1. K.Amulya Swapna, B.Vinod Naik / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 6, November- December 2012, pp.1569-1572
Performance Analysis of MUSIC Algorithm for Various Antenna
Array Configurations
K.Amulya Swapna*, B.Vinod Naik**
* Dept of ECE, PVP Siddhartha Institute of Technology, Vijayawada-7, India,
**Dept of ECE, PVP Siddhartha Institute of Technology, Vijayawada-7, India,
Abstract
Smart antenna involves the array uniform linear array (ULA) for applications such as
processing to manipulate the signals induced on radar, sonar and wireless communications [1]. The
various antenna elements in such a way that the direction of arrival (DOA) estimation of multiple
main beam directing towards the desired signal narrowband signals is a classical problem in array
and forming nulls towards the interferers. To signal processing. An array antenna system with
locate the desired signal various DOA estimation innovative signal processing can enhance the
algorithms are used. High resolution algorithms resolution of DOA estimation. An array sensor
take the advantage of array geometries to better system has multiple sensors distributed in space.
process the incoming signals. This paper explores This array configuration provides spatial samplings
the high resolution MUSIC algorithm by using a of the received waveform [2]. Generally the choice
Uniform Linear Array (ULA) and Uniform of DOA estimator is made adequately in accordance
Circular Array (UCA). MUSIC algorithm is with the array geometries used. In this paper,
modeled and simulated in a new proposed array computer simulation programs using Matlab were
geometry. In this paper, the performance developed to evaluate the direction-of-arrival
obtained in both situations is analyzed through performance of MUSIC algorithm based on uniform
computer simulation. linear array (ULA) and uniform circular array
(UCA) geometries.
Keywords: Smart antenna, Direction of arrival,
MUSIC. II.MUSIC ALGORITHM
MUSIC is an acronym which stands for
I.INTRODUCTION “Multiple Signal Classification”. This approach
Over the past few decades, as demand for was proposed by Schmidt. It is one of the high
increased capacity and quality grow, improved resolution subspace DOA algorithm. MUSIC deals
methods for harnessing the multi-path channel must with the decomposition of covariance matrix into
be developed. The use of adaptive antenna array is two orthogonal matrices, i.e., signal sub-space and
one area that shows promise for improving capacity noise sub-space. Estimation of DOA is performed
for wireless systems and providing improved safety from one of the subspaces, assuming that noise in
through position location capacities. Estimating the each channel is highly uncorrelated. This makes the
direction of arrival (DOA’s) of electromagnetic covariance matrix diagonal. The direction of sources
waves impinging on antenna arrays is an important is determined from steering vectors that are
issue in array signal processing for wireless orthogonal to the noise subspace, which is by
communication. It is indeed in determining the finding the peak in spatial power spectrum.
location of the mobile with high accuracy. Different Consider a uniform linear array with D
direction finding techniques and algorithms have arriving signals impinging on a M element array.
been developed leading to significant improvements The signals are received by an array of M elements
in DOA estimation over the last decades. Subspace with M potential weights. Many of the DOA
based methods provide high resolution DOA algorithms rely on the array correlation matrix.
estimation. The various DOA estimation algorithms M×M array correlation Matrix 𝑅 𝑥𝑥 with
are Bartlett, Capon, Min-norm, MUSIC and uncorrelated noises and equal variances can be
ESPRIT. The MUSIC algorithm is one of the most defined as [3]
popular and widely used subspace-based techniques 𝑅 𝑥𝑥 = 𝐸[𝑥 . 𝑥 𝐻 ] (1)
for estimating the DOA’s of multiple signal sources.
The Conventional MUSIC algorithm involves a 𝑅 𝑥𝑥 = 𝐴 𝑅 𝑠𝑠 𝐴 𝐻 + 𝜎 2 𝐼
𝑛 (2)
computationally demanding spectral search over the
angle and therefore its implementation can be Where 𝐴 = [𝑎 𝜃1 𝑎 𝜃2 𝑎 𝜃3 . 𝑎 𝜃 𝐷 ] is an M×D
prohibitively expensive in real-world applications. array steering matrix.
