 trials.
17 In the first experiment, we consider relatively fast mov-ing sources. The true parameters are given by θ0 = [10◦, 60◦, 66◦],θ1=[0.6◦,−1.0◦, 0.4◦] whereθ1is measured by degrees per time unit. In order to get a good insight into
90 80 70 60 50 40 30 20 10 0 DO A (d egr ee ) 5 10 15 20 25 30 35 40 45 50 Time index True trajectory Estimated trajectory
Figure2: True trajectory and estimated trajectory by REM II for the narrowband case. SNR=20 dB.
the tracking behavior, the same initial values are used in all trials. We applied LPA-based beamforming to 20 snap-shots to obtain the initial estimatesθ00=[10.5◦, 59.5◦, 68.5◦], θ01 =[0.58◦,−0.99◦, 0.38◦]. The initial estimate for REM I is given byθ00. Both algorithms use a constant step size=0.6. Figures1and2present the true values ofθand an example of estimated trajectories. As shown in both figures, two source directions cross with each other att=32. Obviously, the re-cursive procedure designed for the most general case cannot follow fast moving sources at all. In contrast, the estimated trajectory obtained by REM II is very close to the true one. Figures3and4show the root mean square errors (RMSEs) of the DOA estimates, defined as
θt−θ2, averaged over 200 trials. Since REM I fails to track the moving sources, the cor-responding RMSE grows with increasing time. On the other hand, the RMSE associated with REM II decreases slightly at the beginning of the recursion and then remains almost constant. Comparing Figures3and4, one can observe that SNR=20 dB has a slightly lower RMSE than SNR=10 dB.
The second experiment involves three slowly moving sources. The true parameter values are given by θ0 = [30◦, 50◦, 62◦],θ1 =[0.06◦,−0.1◦, 0.05◦]. Note that the an-gular velocityθ1is approximately 1/10 of that considered in the previous experiment. We applied the ML method to ob-tain the initial estimatesθ00 = [30.1◦, 50.8◦, 60.9◦]. Because the angular velocity is very small compared to that in the previous experiment, we takeθ01 =[0◦, 0◦, 0◦] as the initial value forθ1. The initial estimate for REM I is given byθ00. Both algorithms use a constant step size =0.6. Figures5 and6 present the true and estimated trajectories obtained by REM I and REM II. Similarly to the first experiment, two source directions cross with each other att=126. The esti-mated trajectory by REM I is close to the true one when no crossing happens. Betweent=100 andt=230, where two
102 101 100 10−1 RMSE o f θ 0 5 10 15 20 25 30 35 40 45 50 Time index REM I REM II
Figure3: RMSE of θversus time for the narrowband case. SNR =20 dB. 102 101 100 10−1 RMSE o f θ 0 5 10 15 20 25 30 35 40 45 50 Time index REM I REM II
Figure4: RMSE of θversus time for the narrowband case. SNR =10 dB.
source directions cross with each other, the estimated trajec-tories associated with the first two sources do not get close to each other. Instead, they just depart in the vicinity oft=126. For the same scenario, REM II provides a more accurate es-timate.Figure 6shows that the crossing point causes a larger deviation from the true trajectory. Due to a higher sensitivity to the variation of angular velocity at the crossing point, the estimated trajectory inFigure 6is slightly worse than that in
Figure 2. Comparison of Figures7and8with Figures3and
4 shows an overall lower RMSE in this scenario. Although REM I provides more reliable estimates than in the first ex-periment, REM II still outperforms REM I.
90 80 70 60 50 40 30 20 10 0 DO A (d egr ee ) 50 100 150 200 250 300 Time index True trajectory Estimated trajectory
Figure5: True trajectory and estimated trajectory by REM I for the narrowband case.θ0 =[30◦, 50◦, 62◦],θ1 =[0.06◦,−0.1◦, 0.05◦]. SNR=20 dB. 90 80 70 60 50 40 30 20 10 0 DO A (d egr ee ) 50 100 150 200 250 300 Time index True trajectory Estimated trajectory
Figure6: True trajectory and estimated trajectory by REM II for the narrowband case. SNR=20 dB.
In the third experiment, three sources move slowly with different speeds but do not cross with each other. The true parameters are given by θ0 = [10◦, 30◦, 62◦], θ1 = [0.08◦, 0.1◦, 0.06◦]. The initial estimates are θ00 = [10.04◦, 30.04◦, 62.05◦],θ01 =[0◦, 0◦, 0◦]. We use a constant step size = 0.6. Both algorithms have good tracking abil-ity. Figures9and10show that RMSE is the lowest among all three scenarios. REM II has a better performance than REM I. While REM II has a better performance at higher SNR, REM I seems to be less sensitive to SNRs in all three
102 101 100 10−1 RMSE o f θ 0 50 100 150 200 250 300 Time index REM I REM II
Figure7: RMSE of θversus time for the narrowband case. SNR =20 dB. 102 101 100 10−1 RMSE o f θ 0 50 100 150 200 250 300 Time index REM I REM II
Figure8: RMSE of θversus time for the narrowband case. SNR =10 dB.
scenarios.
