Hindawi Publishing Corporation

International Journal of Antennas and Propagation

Volume 2013, Article ID 104848, 9 pages
http://dx.doi.org/10.1155/2013/104848

Research Article
Direction-of-Arrival Estimation of Closely Spaced Emitters Using
Compact Arrays

Hoi-Shun Lui1, 2 and Hon Tat Hui3

1
Department of Signals and Systems, Chalmers University of Technology, 412 96 Gothenburg, Sweden
2
School of Information Technology and Electrical Engineering, The University of Queensland, Brisbane, QLD 4072, Australia
3
Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576

Correspondence should be addressed to Hoi-Shun Lui; h.lui@uq.edu.au

Received 14 May 2012; Accepted 1 November 2012

Academic Editor: Charles Bunting

Copyright © 2013 H.-S. Lui and H. T. Hui. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.

Performance evaluation of direction-of-arrival (DOA) estimation algorithms has continuously drawn significant attention in the
past years. Most previous studies were conducted under the situation that antenna element separation is about half wavelength in
order to avoid the appearance of grating lobes. On the other hand, recent developments in wireless communications have favoured
the use of portable devices that utilize compact arrays with antenna element separations of less than half wavelength. Performance
evaluation of DOA estimation algorithms employing compact arrays is an important and fundamental issue, but it has not been
fully studied. In this paper, the performance of the matrix pencil method (MPM) that applies to DOA estimations is investigated
through Monte Carlo simulations. The results show that closely spaced emitters can be accurately resolved using linear compact
array with an array aperture as small as around half wavelength.

1. Introduction Another well-known subspace method is Estimation of

Signal Parameters via Rotational Invariance Techniques
Direction-of-arrival (DOA) estimation is one of the most (ESPRIT) [9, 10], which exploits the array structure such that
important applications in array signal processing. In the the specific knowledge of the array manifold is not required.
context of array signal processing, these algorithms can be Compared to MUSIC, it is computationally more efficient
grouped into three categories: conventional methods, sub- without the need of searching peaks throughout the entire
space methods, and maximum likelihood techniques [1–4]. range of angles. A review about subspace methods from the
Throughout the last two decades, subspace methods have statistical prospective can be found in [4]. In addition, there
achieved great success due to their low complexity and rea- is another subspace method, known as the matrix pencil
sonable performance. Out of all the subspace methods, Pisa- method (MPM), which was first applied to damped exponen-
renko’s method [4] was one of the first methods that exploited tial extractions from transient electromagnetic signals [11–
the structure of the data model. Schmidt [5] then studied the 13] and later to DOA estimations [14–20]. One significant
signal properties of a sensor array and introduced the Mul- advantage of MPM over MUSIC and ESPRIT is that DOA
tiple Signal Classification (MUSIC) algorithm. The MUSIC estimation can be done using a single snapshot instead of the
algorithm exploits the geometrical structure of the signal. covariance matrix that requires a large number of snapshots
Performance evaluations of the MUSIC algorithm have been [15, 16]. The incoming directions can be determined directly
well studied in the 1980s to mid-1990s, for example [6, 7]. The without the need for searching among all the directions as in
main advantage of subspace methods over the conventional the MUSIC. Even with coherent signals, spatial smoothing is
beamforming method is that the resolution is not limited by not required.
the aperture of the entire array, and subspace methods are The DOA estimation problem is about estimation of
therefore also known as super resolution techniques [8]. undamped exponentials based on the measured samples of
2 International Journal of Antennas and Propagation

