DIRECTION OF ARRIVAL ESTIMATION OF UNKNOWN NUMBER OF WIDEBAND SIGNALS IN UNATTENDED GROUND SENSOR NETWORKS
G. Mathai∗, A. Jakobsson†, and F. Gustafsson∗
∗Dept. of Electrical Engineering, Link¨oping University, Sweden †Dept. of Mathematical Statistics, Lund University, Sweden
ABSTRACT
Unattended Ground Sensor Networks (UGSN) are be-coming increasingly popular for surveillance and situ-ational awareness applications. Acoustic sensors can be used in UGSN to detect and to classify targets, and these sensors are cost efficient, easy to deploy, and above all, non-jammable since they are passive. An array of acoustic sensors can detect multiple sound sources and determine the direction of arrival (DOA), and the network can deal with the sensor multi-target tracking. This contribution focuses on DOA estimation of wideband sources, such as vehicles. We develop a coherent DOA estimation method by taking advantage of the spatial sparsity of the wideband acous-tic sources as a prior information, as an extension to the recently proposed SPICE method for narrowband sources. The method has been tested on both simulated data and field test data with different vehicles with very good performance compared to other state of the art methods.
Index Terms— Sensor networks, target tracking, direction of arrival, estimation, acoustic sensors
I. INTRODUCTION
Direction of Arrival (DOA) estimation is an area which has attracted a notable interest in the literature during recent decades, partly because of its potential applications such as, for example, surveillance, track-ing, speech recognition, seismic exploration (see, e.g., [1]–[3]). Tracking in networks with group microphones can be accomplished by the fusion of DOA estimate from each node using the general methodology of trian-gulation. This contribution focuses on the challenging DOA estimation problem with poor signal to noise and several targets. Problem of detection and tracking of
acoustic signals in an open field is extremely chal-lenging because of several factors such as the non-stationarity of the sources, lack of knowledge about the number of sources, presence of multiple correlated targets, environmental noise, configuration as well as number of sensors of the acoustic array, and mutual coupling between the sensors. Several wideband DOA estimation algorithms for resolving moving ground tar-gets from their acoustic sources have been developed. Typical approaches for DOA estimation include para-metric maximum likelihood (ML) subspace-based esti-mation such as the ones developed in [4], [5], as well as non-parametric and data-adaptive methods, often based on the concepts of Capon’s MVDR beamformer, as examined in [6], [7]. The former approaches typically suffer from requiring notable a priori information on the number of sources impinging on the array and/or the structure of such sources. The non-parametric meth-ods on the other hand generally offer notably lower resolution than their parametric counterparts. A recent development in the area of compressed sensing (CS) has attracted a lot of interest in signal processing and related applications. The basic idea in CS is to exploit the sparsity of the signals in some domain to reconstruct the unknown signal. It exploits the sparsity of the spatial spectrum to produce the DOA estimate thus forming semi-parametric class of estimators such as the ones developed in [8]–[10]. One notable DOA algorithm is the sparse narrowband SPICE covariance fitting algorithm presented in [8], where it was shown to offer a notable performance gain as compared to many other DOA estimation algorithms. It has since been found to be equivalently be expressed using an `1 penalized least squared formulation, for a particular
choice of penalty [?]. In this work, we extend the SPICE algorithm to allow also for wideband sources. One option for extending the algorithm would be to
incoherently add the outputs of narrowband SPICE at each and every frequency bin to form the wideband output, but such a solution would be computationally intensive and will fail when multiple correlated sources are present. In this work, we instead resort to transform the wideband signals using the aid of focusing matri-ces. This is done by exploiting the ideas behind the recent WINGS (Wavefield-Interpolated Narrowband-Generated Subspace) algorithm [11], introducing a novel data independent transformation method having the same features as WINGS, but without resorting to Bessel decomposition and mode selection. The de-veloped method in conjunction with SPICE has been tested both on simulated and real wideband signals and as will be demonstrated it compares favorably to alternative methods in the area.
The organization of the paper is as follows. The signal model is explained in Section II. The extension of narrowband signal model to a wideband model by deriving the transformation matrices are discussed in Section III. Our proposed method, as a wideband extension to SPICE, is presented in Section IV. The performance comparison experiments using both sim-ulated and real data are given in Section V. Finally, the conclusion and future directions of research are provided in section VI.
