International Journal of Engineering Trends and Technology (IJETT) – Volume 34 Number 7- April 2016
Performance analysis of MUSIC algorithm for
AOA estimation in wireless communication
Padma S Shrimandal1, Dr. Rajendra R Patil2
1
2
Department of ECE, APPA Institute of Engineering& Technology, Kalaburgi, Karnataka, India.
Associate Professor, Department of ECE, APPA Institute of Engineering& Technology, Kalaburgi, Karnataka,
India.
Abstract—: Space division multiple access (SDMA) is
a satellite communications mode or the channel access
method that increases the capacity of the system and
transmission quality by directing the signal into narrow
transmission beams through the use of smart antennas.
Generally co-located with a base station, a smart
antenna system combines an antenna array with a
digital signal-processing capability to transmit and
receive in an adaptive, spatially sensitive manner. That
is, such a system can automatically alter the
directionality of its radiation patterns in response to its
signal environment. This paper involves the
implementation of AOA algorithms for smart antenna
systems. Here we will see one of the important signal
processing algorithm namely MUSIC algorithm.
II. BACKGROUND FOR ANGLE OF ARRIVAL
PROBLEM
In the propagation channel of wireless systems, it is
apparent that even for one source signal there are several
possible propagation paths and angles of arrival [8]. If
various transmitters are operating at the same time, each
source will potentially create many multipath
components at the receiver end. Thus, it is important for
a receiver antenna array to estimate the angles of arrival
in order to decipher which emitters are present and what
are their possible angular locations.
Keywords: SDMA, Smart antennas, AOA, MUSIC.
I. INTRODUCTION
Figure 2.1: Uniform Linear Antenna array
Smart antenna is one among the possible solutions to
increase the channel capacity due to an increase in the
number of mobile units and the need for high-speed
digital communication in mobile communication. Smart
antenna utilizes the beamforming technique to spatially
direct the electromagnetic power to an intended mobile
unit while spatially suppressing the signal power along
other mobile units [1]. The system needs the process of
angle of arrival estimation to locate the mobile units
before beamforming can be performed. Angle of arrival
(AOA) estimation is the method of determining the
direction of an incoming signal from a mobile user to a
base station. In this paper we will study MUSIC
algorithm. MUSIC, which is abbreviated as multiple
signal classification, was first proposed Schmidt and is a
popular and efficient Eigen value decomposition method
[2].
Consider a uniform linear antenna array
geometry with L elements say 0, 1,…, L- 1 with a
spacing of half the wavelength spacing ( d ) between
them. Let b(t ) be the baseband signal that is received
by each of the array element, but at a different instant of
time. If the phase of baseband signal b(t ) received at
element 0 is zero. By analyzing the geometry from
figure 2.1, using basics of trigonometry and facts from
the wave propagation, the time delay of arrival can be
computed as follows
t k 
k D sin 
c
(2.1)
Where, c is the speed of light,
k
is an integer
and  is the direction from which plane wave is
striking on antenna array
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International Journal of Engineering Trends and Technology (IJETT) – Volume 34 Number 7- April 2016
Suppose b(t )
is a narrowband digitally
modulated signal with low pass equivalent bl (t ) ,
carrier frequency f c , and symbol period
T . Narrow
band signal can be written as

b(t )  Re bl (t ) e j 2 
f ct
d
D

(2.9)
Using the equations (2.1) and (2.8) in equation (2.7)
one can arrive at equation as follows

(2.2)
th
The signal received by the k element is given by
xk (t )  Rebl (t  t k )e
j 2 f c (t  tk )
xk (nT )  bl (nT ) e
 j 2 f c t K
 bl (nT ) e
 j 2
c kD sin( )
c

(2.10)
Substituting the value of „D‟ from equation (2.9) in
equation (2.10) gives
(2.3)
Now that suppose the received signal at the kth
element is down converted to the baseband. In that
case, the baseband received signal is defined as
xk (nT )  bl (nT ) e
 j 2
c k d  sin( )
c

(2.11)
x k (t )  bl (t  t k ) e
 j 2 f c t k
After simplification, we get
(2.4)
x k (nT )  bl (nT ) e  j 2
2.1 Narrowband Approximation
k d sin ( )
(2.12)
The received baseband signal after sampling
with a sampling period of T seconds is given by
xk (nT )  bl (n T  t K ) e
 j 2 f c t K
When discrete time notation is used with time index n,
equation (2.12) can be written as follows
x k ( n )  b ( n ) e  j 2
(2.5)
(2.13)
In a wireless digital communication system,
the symbol period will be much greater than each of the
propagation delays across the antenna array given by
Where,
the range
T   t K , k  0,1,......., L  1
a K ( )  e  j 2
 b(n) a K ( )
k d sin ( )
and k is an integer in
0  k  L 1
Let the nth sample of the baseband signal at the
kth element be referred as xk (n ) .When there are M
(2.6)
This allows the following approximation to be made as
xk (nT )  bl (nT ) e
k d sin ( )
signals present, the nth symbol of the ith signal will be
denoted by bi (n) for i  0,1,2,....,M  1 . The
baseband sampled signal at the kth element can be
expressed as
 j 2 f c t K
(2.7)
M 1
The constants c and f c can be related through the
basic equation given by
x k ( n)   bi ( n) a ( i )
i 0
(2.14)
c  fc 
(2.8)
Where, λ is the wavelength of the propagating wave
and f c is the carrier frequency. The element spacing
can be computed in wavelengths by using
ISSN: 2231-5381
2.2 Formulation of Antenna array Data Matrix
By considering all the antenna array elements,
i.e k  0,1,2,....L  1 , equation (2.14) can be written
in a matrix form as
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International Journal of Engineering Trends and Technology (IJETT) – Volume 34 Number 7- April 2016
 x0 [n] a0 ( 0 ) a0 (1 ) .......................a0 ( M 1 )  b0 [n]  n0 [n]
 x [n]  a ( ) a ( ) ........................a ( )  b [n]  n [n] 
1 M 1   1
1  1 0 1 1
 1 
.   .
.
.  .   . 