the signal from an array. Performance evaluations of subspace performance of subspace methods, however, cannot be easily
methods and the corresponding Cramer-Rao Lower Bound observed.
(CRLB), which gives the lower bound on the variance of any In view of this, this paper aims to characterize the effect
unbiased estimator, have been well studied for the estimation of (i) changes in the array aperture with a fixed number
of a single damped/undamped exponential (e.g., [21–28]). of elements and (ii) changes in the number of elements in
Attempts have also been made for the case with two expo- a fixed aperture on the accuracies DOA estimations with
nentials embedded in the signal. For instance, Stoica and signals corrupted by Additive Gaussian White Noise
Nehorai [26] have studied the DOA estimation performance (AGWN) using a Monte Carlo approach. In particular,
of MUSIC for the case of two equal powered signals, and the MPM will be considered. The focus of this study is on
the results showed that the variance of MUSIC algorithm is the effect of variations of the array aperture on the MPM
smaller when (i) the number of sensors (the length of data) algorithm under the ideal sensor condition. A brief review of
increases, (ii) the signals are not highly correlated, and (iii) MPM is first given followed by the Monte Carlo simulations
the two signals are not closely spaced. Steedly and Moses and discussions towards the end of the paper.
[27] derived the complete expressions for the CRLB of the
parameters of an exponential model with one set of damped/
undamped exponentials and multiple sets of amplitude coef- 2. Matrix Pencil Method (MPM) for
ficients. For the case of two exponentials, numerical examples DOA Estimation
have shown that the angle CRLB is lower as the data length is
large enough, and the signals are not closely spaced. In these Consider a space where there are 𝑃 + 1 signals coming from
studies, multiple snapshots of the signals or a single snapshot different directions and a uniform linear array (ULA) of
with long data length [27] are considered, and the separation 𝑁 + 1 identical sensors. The sensors are located along the
between the sensors is usually fixed to the half wavelength. 𝑥-axis with element separation between adjacent sensors
This is equivalent to the case where the array aperture is denoted by Δ. Then, the signal 𝑥(𝑛) received by the sensors
large (more than a wavelength if more than 3 elements). Such can be written as [11, 14–22]
arrays are sometimes known as “standard arrays.” The half
wavelength separation is essentially the upper limit of Nyquist 𝑃 𝑗2𝜋Δ𝑛 cos 𝜙𝑝
sampling rate [4] that avoids the appearance of grating lobes 𝑥 (𝑛) = ∑ 𝐴 𝑝 exp (𝑗𝛾𝑝 + ), 0 ≤ 𝑛 ≤ 𝑁,
in the visible region of an array. At the same time, it provides 𝑝=0 𝜆
the largest possible array aperture potentially resulting in the (1)
highest resolution, for instance, in conventional beamformers
[1]. In addition, another practical consideration is the mutual where 𝐴 𝑝 , 𝛾𝑝 , and 𝜙𝑝 are, respectively, the amplitude, phase,
coupling effect between antenna elements. In most array and DOA of each of the 𝑃 + 1 plane wave sources incident
signal processing algorithms, the algorithms are developed on the ULA. 𝜆 is the wavelength of the operation frequency.
under the assumption that interactions between array ele- Equation (1) can also be written as
ments can be ignored. Mutual coupling is usually treated as
a source of noise. Calibrations and preprocessing procedures 𝑃
are required to compensate such an undesirable effect [29– 𝑥 (𝑛) = ∑ 𝑐𝑝 𝑦𝑝𝑛 , (2a)
31]. Recent work has demonstrated that the undesirable 𝑝=0
mutual coupling effect can be effectively compensated for
arrays down to ≈ 0.1𝜆, both numerically [32, 33] and experi-
where
mentally [34].
In the last two decades, there has been significant
interest in reducing the physical dimensions of existing 𝑐𝑝 = 𝐴 𝑝 exp (𝑗𝛾𝑝 ) , (2b)
electronic devices with the rapid development of the elec-
tronics industry. In line with this development, recent (𝑗2𝜋Δ cos 𝜙𝑝 )
𝑦𝑝 = exp [ ]. (2c)
applications in antenna arrays have been focused on com- 𝜆
pact array design [35–37] with less than a half wave-
length element separation. To our knowledge, the perfor-
In matrix form, this can be written as
mance of subspace methods under the situations where
(i) the array aperture is significantly reduced to less than
a wavelength and (ii) the increment of the number of 𝑥 (0) 1 1 ... 1 𝑐0
[ 𝑥 (1) ] [ 𝑦0 𝑦1 . . . 𝑦𝑝 ] [ 𝑐1 ]
sensors is within such a small array aperture has not been [ ] [ ][ ]
[ .. ] = [ .. .. . . .. ] [ .. ] (2d)
well exploited. Perturbation analysis for subspace meth- [ . ] [ . . . . ][ . ]
ods and the corresponding CRLB are well studied for 𝑁 𝑁 𝑁
the single exponential process. When there is more than [𝑥 (𝑁)] [𝑦0 𝑦1 ⋅ ⋅ ⋅ 𝑦𝑝 ] [𝑐𝑝 ]
one mode embedded in the signals, however, the variance
or
[21] and the CRLB [27] of the estimation problem are
given in the form of matrix expressions. The results of how
array aperture and sampling rates change can affect the x = Yc. (2e)
International Journal of Antennas and Propagation 3

