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, Linköping University, Sweden
†
Dept. of Mathematical Statistics, Lund University, Sweden
ABSTRACT
Unattended Ground Sensor Networks (UGSN) are becoming increasingly popular for surveillance and situational 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 multi-sensor multitarget 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 acoustic 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, tracking, 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 triangulation. 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 challenging because of several factors such as the nonstationarity 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 targets from their acoustic sources have been developed.
Typical approaches for DOA estimation include parametric maximum likelihood (ML) subspace-based estimation 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 methods 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 matrices. This is done by exploiting the ideas behind the
recent WINGS (Wavefield-Interpolated NarrowbandGenerated 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 developed 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 simulated and real data are given in Section V. Finally,
the conclusion and future directions of research are
provided in section VI.
Algorithm 1 Wideband SPICE
1: Choose ω0 < ωmax and form aθk = aθk (ω)|ω=ω0
i=ω
2: Compute {Ti }i=1 max
Pωmax
∗
3: Compute R̂ = ω 1
i=1 Ti R(ωi )Ti
max
4:
5:
6:
7:
8:
aθ R̂−1 aH
Compute {wk } = k M θk
(0)
Set i = 0 and initialize {pk }K+M
k=1
repeat
P
(i)
∗
R(i) = K+M
k=1 pk aθk aθk
P
1/2 (i)
(i) −1 R̂1/2 k
ρ(i) = K+M
k=1 wk pk k aθk R
R(i)
(i) ka
(i+1)
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 experience 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
xk (t − τ (θk )) + w(t)
R̂1/2 k
stationary during N snapshots, the Fourier transform
of the received signal may be expressed as
x̃(ω) =
K
X
e−jωτ (θk ) x̃k (ω) + w̃(ω)
(2)
k=1
= Aω X̃(ω) + w̃(ω)
(3)
where x̃k (ω) and w̃(ω) denote the Fourier transform
of the k :th signal and noise, respectively, and
Aω = [aθ1
aθ2 . . . aθK ]
(4)
T
X̃ω = [x̃1 (ω)
II. SIGNAL MODEL
−1
pk
= pk θk 1/2 (i)
wk ρ
10:
i=i+1
(i)
(i−1) K+M
11: until {pk − pk
}k=1 ≤ 9:
x̃2 (ω) . . . x̃K (ω)]
(5)
where (·)T is the transpose operator, and
T
(6)
aθk = e−jωτ1 (θk ) e−jωτ2 (θk ) . . . e−jωτM (θk )
Assuming the incoming signals are spatially uncorrelated with each other, the covariance matrix may be
written as
∗
R = Et [x(t)x(t) ] =
K
X
pk aθk a∗θk + σI
k=1
where Et [·] is the expectation operator over t, (·)∗ the
Hermitian, σ the noise power, and pk the power emitted
by source k forming the spatial spectrum.
t = 1, 2 . . . , N
III. EXTENSION TO WIDEBAND
k=1
(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
In case of narrowband signals whose bandwidth is
small in comparison to the central frequency, ωc , the
time domain signal may be expressed as
x(t) = Aωc X(t)
(7)
the covariance matrices R(ωi ) and R(ω0 ) such that
2.5
wideband MUSIC
Capon
wideband SPICE
(10)
Aω0 Rs (ωi )A∗ω0
(11)
R(ω0 ) =
2
10log(RMSE)
R(ωi ) = Aωi Rs (ωi )A∗ωi
where Rs (ωi ) = X̃(ωi )X̃(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
1.5
1
Aω0 Rs (ωi )A∗ω0 = Ti Aωi Rs (ωi )A∗ωi Ti∗
0.5
0
−5
(12)
The matrix Ti forms the sought focusing matrix. Using
the identity [17]
0
5
SNR(dB)
10
15
20
vec(ABC) = (AT ⊗ B)vec(C)
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.
where vec and ⊗ denote the vectorization operator
and Kronecker product, respectively. Now, (12) may
be expressed as
0 = {Āω0 ⊗ Aω0 − T̄i Āωi ⊗ Ti Aωi }vec{Rs (ωi )}
¯ i ⊗ Ti )(Āω ⊗ Aω )}vec{Rs (ωi )}
= {Āω ⊗ Aω − (T
0
0
i
i
(13)
allowing the covariance matrix of x(t) to be expressed
as
R = Aωc Et [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 [Aω X̃(ω)X̃(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 signals need to be preprocessed for instance using a wideband 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
where (·) denotes the complex conjugate, implying that
in order for (13) to hold independent of frequency
specific covariance matrix, Rs (ωi ), T̄i ⊗ Ti needs to
satisfy
Āω0 ⊗ Aω0 = (T̄i ⊗ Ti )(Āωi ⊗ Aωi )
(14)
or, approximately,
(T̄i ⊗ Ti ) ≈ (Āω0 ⊗ Aω0 )(Āωi ⊗ Aωi )†
(15)
where (·)† is the pseudo inverse. From (15), Ti may be
computed by utilising the block structure of (T̄i ⊗ Ti )
as
1
(16)
Ti = √ B 1
b11
where B1 ∈ MM ×M is the first block matrix of (T̄i ⊗
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.
150
GPS
Target 1
Target 2
DOA Estimate (degrees)
100
50
0
−50
−100
−150
0
Fig. 2. The 3-sensor uniform circular microphone array.
2
4
6
8
Time (second)
10
12
14
Fig. 3. DOA estimation performance of wideband
Capon on real measurements.
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 covariance matrix for each spatial frequency of interest,
Rωi . These matrices are then refocused using the
transformation matrices Ti , given in (16), such that
Ti R(ωi )Ti∗ = R(ω0 )
(17)
allowing the sample covariance matrix to be formulated
as
ωmax
1 X
R̂ =
Ti R(ωi )Ti∗
ωmax
i=1
ωX
max
Rs (ωi )
= A ω0
A∗ω0
(18)
ωmax
i=1
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 presented, using both simulated data and real data.
V-A. Simulated data
We analyze the performance of the proposed wideband SPICE algorithm and compare its performance
with two other existing standard wideband algorithms,
namely the wideband MUSIC [2] and the Capon beamformer [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 σ 2 . The signal to noise ratio (SNR),
defined as 10 log(pk /σ 2 ) (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 θ̂ = [θ̂1 θ̂2 ] are computed using the three different methods. The Root Mean Square Error (RMSE)
defined as {(θ − θ̂)2 /N }1/2 of the estimates are plotted
−3
4.5
150
x 10
GPS
Target 1
Target 2
4
3.5
3
Signal Power Estimate
DOA estimate (degrees)
100
50
0
−50
2.5
2
1.5
−100
1
0.5
−150
0
2
4
6
8
Time (second)
10
12
14
0
−200
−150
−100
−50
0
50
100
150
200
DOA Estimate (degrees)
Fig. 4. DOA estimation performance of wideband
SPICE on real measurements.
Fig. 5. DOA estimate of the two targets (pk v/s θk )
using wideband SPICE at an arbitrary time shot.
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 beamformer’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.
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.
V-B. Field trial data
Real data is collected by conducting field trials in
an area close to Skövde, 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
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 standard 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 transformation 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 correlated 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 Löfberg for their
valuable comments.
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