Proceedings of the IASTED Internatioanl Conference
Signal and Image Processing (SIP’99)
October 18-21, 1999 - Nassau, Bahamas
TIME-DELAY SELECTION FOR ROBUST ANGLE OF ARRIVAL
ESTIMATION
JARI YLI-HIETANEN, KONSTA KOPPINEN, JAAKKO ASTOLA
Tampere University of Technology
Signal Processing Laboratory
P.O. Box 553, FIN-33101 Tampere, Finland
email {jari.yli-hietanen, konsta.koppinen, jaakko.astola}@cs.tut.fi
ABSTRACT
2. METHOD
The differences in the arrival times of a signal at spatially separated sensors can be used to estimate the angle
of arrival of the signal. Due to noise or sensor malfunction, the time difference estimates may be erroneous and
thus the angle of arrival estimate may be inaccurate. In this
paper, an angle of arrival estimation method which is
robust with respect to outliers caused by large errors in
time delay estimates is presented. The method utilize the
dependencies in the arrival times to reject the time delay
estimates which are outliers. The computational load of
the method is also considered.
Let us assume that the sensor array is receiving a wave
propagating signal caused by a distant event. Thus the
time delay ( τ im ) between the signals received by the ith
and the mth sensor is
T
τ im = k ( x i – x m )
where xi and xm are three-dimensional vectors representing the sensor locations in an orthogonal coordinate system and k is the propagation vector of the plane wave
which determines the direction and the velocity (c) of
propagation of the plane wave, i.e.
1
(2)
k = --- .
c
Throughout this text, x denotes the L2-norm of the vector x . The vector
(3)
x im = x i – x m
1. INTRODUCTION
The angle of arrival of a signal can be determined using
the differences in the arrival times of the signal at spatially
separated sensors. If the signal to noise ratio is high
enough, these differences, i.e. time delays, can be estimated by simple methods, e.g. polarity coincidence correlation [1]. This is a well-established discipline and many
articles have been published on it [2].
is called a sensor vector.
The estimate of k is obtained as a solution of the matrix
equation
(4)
Vk = d
where
Noise can cause a time delay estimate to be very inaccurate, especially when the noise received by the sensors
is correlated, due to some other signal source. Malfunctions in one or more sensors can have the same effect. In
this text we define an outlier to be an estimate which deviates from the correct value by at least a relatively large
predefined constant. The use of the time delay estimates
that are outliers leads to an inaccurate estimate of the angle
of arrival. For example, only one time delay estimate
which is an outlier can cause the angle of arrival to be an
outlier too when the method presented in [3] is used. An
robust time-delay based angle of arrival estimation method
was presented in [4]. The method is based on the subgroups of three time delay estimates and the search of the
subgroup giving the best fitting estimate of propagation
vector. An threshold-based distance measure was used to
select the estimate of propagation vector which best fits to
the time delay estimates.
V = xi m xi m … xi
1 1
2 2
d = di m di m … di
1 1
2 2
T
M mM
T
M
,
(5)
mM
d im is the estimate of the time delay τ im and M is the
number of the time delay estimates used. The least squares
solution of equation (4) was used and
T
–1 T
k LS = ( V V ) V d
(6)
was taken as the estimate of k in [3]. This estimate is not
robust because only one time delay estimate that is an outlier causes remarkable error in the estimate of k.
Due to the planewave assumption (equation (1)) the
three time delays ( τ il , τ im , and τ lm ) between three sensors (i, l, and m) are linearly dependent,
(7)
τ il + τ lm – τ im = 0 .
In this paper, a robust time-delay based angle of arrival
estimation method which does not require the search over
the propagation vector estimates is presented. The method
is based on the selection of the time delay estimates to be
included in the estimation of the angle of arrival, i.e. the
time delay estimates that are assumed to be outliers are
detected and rejected. This kind of selection is possible
due to the dependencies between the arrival times of the
signal at the sensors.
296-196
(1)
This gives the possibility to detect the time delay estimates
which are outliers and due to that a test statistic is defined:
 1, if d il + d lm – d im < THR ilm
.
(8)
∆ ilm = 
 0, otherwise
-1-
higher than that of the least squares method. The additional operations that need to be performed when compared to the least squares method are given by equations
(8) and (11) and the search over the sensors to find the
‘master sensor’ î .
The selection of the thresholds THR ilm is based on the
norm of the maximum tolerable error (denoted by D ( k ) ) in
the propagation vector estimate:
α il + α lm – α im
(9)
THR ilm = 2D ( k ) sin ( α il ) cos  -------------------------------------


2
where α il is the angle between the lines which coincide
3. SIMULATION RESULTS
with the vectors x mi and x ml :
The section presents simulation results to demonstrate
the performance of the proposed method. The simulations
consisted of 10000 realisations of a situation with an small
sensor array consisting of six omnidirectional sensors, a
signal source with a fixed position and an interfering signal source at a random position. Independent Gaussian
noise was added to the signals received by the sensors to
model measurement and background noise. The possible
malfunctioning of the sensors was also modelled.