II. SIGNAL MODEL
Consider a coplanar sensor array consisting of M omnidirectional sensors, and let θ ∈ CK×1 be a dense grid covering all possible directions of interest. It is here assumed that this grid has been selected fine enough, so that the sources may be viewed as being reasonably close to some of those grid points. This is clearly a restrictive assumption, but practical experi-ence shows that the developed estimator is quite robust to this assumption (see also the related discussion in [12], [13]). A signal impinging on the array at time t may be modelled as x(t) = K X k=1 xk(t − τ (θk)) + w(t) t = 1, 2 . . . , N (1) where xk(t) denotes the signal impinging from k:th
source, τ (θk) ∈ CM ×1 are the propagation delays
between source k and the sensor array, and w(t) ∈ CM ×1 is additive noise. Assuming the sources to be
Algorithm 1 Wideband SPICE
1: Choose ω0 < ωmax and form aθk = aθk(ω)|ω=ω0
2: Compute {Ti}i=ωi=1max
3: Compute ˆR = ω1 max Pωmax i=1 TiR(ωi)Ti∗ 4: Compute {wk} = aθkRˆ−1aHθk M
5: Set i = 0 and initialize {p(0)k }K+Mk=1
6: repeat 7: R(i)=PK+M k=1 p (i) k aθka ∗ θk 8: ρ(i) =PK+M k=1 w 1/2 k p (i) k k aθkR (i)−1Rˆ1/2k
9: p(i+1)k = p(i)k kaθkR(i) −1Rˆ1/2k
w1/2k ρ(i)
10: i = i + 1
11: until {p(i)k − p(i−1)k }K+Mk=1 ≤
stationary during N snapshots, the Fourier transform of the received signal may be expressed as
˜ x(ω) = K X k=1 e−jωτ (θk)x˜ k(ω) + ˜w(ω) (2) = AωX(ω) + ˜˜ w(ω) (3)
where ˜xk(ω) and ˜w(ω) denote the Fourier transform
of the k:th signal and noise, respectively, and
Aω= [aθ1 aθ2. . . aθK] (4)
˜
Xω= [˜x1(ω) x˜2(ω) . . . ˜xK(ω)]T (5)
where (·)T is the transpose operator, and aθk =e
−jωτ1(θk) e−jωτ2(θk) . . . e−jωτM(θk)T (6)
Assuming the incoming signals are spatially uncorre-lated with each other, the covariance matrix may be written as R = Et[x(t)x(t)∗] = K X k=1 pkaθka ∗ θk+ σI
where Et[·] is the expectation operator over t, (·)∗ the
Hermitian, σ the noise power, and pkthe power emitted
by source k forming the spatial spectrum.
III. EXTENSION TO WIDEBAND
In case of narrowband signals whose bandwidth is small in comparison to the central frequency, ωc, the
time domain signal may be expressed as
−50 0 5 10 15 20 0.5 1 1.5 2 2.5 SNR(dB) 10log(RMSE) wideband MUSIC Capon wideband SPICE
Fig. 1. Estimation performance by MUSIC, Capon and wideband SPICE using synthetic data for two correlated wideband sources placed at 145◦ and −88◦ as a function of the SNR, using 200 Monte Carlo simulations.