The uniform circular array (UCA) is able to provide
360° of coverage in the azimuth plane and has 𝑅 𝑠𝑠 = [𝑆1 𝑘 𝑆2 𝑘 𝑆3 𝑘 … 𝑆 𝐷 𝑘 ] is D×D source
uniform performance regardless of angle of arrival. correlation matrix.
Thus, sometimes UCA is more suitable than
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2. K.Amulya Swapna, B.Vinod Naik / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 6, November- December 2012, pp.1569-1572
𝑅 𝑛𝑛 = 𝜎 2 𝐼 represents the noise correlation matrix
𝑛 number. The array steering matrixes of the ULA and
(M×M elements), and I represents the identity PA have dimensions of N×D and (N+2) ×D,
matrix (N×N elements).The array correlation matrix respectively.
has M eigen values (𝜆1 , 𝜆2 , . . . , 𝜆 𝑀 ) along with
the associated eigen vectors 𝐸 = [𝑒1 𝑒2 . . . 𝑒 𝑀 ]. If the
eigen values are sorted from smallest to largest, we
can divide the matrix 𝐸 into two subspaces
Figure 1: Uniform Linear Geometry (ULA)
E = [EN ES ] (3)
The first subspace 𝐸 𝑁 is called the noise
subspace and is composed of M-D eigen vectors Figure 2: Proposed Array Geometry (PA)
associated with the noise, and the second subspace
𝐸 𝑆 is called the signal subspace and is composed of If 𝑎 𝑈𝐿𝐴 (𝜃 𝑚 ) represents the array steering vector for
D eigen vectors associated with the arriving signals. each of the input signals on the linear array, then for
The noise subspace is an M× (M-D) matrix, and the the symmetrical linear array 𝑎 𝑈𝐿𝐴 (𝜃 𝑚 ) can be
signal subspace is an (M×D) matrix. The MUSIC written as an N×1 vector expressed as
algorithm is based on the assumption that the noise
subspace Eigen vectors are orthogonal to the array 𝑁 −1
steering vectors 𝑎 θ at the angles of 𝑒 −𝑗 ( 2
)𝑘𝑑𝑐𝑜𝑠 𝜃 𝑚
𝑁 −3
−𝑗 𝑘𝑑𝑐𝑜𝑠 𝜃 𝑚
arrivals𝜃1 , 𝜃2 , … , 𝜃 𝐷 .Because of this orthogonality 𝑒 2
condition, one can show that the Euclidian distance 𝑎 𝑈𝐿𝐴 𝜃 𝑚 = ⋮ (5)
𝑑2 = 𝑎 𝐻 𝜃 𝐸 𝑁 𝐸 𝑁𝐻 𝑎 𝜃 = 0. Placing this distance 𝑁 −3
𝑒𝑗
𝑘𝑑𝑐𝑜𝑠 𝜃 𝑚
2
expression in the denominator create sharp peaks at 𝑁 −1
the angles of arrival. The MUSIC pseudospectrum is 𝑒𝑗 2
𝑘𝑑𝑐𝑜𝑠 𝜃 𝑚
given as
Where d is the inter-element space and k=2𝜋/𝜆. The
1 steering vector of the proposed array is represented
𝑃 𝑀𝑈 𝜃 = 𝐻 (4)
𝑎 (𝜃) 𝐻 𝐸 𝑁 𝐸 𝑁 𝑎 (𝜃) with 𝑎 𝑃𝐴 (𝜃 𝑚 ) that is an (N+2) ×1 vector and it can
be written as
𝑁 −1
III.MODELLING THE NEW ARRAY 𝑒 −𝑗 2
𝑘𝑑𝑐𝑜𝑠 𝜃 𝑚
The standard array geometry that is used in 𝑁 −3
𝑒 −𝑗
𝑘𝑑𝑐𝑜𝑠 𝜃 𝑚
the smart antenna system is the Uniform Linear 2
Array (ULA).The main advantage of using a ULA is ⋮
𝑁 −3
the simplicity, excellent directivity, and production 𝑎 𝑃𝐴 𝜃 𝑚 = 𝑗 𝑘𝑑𝑐𝑜𝑠 𝜃 𝑚 (6)
𝑒 2
of the narrowest main lobe in a given direction in 𝑁 −1
𝑒𝑗
𝑘𝑑𝑐𝑜𝑠 𝜃 𝑚
2
comparison to the other array geometries. However
a ULA does not work equally well for all azimuth 𝑒 𝑗𝑘𝑑𝑐𝑜𝑠 𝜃𝑚
directions and the DOA estimation accuracy and the 𝑒 −𝑗𝑘𝑑𝑐𝑜𝑠 𝜃 𝑚
resolution are low at directions close to the array
endfire. This major drawback can be resolved by The first N rows of 𝑎 𝑃𝐴 (𝜃 𝑚 )are related to the linear
employing other array geometries, such as circular part of the array and the two remaining rows show
and hexagonal, but these geometries may lead to the effect of the top and the bottom elements in the
complexity of array structure and calculations and proposed array.