Next, we define the 𝐿 × (𝑁 − 𝐿 + 2) matrix X󸀠 using 𝑥(𝑘), the where 𝑝 = 1, 2, . . . , 𝑃 + 1 (𝑃 + 1 singular values from (6a),
elements of X, (6b) and (6c)). It is well known that if the argument of the
𝑥 (0) 𝑥 (1) ⋅ ⋅ ⋅ 𝑥 (𝑁 − 𝐿 + 1) function cos−1 in (7) is greater than 1 or a complex number,
[ 𝑥 (1) 𝑥 (2) ⋅ ⋅ ⋅ 𝑥 (𝑁 − 𝐿 + 2)] the resultant angle will become complex which is considered
[ ]
X󸀠 = [ .. .. .. .. ], (3) as a fail estimate for MPM. The resultant complex estimation
[ . . . . ] angle can be used as a sanity check which indicates that MPM
[𝑥 (𝐿 − 1) 𝑥 (𝐿) ⋅ ⋅ ⋅ 𝑥 (𝑁) ] fails to resolve all the signals in that particular snapshot.
where 𝐿 is the pencil parameter which satisfies the criteria of
(𝑁 + 1 − 𝑃) ≥ 𝐿 ≥ (𝑃 + 2). Applying a singular value decom- 3. Monte Carlo Simulations
position (SVD) to X󸀠 , we have
X󸀠 = UΣV𝐻. (4) This section presents Monte Carlo simulations of DOA esti-
mations using ULAs with different array apertures and dif-
The superscript “𝐻” denotes the complex conjugate transpose ferent number of elements. Three cases are presented to illus-
of a matrix. If the data is not contaminated by noise, the trate our findings. First, a four-element ULA with different
first 𝑃 + 1 singular values are nonzero, and, hence, 𝑃 can be apertures under the illumination of two closely spaced emit-
determined. If the data is noise contaminated, the parameter ters is studied. Then, ULAs with fixed apertures but different
𝑃 is estimated by observing the ratio of the various singular numbers of elements are considered. Lastly, DOA estimations
values to the largest one as defined by [11] of three closely spaced incoming signals using arrays with a
𝜎𝑐 fixed aperture but different numbers of elements are studied.
≈ 10−𝑤 , (5)
𝜎max Case 1 (Four-Element ULAs with Different Apertures (2 Incom-
where 𝑤 is the number of accurate significant decimal digits ing Signals)). A ULA with four ideal isotropic sensors is
of the data 𝑥(𝑛). The singular values for which the ratio in studied. The objective is to investigate how the accuracy of
(5) is below 10−𝑤 are essentially noise singular values, and DOA estimation is affected by the reduction of the array
they should be discarded. It should be noted that when the aperture. The separation of the array elements is changed
incoming signals are spatially well resolved, these results in from Δ = 0.5𝜆 down to 0.05𝜆, with a step size of 0.05𝜆. This
larger singular values are easier to be solved. For instance, if corresponds to a reduction of the array aperture (defined as
we have two cases with incoming signals from the directions the physical length of the array) from 1.5𝜆 to 0.15𝜆. The case