In addition to the narrowband signals, we also applied REM I and REM II to broadband signals with 3 frequency bins. The scenario similar to the second experiment leads to the results presented in Figures11and12. The estimates be-have similarly to the narrowband case. Comparison of RM-SEs shows that more frequency bins leads to higher accuracy.
5.2. Comparison with LPA beamforming
We compare REM II with the LPA-based beamforming ap-proach suggested by Katkovnik and Gershman [12]. Both algorithms assume the motion model (16). In the first
ex-102 101 100 10−1 RMSE o f θ 0 50 100 150 200 250 Time index REM I REM II
Figure 9: RMSE ofθversus time for the narrowband case. SNR =20 dB.θ0=[10◦, 30◦, 62◦],θ1=[0.08◦, 0.1◦, 0.06◦]. 102 101 100 10−1 RMSE o f θ 0 50 100 150 200 250 Time index REM I REM II
Figure10: RMSE ofθversus time for the narrowband case. SNR =10 dB.
periment, the narrowband signals are generated by the fol-lowing parameter set θ0 = [10◦, 60◦], θ1 = [0.6◦,−1.0◦], SNR=0, 10 dB. In the second experiment, we consider mov-ing sources with lower angular velocities θ0 = [30◦, 50◦], θ1=[0.06◦,−0.1◦]. A sliding window of 25 snapshots is used in the LPA beamforming. The REM II is initialized by the LPA beamforming in the first scenario and ML method in the second one. To ensure the same data length in each time interval, we use additional (W−1) samples in the LPA beam-forming processing.
The estimated trajectories presented in Figures13and14 are very close to the true ones. The RMSEs ofθ0andθ1
corre-102 101 100 10−1 RMSE o f θ 0 50 100 150 200 250 300 Time index REM I REM II
Figure 11: RMSE ofθversus time for the broadband case. SNR =20 dB.θ0=[30◦, 50◦, 62◦],θ1=[0.06◦,−0.1◦, 0.05◦]. Number of frequency bins=3. 102 101 100 10−1 RMSE o f θ 0 50 100 150 200 250 300 Time index REM I REM II
Figure 12: RMSE ofθversus time for the broadband case. SNR =10 dB.
sponding to the first source are plotted in Figures15and16. Using the initial value provided by LPA beamforming, RMSE associated with REM II changes slowly over time. While es-timates of θ0 remain constant, the estimates ofθ1 become more accurate with increasing recursions. Also, we can ob-serve that while LPA beamforming provides an overall better θ0 estimates and better angular velocity estimates at begin-ning of the recursion, REM II improvesθ1estimates with in-creasing time and has less fluctuations.
Compared with the Cram´er-Rao bounds (CRBs) [23], one realizes that an REM II is certainly not an efficient
es-90 80 70 60 50 40 30 20 10 0 DO A (d egr ee ) 10 20 30 40 50 60 Time index True trajectory Estimated trajectory
Figure13: True trajectory and estimated trajectory by LPA beam-forming. SNR=10 dB.θ0=[10◦, 60◦],θ1=[0.6◦,−1.0◦]. 90 80 70 60 50 40 30 20 10 0 DO A (d egr ee ) 10 20 30 40 50 60 Time index True trajectory Estimated trajectory
Figure14: True trajectory and estimated trajectory by REM II. SNR =10 dB.
timator. However, the ML approach suggested in [23], whose estimation accuracy is close to CRB, is a batch processing and requires a complicated multidimensional search procedure.
In the second experiment, REM II provides much more accurate estimates than LPA beamforming.Figure 17shows that LPA beamforming even fails to follow the moving sources. We can observe inFigure 19that REM II has lower RMSE in bothθ0andθ1estimation. Consequently, as shown
in Figure 20the resulting DOA estimates are much better
than LPA beamforming. In both experiments, the compu-tational time needed for LPA beamforming is about 800
102 101 100 10−1 10−2 RMSE o f θ0 15 20 25 30 35 40 45 50 55 60 Time index REM II LPA beamforming CRB (a) 100 10−1 10−2 10−3 RMSE o f θ1 15 20 25 30 35 40 45 50 55 60 Time index REM II LPA beamforming CRB (b)
Figure15: (a) RMSE ofθ0corresponding to the first source versus time. (b) RMSE ofθ1versus time. SNR=10 dB.
times as high as that required by REM II due to the two-dimensional search procedure.
We conclude that REM I is suitable for tracking slowly time-varying DOA parameters, REM II performs well for both slowly and fast moving sources. Both procedures gener-ate accurgener-ate estimgener-ates when there is no crossing point. When two source directions coincide with each other, the steering matrix H(θ) becomes rank deficient. The signal waveform s(t) cannot be determined properly. Consequently the DOA parameter cannot be estimated accurately. In this case, regu-larization is needed [23]. Since REM II incorporates a linear polynomial model, it has a better tracking ability than REM I when this critical situation occurs. Compared to LPA beam-forming, our method has a clear computational advantage. It provides comparable results with LPA beamforming in the fast moving sources case and outperforms LPA beamforming in the slow moving source case. In addition, REM is applica-ble to both narrowband and broadband signals.
