Content uploaded by Mohammad Abdul Matin
Author content
All content in this area was uploaded by Mohammad Abdul Matin on Oct 20, 2017
Content may be subject to copyright.
Abstract—The performance of smart antennas relies heavily
on the accurate estimation of the direction of arrival (DOA).
This paper analyzed various techniques for DOA estimation
and investigated their performance. Simulation results showed
that MUSIC algorithm outperforms in terms of root mean
square error (RMSE), angular resolution and the number of
elements in an antenna array. Profound analysis of these
algorithms can be used to determine the direction of arrival of
the signals at uniform linear array (ULA).
Index Terms—Conventional beamforming, DOA, MUSIC,
MVDR, ULA.
I. INTRODUCTION
Using smart antennas is one of the most promising
technologies that can increase the capacity in wireless
networks by effectively reducing multipath and co-channel
interference. Two major issues are concerned with the smart
antenna - DOA estimation and the beamforming [1]. The
accurate estimation of the DOA of all signals transmitted to
the adaptive array antenna contributes to the maximization of
its performance with respect to recovering the signal- of-
interest (SOI) and suppressing any presence of interfering
signals. The beamforming technique also ensures less
interference to the system and thus increasing the
performance.
The DOA algorithms can be categorized into two groups:
the conventional algorithm (Conventional Beamforming
method, MVDR Beamforming) and the subspace algorithm
(MUSIC). The conventional (Barlett) beamformer dates
back to the second world-war, and is a mere application of
Fourier-based spectral analysis to spatio-temporally sampled
data. However, this technique works well with only one
incoming signal. To overcome this problem, in 1969, the
Capon‟s beamforming technique (Minimum Variance
Algorithm) was proposed in which the array output power
contains signal contributions from the desired angle as well
as the undesired angles. The introduction of subspace – based
estimation techniques marked the beginning of a new era in
the sensor array signal processing literature [2]. The Music
(Multiple Signal Classification) algorithm proposed by
Schmidt in 1986, is a very popular subspace- based method
for super-resolution direction finding [3].In this paper, a
detailed study of different DOA algorithms has been
provided and observed that MUSIC algorithm, has the least
RMSE as a function of snapshots compared to the other
mentioned algorithms, according to our simulation results.
II. SYSTEM DESCRIPTION
The system consists of Uniform Linear Array (ULA) with
M number of antenna elements that are spaced linearly with
equal distance.
Fig. 1. System model for DOA estimation using a uniform linear array of M
elements.
The elements of ULA collect signals form a remote source,
i. The spacing between the lines of elements, , is restricted to
half wavelength of received signals. Suppose that a plane
wave signal generated by the source i impinges on the array
at an angle and the signal generated by the source i is the
narrowband signal,(). Then it travels at a speed of c over a
distance and reaches the first rightmost elements. If we
consider all the signals generated by all the d sources, ,
1≤i≤ d, , the total signal and noises received by the mth array
element at time t can be expressed as:
= (1)
=1 + ,= 1 (1)
This equation can be written as:
= 1,(2) () 1()
2()
()+ ()
=+ () (2)
where
= [1 2 ()] (3)
is the data column vector received by the array,
= [1 2 ()] (4)
is the signal column vector generated by the sources,
= [ 2 ()] (5)
Analysis of Direction of Arrival Techniques Using
Uniform Linear Array
S. N. Bhuiya, F. Islam, and M. A. Matin
International
Journal of Computer Theory and Engineering, Vol. 4, No. 6, December 2012
931
Manuscript received August 2, 2012; revised September 30,2012.
The authors are with the Dept. of Electrical Engineering and Computer
Science North South University, Dhaka, Bangladesh (e-mail:
shayla_298@hotmail.com).
is a zero mean spatially uncorrelated additive noises with
covariance matrix equal to
2. The array steering column
vector is defined as:
= [1 2 (1)] (6)
The columns of × steering matrix
= (1) () ()
= 1 1 1
12
(1)1(1)2 (1) (7)
where, = 2
sin , is called spatial frequency for
the ith source that generates the signals of incident angle, .