T
x mi x ml


α il = acos  --------------------------
x
x
 mi ml 
(10)
An indication of the reliability of a time delay can be
(τ)
obtained by calculating the confidence factor ( conf im ) of
the time delay τ im :
The array used in the simulations consisted of six sensors located at corners of the cubic of size l x l x l except
corners (l,0,0) and (0,l,0) where l is 10c times the sampling interval. The array is shown in Fig. 1. The source
signal used was a sum of 5 sinusoids with frequencies of
0.02π, 0.06π, 0.1π, 0.26π, and 0.46π with respect to the
sampling frequency 2π and with a common amplitude.
The source signal was set to propagate as a plane wave
from the direction θ = 0.25π, φ = 0.25π . The polar coordinate system used is shown in Fig. 2. The same signal but
propagating as a plane wave to a different direction was
used as an interfering signal. For each of 10000 realisations, the direction of propagation was randomly selected
such that θ was chosen from a uniform distrubution on
[0,π] and φ from a uniform distrubution on [0,2π] . The
variance of the the interfering signal was 0.49vs where vs
was the power of the test signal.
K
(τ)
∑
conf im =
∆ ilm
(11)
l = 1
l ≠ i, m
where K is the number of sensors in the array. In order to
achieve a robust estimate of the propagation vector the
search is done to find the ‘master sensor’ î and the time
delays τ , m ∈ I which maximize the minimum of the
îm
î
confidence factors conf
(τ)
,
îm
m ∈ I with the condition that
î
there exist three linearly independent vectors among the
sensor vectors x , m ∈ I . After that the estimate of the
îm
î
propagation vector is simply the least-squares solution of
the equation (4) formed of these selected time delays and
sensor vectors.
The signals received by the sensors were the sum of the
source signal and the interfering signal delayed according
to their respective direction of propagation and white independent gaussian noise with variance 0.1vs. Malfuctions in
the sensors were simulated by setting with probability 0.15
the signal received by a sensor to white independent gaussian noise with variance 0.1vs. Each sensor either func-
The computational load of the method is only slightly
z
z
θ
y
k
y
φ
x
x
Fig. 2. The polar coordinate system
Fig. 1. The array
296-196
-2-
tioned or malfunctioned for the whole duration of the
realisation.
4. CONCLUSION
In this paper we presented a time delay selection
method for angle of arrival estimation that is more robust
than the least squares method and computationally significantly less demanding than method of [4].
All the 15 possible time delays were estimated using
polarity coincidence correlation [1]. The length of each
realisation was 5000 samples. For each realisation, the
estimate of the propagation vector k was formed using the
least squares solution, the method proposed in [4], and the
method proposed in this paper. The thresholds THR ilm
REFERENCES
[1]
R. Cusani, Performance of fast time delay estimators, IEEE Trans. Acoust., Speech, Signal Processing,
vol. 37, no 5, pp. 757-759, May 1989.
[2] Special issue on time-delay estimation, IEEE Trans.
Acoust., Speech, Signal Processing, vol. ASSP-29, no
3, June 1981.
[3] J. Yli-Hietanen, K. Kalliojärvi, J. Astola, Low-complexity angle of arrival estimation of wideband signals using small arrays, Proc. 8th IEEE Signal
Processing Workshop on Statistical Signal and Array
Processing, Corfu, Greece, June 1996, pp. 109-112.
[4] J. Yli-Hietanen, K. Kalliojärvi, J. Astola, Robust
Time-Delay Based Angle of Arrival Estimation, Proc.
1996 IEEE Nordic Signal Processing Symposium,
Espoo, Finland, September 1996, pp. 219-222.
were selected using equation (9) with D ( k ) set to 0.1/c.
The error criteria used was the norm of the difference
between the estimated and the true propagation vectors.
Both the estimated and the true propagation vectors were
scaled to have unity norm before calculating the difference
vector.
The cumulative distribution functions of the difference
vector norms estimated from the 10000 realisations for the
different methods are shown in Fig. 3. The bottom solid
line represents the least squares method, the middle
dashed line represents the proposed method and the top
dotted line represents the method proposed in [4]. As seen
from the figure the probability of errors larger than any
threshold is greater for the least squares method than for
the two other methods. The error probability of the method
of [4] is smaller than that of the proposed method. However the computational complexity of the proposed
method is significantly less than the method of [4].
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Fig. 3. Estimated cumulative distributions of errors. Solid line: Least squares, dotted line: Method of [4], dashed line: Proposed method
296-196
-3-