array aperture may become larger. Thus it is
desirable to develop simple array configurations IV.UNIFORM CIRCULAR ARRAY (UCA)
which performs equally well for all azimuth Consider a UCA consisting of M identical
directions. Displaced Sensor Array (DSA) is such a elements uniformly distributed over the
configuration which has equally improved circumference of a circle of radius r. Assume that
performance for all azimuth angles [4]. Another narrowband sources centered on wavelength λ,
simple array based on ULA is proposed and impinge on the array from directions 𝜙 𝑖 (i=1,2…D)
illustrated here to improve DOA estimation results respectively where 𝜙 𝑖 𝜖[0 2𝜋] is the azimuth angle
at array endfire directions. The array configuration measured from the x axis counter-clockwise. Fig. 3
affects the array steering vectors and the dimension depicts a receiver formed by an UCA with incident
of signal vector. In order to investigate the proposed plane waves from various directions.
array performance in DOA estimation of narrow
band signals, a ULA with N elements and the PA
with N+2 elements, as depicted in Fig. 1, 2 are
compared. Both of the arrays are assumed
symmetric about the origin. So, N is assumed as odd
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3. K.Amulya Swapna, B.Vinod Naik / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 6, November- December 2012, pp.1569-1572
VI.SIMULATION RESULTS
The MUSIC DOA estimation is simulated
using Matlab. An uniform linear array with M
elements is considered. Fig. 4 shows the MUSIC
spectrum for linear array for direction of arrivals -5,
10, 25 degrees for different antenna elements. When
the elements in array are increased to 11 MUSIC
spectrum takes the form of sharper peaks in which
angular resolution is improved.
Figure 3: Receiver with Uniform Circular M-
element array
Suppose D signals impinging on the UCA
of M elements from directions {𝜃1 , ∅1 },..,{𝜃 𝐷 , ∅ 𝐷 }
the received signal at the antenna array can be
described as
𝑋 𝑘 = 𝐴𝑆 𝑘 + 𝑛(𝑘) (7)
Where 𝑋 𝑘 = [𝑥1 𝑘 , 𝑥2 𝑘 , … 𝑥 𝑀 𝑘 ] 𝑇 is the
𝑘 𝑡ℎ snapshot of the received signal at the antenna
array, T denotes the transpose. The Array steering
matrix is given as
Figure 4: MUSIC spectrum for varying array
𝐴 = [𝑎 𝜃1 , ∅1 , 𝑎 𝜃2 , ∅2 , … , 𝑎 𝜃 𝐷 , ∅ 𝐷 ] (8) elements.
And steering vector Fig. 5 shows the MUSIC spectrum obtained for
snapshots equal to 10 and 100. Increased snapshots
𝑎 𝜃 𝑖 , ∅ 𝑖 = [𝑎1 𝜃 𝑖 , ∅ 𝑖 , 𝑎2 𝜃 𝑖 , ∅ 𝑖 , … , 𝑎 𝐷 (𝜃 𝑖 , ∅ 𝑖 ) is leads to sharper MUSIC spectrum peaks indicating
the array response to the incident signal from more accurate detection and better resolution.