[3]
The idea is to „steer‟ the array in one direction at a time and
measure the output power. The steering direction which
coincides with the DOA of a signal and result in a maximum
output power yields the DOA estimates. According to [3] an
array can be steered electronically just as a fixed antenna can
be steered mechanically. However, the array pattern can
change shape in addition to changing orientation. A weight
vector w can be designed and then used to linearly combine
the data received by the array elements to form a single
output signal Y(t),
= () (8)
The total averaged output power out of an array over K
snapshots can be expressed as [2]
=1
()2
=1 =1
=1 ( )()
= (9)
A. Classical Beamformer
The classical or conventional beamformer method is also
referred to as the delay and sum method or Barlett method.
Suppose an ULA consists of M number of elements, where
A(θ) is defined as the steering vector with a scanning angle θ.
The idea is to scan across the angular region of interest,
where weight vector,
= () (10)
Inserting this equation of the weight vector into equation
(9), we can calculate the output power of the classical
beamformer as a function of the angle of arrival-
() == (11)
The direction that produces the largest output power is the
estimate of the desired signal‟s direction. [3],[4]
B. Minimum Variance Distortionless Response
MVDR is also known as the Capon‟s Minimum Variance
method [5]. This method was proposed to overcome the poor
resolution problem associated with the delay-and-sum
method [6], [7]. It uses the array weights which are obtained
by minimizing the mean output power with the constraint that
the gain remains unity in the look direction, θ of the SOI.
The optimized weight vector is given by –
=
1()
1() (12)
This gives the expression for the MVDR Spatial Spectrum-
==1
1() (13)
The angle θ in spectrum corresponding to the peak value
gives the true DOA estimation.
C. Multiple Signal Classification Algorithms
Music is a high resolution subspace DOA algorithm, which
stands for multiple signal classification. An estimate of
the covariance matrix is obtained and its eigenvectors are
separated into signal and noise subspace and the DOA is
estimated from one of these subspaces. It is assumed that the
noise in each channel is uncorrelated. This leads to a diagonal
covariance matrix.
=()+ 2 (14)
where, = [1,2,3,] is a ×
array steering matrix. σ² is the noise variance and I is an
identity matrix of size ×
=()+ 2= (15)
with Q unitary and a diagonal matrix ={1,2,,} ,
of real eigenvalue ordered as 120.The
vector that is orthogonal to A is the eigenvector of R having
the eigenvalue σ² [8].
The number of such linearly independent vectors are L − M.
Now, the rest of the eigenvalues are greater than σ², we can
divide the eigenvalue and corresponding vector pairs into the
noise and signal eigenvectors.
Now we can write it as follows-
=+ (16)
where, =2. All noise eigenvectors are orthogonal to A.
Therefore the columns of should span the range space of A
while the columns of span its orthogonal complement.
The signal and noise subspace‟s projection operators are
defined as -
==()1 (17)
== 1 ()1 (18)
MUSIC “Spatial Spectrum” is defined as [9], [10]
=()()
()() (19)
III. SIMULATION RESULTS
The classical, MVDR and MUSIC techniques for DOA
estimations are simulated in MATLAB and their
performances are demonstrated in terms of the number of
elements, root mean square error as a function of snapshots
and angular resolution.
The number of elements, M, is an important parameter that
impacts on the accuracy of the DOA estimation done by three
algorithms. Here, the simulation has been carried out for two
different cases: M=10 and M=50 for two differentsignals
arriving at -5M and 5M.
International
Journal of Computer Theory and Engineering, Vol. 4, No. 6, December 2012
932
a)
b)
Fig. 2. DOA estimation analysis for varying number of elements for: (a)
M=10 and (b) M= 50.
The simulated results show that increasing number of
elements improves the performance of all three algorithms.
However, increasing the number of elements leads to an
increase in number of receivers as well as the storage
capacity for the data. For less number of elements, only
MUSIC algorithm gives the expected result.
Next, RMSE parameter has been used to measure the
performance of the above three kinds of algorithms as a
function of varying number of snapshots and it can be defined
as- = 1
()2
=1 (20)
where K is the total number of snapshots, is the true
angle and is the angle estimated by the algorithm. For
different values of K we get different values of RMSE.
Fig. 3. RMSE for three different algorithms as a function of snapshots, for
θ=-30M
The Fig depicts that as the number of snapshots increases,
the resolution capability of the three algorithms increases.
However, for all values of K, the MUSIC algorithm shows
the least RMSE compared to the conventional
algorithms-Classical and MVDR methods. This clearly
indicates that the MUSIC algorithm provides the closest
estimate of the true angle and therefore results in the lowest
RMSE for different number of snapshots.