6. CONCLUSION
We addressed the problem of tracking multiple moving sources. Two recursive procedures are proposed to estimate the time-varying DOA parameter. We applied the recursive EM algorithm to a general case in which the motion of the
102 101 100 10−1 10−2 RMSE o f θ0 15 20 25 30 35 40 45 50 55 60 Time index REM II LPA beamforming CRB (a) 100 10−1 10−2 10−3 RMSE o f θ1 15 20 25 30 35 40 45 50 55 60 Time index REM II LPA beamforming CRB (b)
Figure16: (a) RMSE ofθ0corresponding to the first source versus time. (b) RMSE ofθ1versus time. SNR=0 dB.
90 80 70 60 50 40 30 20 10 0 DO A (d egr ee ) 10 20 30 40 50 60 Time index True trajectory Estimated trajectory
Figure17: True trajectory and estimated trajectory by LPA beam-forming.θ0=[30◦, 50◦],θ1=[0.06◦,−0.1◦]. SNR=10 dB.
sources is arbitrary and a specific case in which the motion of sources is described by a linear polynomial model. Because
90 80 70 60 50 40 30 20 10 0 DO A (d egr ee ) 10 20 30 40 50 60 Time index True trajectory Estimated trajectory
Figure18: True trajectory and estimated trajectory by REM II. SNR =10 dB. 102 101 100 10−1 10−2 RMSE o f θ0 15 20 25 30 35 40 45 50 55 60 Time index REM II LPA beamforming CRB (a) 100 10−1 10−2 10−3 RMSE o f θ1 15 20 25 30 35 40 45 50 55 60 Time index REM II LPA beamforming CRB (b)
Figure19: (a) RMSE ofθ0corresponding to the first source versus time. (b) RMSE ofθ1versus time. SNR=10 dB.
of the simple structure of the gain matrix, the suggested pro-cedures are easy to implement. Furthermore, extension of
102 101 100 10−1 RMSE o f θ 15 20 25 30 35 40 45 50 55 60 Time index REM II LPA beamforming
Figure20: RMSE ofθversus time. SNR=10 dB.
our approaches to broadband signals is straightforward. Nu-merical experiments showed that our approaches provide ex-cellent results in a slowly changing environment. When the DOA parameter changes fast or two source directions cross with each other, the procedure derived for a linear polyno-mial model has a better performance than the general pro- 18 cedure. Important issues such as step size design and conver-gence analysis are still under investigation.
ACKNOWLEDGMENTS
The authors thank the anonymous reviewers for their constructive comments that significantly improved the manuscript and also thank Associate Editor J. C. Chen for coordinating a speedy review.
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Pei-Jung Chung received the Dr.-Ing. degree in 2002 from Ruhr-Universit¨at Bochum, Germany. From 1998 to 2002, she was with the Signal Theory Group in the Department of Electrical Engineering and Information Sciences, Ruhr-Universit¨at Bochum, Germany. From May 2002 to January 2004, she held a postdoctoral position at Carnegie Mellon University, USA. Since February 2004, she has been a
Postdoctoral Researcher at the University of Michigan, Ann Arbor, USA. Her research interests include statistical signal processing and sensor array processing.
Johann F. B¨ohmereceived the Diplom in mathematics in 1966, the Dr.-Ing. degree in 1970, and the Habilitation degree in 1977, both in computer science, from the Tech-nical University of Hannover, Germany, the University of Erlangen-Nuremberg, many, and the University of Bonn, Ger-many, respectively. From 1967 to 1974, he was with the Sonar Research Laboratory Krupp Atlas Elektronik, Bremen, Germany.
He then joined the University of Bonn till 1978 and the FGAN in Wachtberg-Werthhoven. Since 1980, he has been a Professor of sig-nal theory in the Department of Electrical Engineering and Infor-mation Sciences, Ruhr-Universit¨at Bochum, Germany. His research interests are in the domain of statistical signal processing and its applications. He is an IEEE Fellow and recipient of the 2003 IEEE Signal Processing Society Technical Achievement Award.
Alfred O. Heroreceived the Ph.D. degree from Princeton University, in (1984), in electrical engineering. Since 1984, he has been a Professor with the University of Michigan, Ann Arbor, Mich, USA, where he has appointments in the Department of Electrical Engineering and Computer Sci-ence, the Department of Biomedical En-gineering, and the Department of Statis-tics. His research interests are in the areas
including estimation and detection, statistical communications, bioinformatics, signal processing, and image processing. He is a Fellow of the Institute of Electrical and Electronics Engineers (IEEE) and has received the 1998 IEEE Signal Processing Society Meritorious Service Award, the 1998 IEEE Signal Processing Soci-ety Best Paper Award, and the IEEE Third Millenium Medal.
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