direction (𝜃 𝑖 , ∅ 𝑖 ).The above equation can be
expressed in detail as
𝑎 𝜃, ∅ =
𝑇
𝑒 𝑗𝜂𝑠𝑖𝑛𝜃 cos 𝜙 −𝛾1
, 𝑒 𝑗𝜂𝑠𝑖𝑛𝜃 cos 𝜙 −𝛾2
, . . , 𝑒 𝑗𝜂𝑠𝑖𝑛𝜃 cos 𝜙 −𝛾 𝑀
Where 𝜂 = 𝑘𝑟𝑠𝑖𝑛𝜃 , k is the wave number. The
angular distance between the elements of the array
is given as
𝛾 𝑀 = 2𝜋(𝑚 − 1)/𝑀 (9)
The MUSIC pseudospectrum For the UCA is given
as
1
𝑃 𝑀𝑈𝑆𝐼𝐶 −𝑈𝐶𝐴 𝜙 = 𝑎 (𝜙 ) 𝐻 𝐸 𝐸 𝐻 𝑎 (𝜙 ) (10) Figure 5: MUSIC spectrum for varying snapshots.
𝑁 𝑁
To compare the accuracy of MUSIC
V.NON-UNIFORM CIRCULAR ARRAY
algorithm in both ULA and PA geometries, a ULA
Consider a uniform circular array of M
with N=11 elements is assumed and therefore, the
antenna elements non-uniformly spaced over the
proposed array consists of N=13 elements. Inter-
circumference of a circle of radius r. The array
element spacing is maintained 𝑑 = 𝜆/2, number of
steering vector can be modeled as
snapshots k=1000.
𝑎 𝜃, ∅, 𝛾𝑛
𝑇 Fig. 6, 7 shows the spectrum of MUSIC
= 𝑒 𝑗𝜂𝑠𝑖𝑛𝜃 cos 𝜙 −𝛾1
, 𝑒 𝑗𝜂𝑠𝑖𝑛𝜃 cos 𝜙 −𝛾2
, . . , 𝑒 𝑗𝜂𝑠𝑖𝑛𝜃 cos 𝜙 −𝛾 𝑀
algorithm for both arrays in different DOAs which
are assumed either close to the array boresight (2
Where 𝛾𝑛 = [𝛾𝑛1 , 𝛾𝑛2 , … 𝛾𝑛 𝑀 ],𝛾𝑛 𝑖 represents the sources) or close to the array endfire (2 sources).
angular distance from element i to element i+1 i.e., The spectrum depicted in Fig. 6, the assumed
𝛾1 = 𝛾𝑛1 ;𝛾2 = 𝛾1 + 𝛾𝑛2 ; …; 𝛾 𝑀 = 𝛾 𝑀 −1 + 𝛾𝑛 𝑀−1 . middle DOAs are detected successfully by
individual correct peaks for each of the assumed
sources. The spectrum depicted in Fig. 7 shows that
the MUSIC algorithm has resolved the sources
located at (-2o, 2o) successfully in both array
1571 | P a g e

4. K.Amulya Swapna, B.Vinod Naik / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 6, November- December 2012, pp.1569-1572
configurations. On the other hand the PA has
resolved close sources located at (70o, 75o) while the
ULA failed to resolve these sources.
Figure 9: MUSIC spectrum for the signal with DOA
(θ=50o , Ф =100o).
VII.CONCLUSION
Figure 6: MUSIC spectrum for ULA and PA This paper gives extensive computer
geometries for DOAs (-5o, 5o, 75o, 85o). simulation results to demonstrate the performances
of the ULA, PA, UCA and Non-Uniform circular
antennas obtained in the case of DOA estimation.
Simple array geometry (PA) is proposed which can
resolve the close sources located at close angles to
the array endfire direction accurately when
compared to the ULA. Thus sometimes, UCA is
more suitable than ULA for applications such as
radar, sonar, and wireless communications.
REFERENCES
[1] C.P. Mathews, M. D. Zoltowski, “Signal
Figure 7: MUSIC spectrum for ULA and PA Subspace Techniques for Source
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School of Electrical Engineering, Purdue
An UCA with M=16 antenna elements is considered University, ECE Technical Reports, 1994.
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