allowing the covariance matrix of x(t) to be expressed as
R = AωcEt[X(t)X
∗(t)]A∗
ωc (8)
For a wideband signal this expression is not valid anymore, but may in this case be written as
R = Eω[AωX(ω) ˜˜ X(ω)∗A∗ω] (9)
Unlike in the narrowband case, the expectation is over ω, making us unable to move Aω outside the
expectation. In order to allow for this, one has to reformulate Aω so that it separates the parts dependent
and independent of ω, thereby giving (9) a structure similar to the narrowband case. To enable use of DOA algorithms tailored for the latter case, the measured sig-nals need to be preprocessed for instance using a wide-band focusing technique, such as the ones developed in [14], [15], [16] and [11]. Here, exploiting a foundation reminiscent to the WINGS focusing presented in [11], we construct a focusing matrix, Ti, such that it focuses
a spatial frequency ωi onto a projection frequency ω0,
so that the resultant can be added coherently to form a covariance matrix of narrowband structure. Consider
the covariance matrices R(ωi) and R(ω0) such that
R(ωi) = AωiRs(ωi)A ∗ ωi (10) R(ω0) = Aω0Rs(ωi)A ∗ ω0 (11)
where Rs(ωi) = ˜X(ωi) ˜X(ωi)∗, and note that R(ωi)
and R(ω0) are Hermitian matrices with same inertia
therefore being congruent to each other via some matrix Ti, such that
Aω0Rs(ωi)A ∗ ω0 = TiAωiRs(ωi)A ∗ ωiT ∗ i (12)
The matrix Tiforms the sought focusing matrix. Using
the identity [17]
vec(ABC) = (AT ⊗ B)vec(C)
where vec and ⊗ denote the vectorization operator and Kronecker product, respectively. Now, (12) may be expressed as
0 = { ¯Aω0 ⊗ Aω0− ¯TiA¯ωi⊗ TiAωi}vec{Rs(ωi)}
= { ¯Aω0 ⊗ Aω0− ¯(Ti⊗ Ti)( ¯Aωi ⊗ Aωi)}vec{Rs(ωi)}
(13) where (·) denotes the complex conjugate, implying that in order for (13) to hold independent of frequency specific covariance matrix, Rs(ωi), ¯Ti⊗ Ti needs to
satisfy ¯ Aω0⊗ Aω0 = ( ¯Ti⊗ Ti)( ¯Aωi ⊗ Aωi) (14) or, approximately, ( ¯Ti⊗ Ti) ≈ ( ¯Aω0⊗ Aω0)( ¯Aωi⊗ Aωi) † (15) where (·)†is the pseudo inverse. From (15), Timay be
computed by utilising the block structure of ( ¯Ti⊗ Ti)
as Ti= 1 √ b11 B1 (16)
where B1 ∈ MM ×M is the first block matrix of ( ¯Ti⊗
Ti) and b11 is the first element of B1. In order to
determine Ti, it is required that all the elements B1
are positive and real. It is worth noticing that Ti will
be data independent, and therefore suitable for arbitrary geometry arrays. It should be stressed that the above focusing matrix coincides with the one proposed in [11] but, by using (16), it can be formed in a much simpler way, avoiding the need for Bessel decomposition or mode selection.
Fig. 2. The 3-sensor uniform circular microphone array.
IV. THE WIDEBAND SPICE METHOD The proposed wideband SPICE based algorithm is thus formed by transforming impinging time domain signal into the frequency domain, forming the co-variance matrix for each spatial frequency of interest, Rωi. These matrices are then refocused using the
transformation matrices Ti, given in (16), such that
TiR(ωi)Ti∗= R(ω0) (17)
allowing the sample covariance matrix to be formulated as ˆ R = 1 ωmax ωmax X i=1 TiR(ωi)Ti∗ = Aω0 ωmax X i=1 Rs(ωi) ωmax A∗ω0 (18) This focused covariance matrix is then exploited to construct the SPICE algorithm, such that the wideband SPICE algorithm may be written as in Algorithm 1.
V. RESULTS
In this section, results and comparisons are pre-sented, using both simulated data and real data. V-A. Simulated data
We analyze the performance of the proposed wide-band SPICE algorithm and compare its performance
0 2 4 6 8 10 12 14 −150 −100 −50 0 50 100 150 Time (second)
DOA Estimate (degrees)
GPS Target 1 Target 2
Fig. 3. DOA estimation performance of wideband Capon on real measurements.