of (i) 5∘ and 6∘ and (ii) 5∘ and 46∘ , the second singular value in with Δ = 0.5𝜆 is the usual “standard” interelement separation
case (ii) is larger than that in case (i). As a result, only the cases used in those conventional arrays.
of closely spaced emitters are considered in this work. The Two equal-power and equal-phase coherent sources come
proposition here is that if we could find out how MPM per- from the azimuth angles of 𝜙1 = 5∘ and 𝜙2 = 6∘ , which are
forms under various array configurations for closely spaced close to the endfire direction (𝜙 = 90∘ corresponds to the
emitters, it should also be able to work for spatially well- broadside direction). The signals (both magnitude and phase)
resolved signals. are contaminated with AGWN at a signal-to-noise ratio
Once 𝑃 is determined based on the condition given in (SNR) of 5 dB. To evaluate the performance of the DOA esti-
(5), the submatrix of U with the first 𝑃 + 1 columns can mations, the average bias, 𝑏, is introduced and used as a
be formed and denoted as U𝑠 . To estimate the DOAs, the metric throughout this paper. It is given by
following matrix pencil problem is solved:
𝑄
𝑏𝑖
U2 − 𝜆󸀠 U1 = 0, (6a) 𝑏=∑ , (8)
𝑖=1 𝑄
or equivalently,
where 𝑏𝑖 = |𝜙̂𝑖 − 𝜙𝑖 | is the absolute value of the bias of the
U1 + U2 − 𝜆󸀠 I = 0, (6b) estimated direction (𝜙̂𝑖 being the mean of the estimated
where U1 is U𝑠 with the last row deleted, U2 is U𝑠 with the directions; 𝜙̂𝑖 is determined from 10,000 independent Monte
first column deleted, 𝜆󸀠 is the eigenvalue of the matrix pencil Carlo simulations), 𝑄 is the number of incoming signals
problem, I is the (𝑃 + 1) × (𝑃 + 1) identity matrix, and the which is 2 in this case. For a successful DOA estimation, all
superscript “+” denotes the Moore-Penrose pseudoinverse of the signals should be accurately estimated. A small value of
a matrix [21] defined as average bias is obtained only if all the incoming signals are
−1 accurately estimated. At each value of interelement separation
U1 + = {U1 𝐻 U1 } U1 𝐻. (6c) (Δ) shown in the figures in this paper, the average bias is
obtained by 10,000 independent Monte Carlo simulations.
The superscript “−1” denotes the inverse of a matrix. Once
The results are shown in Figure 1. It is shown that the aver-
(6a), (6b), and (6c) are solved, the DOAs can be obtained by
age biases decrease as the interelement separation increases.
the following formula [16, 18]:
This indicates that the estimated directions are more accurate
−1
𝜆 ln (𝜆 𝑝 󸀠 ) when the interelement separation is large. However, when
𝜙𝑝 = cos [ ], (7) Δ = 0.05𝜆, the average bias is 3∘ when the SNR levels of the
𝑗2𝜋Δ two incoming signals are both 5 dB. This means that MPM
4 International Journal of Antennas and Propagation