Finally, we analysis have done based on the resolution
capability of the DOA estimation techniques considering two
close signals arriving from θ=0M and θ=2M when the number
of elements is 50 and the number of snapshots is 1000.
Fig. 4. DOA estimation analysis for angular resolution.
The comparison shown in Fig. 4, tells that the MUSIC
algorithm has better angular resolution than MVDR which
requires an additional matrix inversion. It can distinguish two
received signals which are at least 2M apart whereas Classical
cannot. Appendix
Symbol Meaning
M Number of elements
() Signal from the ith source
Incident angle of signal
d Number of sources
Total signal & noises received by mth array
Spatial Frequency
() Noise signal
A Steering Matrix
Steering vector
Output signal
w Weight vector
Hermitian of weight vector
Total average output power
K Number of snapshots
Covariance matrix
() Spatial spectrum of classical algorithm
Spatial spectrum of MVDR algorithm
Spatial spectrum of MUSIC algorithm
2 Noise variance
Identity matrix of size ×
Signal subspace
Noise subspace
IV. CONCLUSION
In this paper, a detail analysis of previous algorithms used
to determine the direction of arrival of the signals at ULA is
presented. Simulations have been performed to evaluate the
performance of the classical, MVDR, MUSIC algorithms. It
was observed that although increasing the number of
elements improved the performance of the three algorithms,
MUSIC algorithm gives the expected result with fewer
-15 -10 -5 0 5 10 15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Angle in degrees
Normalized spectrum
MUSIC
MVDR
Classical
-15 -10 -5 0 5 10 15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Angle in degrees
Normalized spectrum
MUSIC
MVDR
Classical
0500 1000
0
0.2
0.4
0.6
0.8
Number of snapshots
RMSE for MUSIC
0500 1000
0
0.2
0.4
0.6
0.8
Number of snapshots
RMSE for MVDR
0500 1000
0
0.2
0.4
0.6
0.8
Number of snapshots
RMSE for Classical
-15 -10 -5 0 5 10 15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Angle in degrees
Normalized spectrum
MUSIC
MVDR
Classical
International
Journal of Computer Theory and Engineering, Vol. 4, No. 6, December 2012
933
numbers of elements. This results in an advantage of
requiring less number of receivers and storage capacity at the
array. These results further show that classical and MVDR
algorithms offer less computational complexity. On the other
hand, the MUSIC algorithm provides the most accurate DOA
estimation.
REFFERENCES
[1] A. Randazzo, M. A. A. Khousa, M. Pastorino, and R. Zoughi,
“Direction of arrival estimation based on Support Vector Regression:
Experimental validation and comparison with MUSIC,” IEEE
Antennas and Wireless Propagation Letters, vol. 6, pp. 379, 2007.
[2] H. Krim and M. Viberg, “Two decades of array signal processing
research,” IEEE Signal Processing Magazine, vol. 13, no. 4, pp. 67,
July 1996.
[3] Z. Chen, G. Gokeda, and Y. Yu, Introduction to Direction-of-Arrival
Estimation, Artech House, vol. 34, no. 37, pp. 46-47, 2010.
[4] L. C. Godara, “Application of antenna arrays to mobile
communications-II: Beam-forming and direction-of-arrival
considerations,” Proc. IEEE, vol. 85, no. 8, pp. 1195–1245, Aug.
1997.
[5] J. Capon, “High resolution frequency-wave number spectrum analysis,”
Proc. IEEE, vol. 57, no. 8, Aug. 1969.
[6] C. A. Balanis and P. I. Ioannides, “Introduction to smart antenna,”
Morgan and Claypool Publishers, 2007, pp.79.
[7] P. Stoica, Z. Wang, and J. Li, “Robust capon beamforming,” IEEE
Signal Processing Letters, vol. 10, no. 6, June 2003.
[8] H. Anton and C. Rorres, Elementary Linear Algebra, 8th ed. John
Wiley and Sons, 2000, pp. 338-360.
[9] R. O. Schmidt, “Multiple emitter location and signal parameter
estimation,” IEEE Trans, vol. AP-34, no. 3, pp. 276-280, Mar. 1986.
[10] R. Kawitkar, “Performance of different types of array structures based
on multiple signal classification (MUSIC) algorithm,” Fifth
International Conference on MEMS, NANO, and Smart Systems, pp. 2,
2009.
International
Journal of Computer Theory and Engineering, Vol. 4, No. 6, December 2012
934