with two other existing standard wideband algorithms, namely the wideband MUSIC [2] and the Capon beam-former [11]. For the synthetic case, we considered an experimental scenario with an isotropic non-coupling uniform circular array (UCA) with three sensors with a radius of 0.15 metres. Two wideband sources were being placed at θ = [θ1 θ2] where θ1 = 145◦ and
θ2 = −88◦. The frequency spectra of the sources are
overlapping and are uniformly distributed between 100 and 300 Hz. The half wavelength criteria on sensor separation puts a limitation on the highest frequency content of the sources to be used for DOA estimation in order to avoid spatial ambiguity in the estimated DOA. The source signals impinging on the sensors are corrupted with additive Gaussian noise of zero mean and variance σ2. The signal to noise ratio (SNR), defined as 10 log(pk/σ2) (in dB), is varied from -5 to
20 by varying the variance of the noise. For wideband SPICE, we have chosen the projection frequency to be the one on the lower side of the frequency spectrum, i.e., ωp = 100 Hz because the contrary will create
numerical instability in the computation of Ti which
in turn could cause errors in final DOA estimate. The field of view is (−180◦, 180◦] and is segmented into K = 360 uniformly spaced points. The estimates of DOA ˆθ = [ˆθ1 θˆ2] are computed using the three
dif-ferent methods. The Root Mean Square Error (RMSE) defined as {(θ − ˆθ)2/N }1/2of the estimates are plotted
0 2 4 6 8 10 12 14 −150 −100 −50 0 50 100 150 Time (second)
DOA estimate (degrees)
GPS Target 1 Target 2
Fig. 4. DOA estimation performance of wideband SPICE on real measurements.
in Fig 1 against various SNRs, where N = 200 is the number of Monte Carlo runs. The plot confirms the inability of wideband MUSIC to deal with correlated sources. The performance of wideband MUSIC does not improve with increased SNR. The Capon beam-former’s performance improved with SNR although the error is rather high at low SNR scenarios. Wideband SPICE outperforms both methods both at low as well as high SNR scenarios which validates its superiority in accuracy.
V-B. Field trial data
Real data is collected by conducting field trials in an area close to Sk¨ovde, in Southern Sweden. The microphone array used for this experiments is a three sensor UCA having radius 0.15 meters as shown in figure 2. More details of the experiment could be found in Section VI in [18]. Target 1 is a four-wheel Nissan Kingcab and target 2 is a two-wheel moped were driven through a predefined track. The combination of the sounds of the sources are collected by the microphone array which is located about 250 m during the start of the experiment. The vehicles come as close as 50 m towards the sensors before they recede away. The signals are sampled at 48 KHz. Vehicles are equipped with GPS sensors whose output serves as the ground truth. The spectra of these wideband non-stationary sources are analyzed and all the major frequencies lie
−2000 −150 −100 −50 0 50 100 150 200 0.5 1 1.5 2 2.5 3 3.5 4 4.5x 10−3
DOA Estimate (degrees)
Signal Power Estimate
Fig. 5. DOA estimate of the two targets (pk v/s θk)
using wideband SPICE at an arbitrary time shot.
between 90 and 300 Hz. Similar to what was done with the synthetic data, we have chosen the projection frequency to be the lowest one of the spectra, i.e., 90 Hz. We decided not to use wideband MUSIC for the DOA estimation since the spectra of the two sources can be correlated. We compared the performance of wideband SPICE with Capon beamformer. Figure 3 shows the plot of of Capon beamformer’s estimate against time. It is able to provide good tracking for one of the targets but the estimates deviate as much as 100◦ from the ground truth for the second target. Meanwhile wideband SPICE as shown in figure 4 was able to provide smooth continuous tracking for both the targets which follow well the true trajectory. Figure 5 shows the instantaneous DOA output of wideband SPICE for the two targets.
VI. CONCLUSION
In this work, we considered the problem of DOA estimation of non-stationary wideband acoustic sources in a ground sensor network. We extended the stan-dard narrowband SPICE DOA estimation algorithm with the aid of frequency transformation matrices to form wideband SPICE which makes it suitable for wideband DOA estimation applications. These trans-formation matrices are data independent which can be computed offline which make them useful for real time applications. Unlike other focusing methods, it does
not require initial values of DOA for the computation. The performance of wideband SPICE is compared with wideband MUSIC and Capon beamformer for corre-lated multiple sources both using synthetic data and real data. The results confirmed the ability of wideband SPICE to simultaneously track multiple targets in low SNR scenario situations. Future research will focus on increasing the spatial resolution as well as increasing the number of targets it could handle.
VII. ACKNOWLEDGMENT
The research leading to these results has received funding from the EUs Seventh Framework Programme under grant agreement no 238710 and the research has been carried out in the MC IMPULSE project. The authors wish to thank FOI (Swedish Defence Agency) FOCUS excellence centre for providing the real data as well as Egils Sviestins and Johan L¨ofberg for their valuable comments.
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