6 the two signals are also included in Figure 2(b). At Δ = 0.05𝜆,

the biases of 𝜙1 = 5∘ and 𝜙2 = 6∘ are at the levels of 0.5∘ and 3∘ ,
5 respectively, which is similar to the results for the two signals
with equal power in Figure 1. Such biases indicate that the two
Average bias/bias (deg)

4 signals cannot be accurately resolved with this array aperture.

To investigate the performance of MPM for DOA esti-
3 mations when the incoming signal direction changes, DOA
estimation problems with the two signals coming from the
2 directions of 𝜙1 = 10∘ to 90∘ and 𝜙2 = 𝜙1 + 1∘ are considered.
Here, only the angular range of 0∘ to 90∘ is considered
1 as the angular range of 90∘ to 180∘ is simply the mirror image
of that of 0∘ to 90∘ . The same DOA estimation calculation
0 procedure is repeated for all cases independently. The step
size of Δ𝜙1 = 5∘ is considered, and the results are shown in
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Figure 3. Similarly, the average biases decrease as Δ increases
Interelement separation for the same incoming signals. The results show that the
SNR = 5 (avg. bias) SNR = 3 (bias 1) average bias is below 10−3∘ when the array element separation
SNR = 3 (avg. bias) SNR = 5 (bias 2) is greater than or equal to 0.2𝜆. Furthermore, we also note
SNR = 5 (bias 1) SNR = 3 (bias 2) from the figure that as the signals are coming from a direction
closer to the broadside direction of 90∘ , the accuracy of esti-
Figure 1: Summary of DOA estimations using MPM for a four- mation increases quite significantly even when the element
element ULA with interelement separation varying from 0.05𝜆 to separation remains the same. For instance, when the element
0.5𝜆. Two equal-power incoming signals are from the directions of separation is 0.05𝜆 with incoming signals of 𝜙1 = 10∘ and
𝜙1 = 5∘ and 𝜙2 = 6∘ with an SNR of 5 dB and 3 dB, respectively. The 𝜙2 = 11∘ , an average bias of ∼ 0.7∘ is resulted. When the
graph shows (i) average bias of the estimated directions and (ii) bias incoming signals become 𝜙1 = 55∘ and 𝜙2 = 56∘ , the average
of the estimation of signals 1 and 2.
bias now becomes < 10−4∘ which is considered as accurate
(with less than 0.01% error).

fails to resolve the two incoming signals as the difference Case 2 (Fixed Aperture ULAs with Different Numbers of
between the two signals is only 1∘ . To probe further, the Elements (2 Incoming Signals)). In the previous case, when
biases of the estimated directions of the two signals are also the two closely spaced emitters approach to the endfire
included. It is interesting to see that for the case of Δ = 0.05𝜆, directions, it was found that the array aperture of the four-
the bias of 𝜙1 is about 0.4∘ (i.e., the estimated direction is element ULA has to be ≥ 0.60𝜆 (Δ ≥ 0.2𝜆) for an
about 5.4∘ ), which is close to the mean of the two incoming accurate DOA estimation. Figure 4 shows the results of DOA
signals (5∘ and 6∘ ). The bias of 𝜙2 is around 6∘ which is far estimations with different fixed array apertures that ranged
from the actual incoming direction, which further confirms from 0.25𝜆 to 1𝜆, but with a different number of array
that MPM fails to resolve the two incoming signals under elements varying from 4 to 20. Note that under a fixed
such an array configuration. To investigate if the performance array aperture, the interelement separation has to be changed
varies at different SNR level, the same DOA problem is with the change in the number of elements in the array. For
repeated except that the SNR level is 3 dB. It is found that there example, for the four-element array with an aperture of 0.5𝜆,
is no significant difference as compared to the case of SNR = the interelement separation is 0.1667𝜆(= 0.5𝜆/(4 − 1)).
5 dB. The array apertures of 0.25𝜆 and 0.5𝜆 are first considered,
Next, we study the effect on the performance of MPM and the results are shown in Figures 4(a) and 4(b). When the
when the incoming signals are not of equal power and equal incoming signals are close to the endfire directions (between
phase. The entire procedure is repeated for the DOA estima- 5∘ and 15∘ ), the average biases decrease when the number
tion of 𝜙1 = 5∘ and 𝜙2 = 6∘ except that the amplitudes of the of elements increases from 4 to 20. As the incoming signals
two signals are different and they are 90∘ out of phase. For are moving towards the broadside direction (between 20∘ and
𝜙2 = 6∘ , four different amplitudes of 𝐴 2 = 2𝐴 1 , 3𝐴 1 , 5𝐴 1 , and 90∘ ), the average biases decrease rapidly with an average bias
10𝐴 1 are considered, where 𝐴 1 and 𝐴 2 are the amplitudes of less than 10−4∘ regardless of the number of elements.
the incoming signals at 𝜙1 = 5∘ and 𝜙2 = 6∘ , respectively. Now, the same estimation procedure is repeated with an
The amplitude of the signal from 𝜙1 = 5∘ remains to be the array aperture of 0.75𝜆. When the aperture is increased from
same for all cases. The average biases for the four cases are 0.5𝜆 to 0.75𝜆, as shown in Figure 4(c), the average biases have
shown in Figure 2(a). The average biases in the range of 1.5∘ been significantly reduced for all cases (below 2.2 × 10−3∘ for
to 2∘ are found when Δ = 0.05𝜆, which is considered as a fail all cases). The maximum average bias is reduced from 1.56 ×
estimate as the difference between the two incoming signals is 10−2∘ (0.5𝜆 aperture) to 2.2 × 10−3∘ (0.75𝜆 aperture), which
only 1∘ . As Δ is larger than 0.2𝜆, the average biases decrease to is almost a factor of 7. This indicates a significant improve-
the level of 10−3∘ which agrees with the results for the equal- ment of the DOA estimations compared with the two cases
power and equal-phase case shown in Figure 1. The biases of of aperture size of 0.25𝜆 and 0.5𝜆.
International Journal of Antennas and Propagation 5

2 3.5

3
1.5
2.5

2
Average bias

Bias
1.5
0.5
1

0.5
0
0

−0.5 −0.5
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Interelement separation Interelement separation

2 times 5 times 2 times (bias 1) 2 times (bias 2)

3 times 10 times 3 times (bias 1) 3 times (bias 2)
5 times (bias 1) 5 times (bias 2)
10 times (bias 1) 10 times (bias 2)
(a) Average Bias (b) Bias of Signal 1 and 2

Figure 2: Summary of DOA estimations using MPM for a four-element ULA with interelement separation varying from 0.05𝜆 to 0.5𝜆. The
two incoming signals are from the directions of 𝜙1 = 5∘ (𝐴 1 = 1, 𝛾1 = 90∘ ) and 𝜙2 = 6∘ (𝐴 2 = 2, 3, 5, and 10, 𝛾2 = 180∘ ) with an SNR of 5 dB.
(a) Average biases of the estimated directions; (b) biases of the estimation of signal 1 and signal 2.

number of elements under a fixed aperture does not neces-

0.6 sary improve the accuracies of the estimation. Comparing
between signals coming from the broadside direction and the
100 endfire directions, better accuracies are expected under the
0.5
same array configuration.
−2
Average bias (deg)

10 When the array aperture is further increased to 1𝜆, as

0.4
shown in Figure 4(d), it is not obvious if increasing the
−4
10 number of elements enhances the DOA performance under
0.3 this aperture. On the other hand, it is worth noting that as the
−6
10 array aperture increases from 0.25𝜆 to 1𝜆, the average biases
0.2 for all cases (different number of elements and incoming
signals) decrease. This indicates that the accuracy of DOA
0.1 20 0.1 estimation depends more on the array aperture than on the
0.2 40 number of elements when the aperture is sufficiently large.
E le 0.3
men
t se
60 gles While not presented here, the results on the array apertures
par 0.4 80 e nt an
atio
n 0.5 Inc
id of 1.25𝜆 and 1.5𝜆 indicate that there is some improvement
of DOA accuracy as the number of elements increases. These
Figure 3: Summary of DOA estimations using MPM for a four- results show that further increase in the number of elements
element ULA with interelement separation varying from 0.05𝜆 to for large aperture arrays (> 1𝜆) only yields minor improve-
0.5𝜆. Two equal-power incoming signals are from the directions of ment to the accuracy of the estimated angles.
𝜙1 = 10∘ to 90∘ and 𝜙2 = 𝜙1 + 1∘ with an SNR of 5 dB. The step size
of Δ𝜙1 = 5∘ is considered. The figure shows the average biases of the
estimated directions. Case 3 (Fixed Aperture ULAs with Different Number of
Elements (3 Incoming Signals)). In the last case, we consider
DOA estimation problems of three closely spaced, equal-
power, and equal-phase signals with an SNR of 5 dB come
The previous results show that for a fixed small array from the directions from 𝜙1 = 45∘ to 90∘ with Δ𝜙1 = 5∘ ,
aperture (especially for the cases of 0.25𝜆 and 0.5𝜆), the 𝜙2 = 𝜙1 + 1∘ , and 𝜙3 = 𝜙1 + 2∘ . First, the array aperture
accuracies of DOA estimation can be improved by increasing of 0.75𝜆 is investigated. The number of array elements varies
the number of elements when the incoming signals are close from 6 to 20 (the minimum number of sensors required for
to the endfire directions. When the incoming signals are resolving 3 incoming signals using MPM is 6). The average
moving towards to the broadside direction, increasing the biases of the DOA estimations are shown in Figure 5(a). It
6 International Journal of Antennas and Propagation

×10−3

0.45
14
0.4
100 10−2 12
0.35

Average bias (deg)

Average bias (deg)

0.3 10
10−2 −4
0.25 10 8

0.2
10−4 6
0.15 10−6
4
0.1 5
5 20
20 2
Nu 10 40 0.05 10 40
mb Nu
mb 60 gles
er o 15 60 gles er o 15
t an
f el
em 20
80 e nt an f ele
me 20
80
iden
ent
s Inc
id nts Inc

(a) (b)

×10−3 ×10−4
5.5
2
5
1.8
4.5
1.6
4
10−4
Average bias (deg)

Average bias (deg)

1.4
−4 3.5
10
1.2
3
1
10−6 2.5
10−6 0.8
2
0.6
1.5
5 0.4
20 5 1
10 20
Nu 40 0.2 Nu 10 40
mb mb 0.5
60
er o 15 gles er o 15 60 gles
f el
em 80 t an f elem 80 t an
ent 20 iden ent 20 i d e n
s Inc s Inc

(c) (d)

Figure 4: Summary of DOA estimations using MPM for a four-element ULA with the number of array elements varying from 4 to 20. Two
equal-power incoming signals are from the directions of 𝜙1 = 10∘ to 90∘ and 𝜙2 = 𝜙1 + 1∘ with an SNR of 5 dB. The step size of Δ𝜙1 = 5∘ is
considered. The figures show the average biases of the estimated direction. The apertures of the arrays are (a) 0.25𝜆, (b) 0.5𝜆, (c) 0.75𝜆, and
(d) 1𝜆.

can be clearly seen that the average biases decrease with the high as the actual separations between the three sources, only
number of the elements. Compared to the case with two 1∘ , and thus it cannot be concluded that the sources are clearly
signals of 𝜙1 = 45∘ , 𝜙2 = 46∘ with the array aperture of 0.25𝜆 resolved.
(Figure 4(a)) and the case of 𝜙1 = 5∘ , 𝜙2 = 6∘ with the Lastly, the incoming signals of the same DOA problems
array aperture of 0.75𝜆 (Figure 4(c)) with 6 array elements, and the array aperture are increased to 1𝜆 and 1.25𝜆, and
the average biases presented in Figure 5(a) (6∘ ) are much the average biases are shown in Figures 5(b) and 5(c),
higher than those in Cases 1 and 2 (10−4∘ and 10−3∘ , resp.). respectively. For the same DOA problem, the average biases
This is because resolving three incoming signals using MPM decrease as the number of array element increases. At an array
and resolving two incoming signals are two different math- aperture of 1𝜆 with 20 array elements, the average biases are at
ematical problems with different CRLB, and they should a level less than 0.1∘ for DOA when the incoming signals are
be considered separately. As expected, when the incoming more than 75∘ . For the DOA problem with incoming signals
signals are moving towards the broadside directions, the (𝜙1 = 45∘ , 𝜙2 = 46∘ , and 𝜙3 = 47∘ ) with 20 array elements,
average biases reduce to 2∘ for the case of 6 antenna elements. the average bias is 0.628∘ . However, when the array aperture is
The average biases are also reduced as the number of elements further increased to 1.25𝜆, the average bias reduces to smaller
increases. However, even with 20 elements, the average biases than 0.1∘ with the 20 array elements (as shown in Figure 5(c)).
are between 0.5∘ and 3∘ . These bias values are considered as The average biases for the same DOA estimation problem
International Journal of Antennas and Propagation 7

2.5
101 5
101

2
Average bias (deg)

Average bias (deg)

4
100
1.5
3
100 10−1
1
2
40 40
50 50 0.5
10 1 10
60 60
Num 15 70 gles Num
70
b er o 80 e nt an ber
of e
15
ngle
s
f elem
20 90 Inc
id lem 80
de nt a
ents ents 20 90 Inci

(a) (b)
0.7 0.16

0.14
0.6
100 100
0.12
0.5
Average bias (deg)
Average bias (deg)

10−1 0.1
0.4
0.08
10−2 0.3
0.06
0.2
10−3 0.04
40 40
50 0.1 10 50 0.02
Nu 10 60 Num 60
mb s
er 15 70 gles ber 15 70 ngle
of e
lem 80 e nt an of elem 80 i d e nt a
d Inc
ent
s 20 90 Inci ents 20 90

(c) (d)

Figure 5: Summary of DOA estimations using MPM for a four-element ULA with the number of array elements varying from 6 to 20. Three
equal-power incoming signals are from the directions of 𝜙1 = 40∘ to 90∘ , 𝜙2 = 𝜙1 + 1∘ , and 𝜙3 = 𝜙1 + 2∘ with an SNR of 5 dB. The step size
of Δ𝜙1 = 5∘ is considered. The figures show the average biases of the estimated directions with array aperture (a) 0.75𝜆, (b) 1.00𝜆, (c) 1.25𝜆,
and (d) 1.50𝜆.

but with an array aperture of 1.5𝜆 are shown in Figure 5(d). two and three incoming signals, respectively (i.e., element
As can be seen, when the signal sources are closer to the separation of ≤ 0.2𝜆). The performance could be improved by
broadside direction, the average biases are smaller. However, increasing the number of elements. In practice, the emitters
an increasing number of array elements do not necessarily are less closely spaced which would be easier to resolve,
increase the accuracy of DOA estimation in this aperture. and potentially, the size of the array and/or the number of
On the other hand, when the signal sources are closer to elements can be further reduced. The finding provides us with
the endfire direction, the accuracy of DOA estimation does the fundamental limit of DOA estimation in compact array.
increase substantially when the number of array elements is Arguably, the conclusions from this study are not limited
increased (see Figure 5(d)). to DOA estimations using compact antenna arrays (with
interelement separation of 0.25𝜆 [35] down to 0.125𝜆 [37])
with MPM but are also applicable to acoustic or sonar arrays,
4. Discussions and Conclusions as well as to other subspace methods.
In this paper, we have investigated the performance of
MPM in DOA estimations using ULAs with different array Acknowledgments
apertures and different numbers of array elements through
Monte Carlo simulations. The results show that minimum This work was supported in part by the National University
linear array apertures of 0.6𝜆 and 1𝜆 are required to resolve of Singapore (NUS) under the Grant no. R-263-000-469-112
8 International Journal of Antennas and Propagation

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