  
 
.
.
.
.
.

  
 . 
.   .
.
.  .   . 

  
  
 xL1  aM 1 ( 0 ) aM 1 (1 ) ...........aM 1 ( M 1 )  bM 1[n] nL1 
(2.15)
The spatial covariance matrix of the array of
antenna elements can be computed as follows. Assume
that bn (signal) and n n (noise) are uncorrelated, and
n n is a vector of White gaussian noise samples with
zero mean. The spatial covariance matrix R is given
by
R  E[ x n x nH ]
(2.18)
Substituting
Where , n k [n ] is additive white Gaussian
noise considered at each element,
x K [n] is the
x n from equation (2.16) in equation
(2.18), one can obtain
R xx  E[( A bn  n n ) ( A bn  n n ) H ]
induced signal, b n is the amplitude of nth source, M is
(2.19)
the number of sources, L is the number of antenna
elements and n n is the amplitude of nth noise sample.
Applying expectation operator (E) to signal bn and
Equation (2.15) can be written in compact form as
noise n n in equation (2.19) gives
xn  [a( 0 ) a(1 ) .......... .... a( L 1 )] bn  nn  A bn  nn
R  A E[bn bnH ] A H  E[nn nnH ]
(2.16)
(2.20)
Where,
a( i )  [ a0 ( i ) a1 ( i ) ......... a L1 ( i )] is called
the steering vector for the angle
 i .These form a
linearly independent set assuming the AOA of each of
the M signals is different. The vector n n represents
the uncorrelated noise present at each antenna element.
Because the steering vectors are a function of the AOA
of the signals, the angles can be computed if the
steering vectors are known or if a basis for the subspace
spanned by these vectors is known. The set of all
possible steering vectors A is known as the array
manifold and is given by
a 0 ( 0 ) a 0 (1 ) .......................a 0 ( M 1 ) 
a ( ) a ( ) ........................a ( ) 
1
1
1
M 1 
 1 0
 .

.
.
A

.
.
 .

 .

.
.


a M 1 ( 0 ) a M 1 (1 ) ...........a M 1 ( M 1 ) 
and
E[nn n ]   I one can obtain antenna array
H
n
2
correlation matrix which is given by
R  A Rss AH   2 I
(2.21)
Where, R is LxL Array Correlation Matrix
or Spatial Correlation matrix,
A is LxM Array
AH is hermitian transpose of A,
 2 is noise variance, I is LxL identity matrix and
Rss is MxM source amplitude matrix given by
Manifold Vector,
b1b1* 0   0 
b1 


b 
*
 0 b2 b2   0 
2 
Rss  E[bn bnH ]   .  * b1* b2* ...........bM*   


 


 


 
 
. 
 0
 bM 
0   bM bM* 
 


(2.17)
2.3 Formation of Array Correlation Matrix or Spatial
Covariance Matrix
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Rss  E[bn bnH ]
Defining

(2.22)
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International Journal of Engineering Trends and Technology (IJETT) – Volume 34 Number 7- April 2016
Where b1 , b2 ,......... bM are amplitudes of M signals
(sources).
2.4 Finding Eigen value and Eigen Vectors of Array
Correlation Matrix
Eigen values and Eigen vectors provide useful
and important information about a matrix. . The
determination of the Eigen values and Eigen vectors of
a system is very much important in engineering, where
it is equivalent to matrix diagonalization and grow up in
common applications associated to stability analysis,
the physics of rotating bodies, and small oscillations of
vibrating systems, to name only a few. It is possible to
determine whether a matrix is positive definite,
invertible, indicate how sensitive determination of
inverse will be to numerical errors. Eigen values and
Eigen vectors are useful in spectrum estimation and
adaptive filtering problems Each Eigen value is paired
with a corresponding so called Eigen vector
The Eigen values of LxL Array Correlation
matrix R is found by solving the characteristic
equation given by
R  I  0
(2.23)
The solution to equation (2.23) gives L Eigen
values
1 , 2 ,......, L .
The Eigen Vector for specific Eigen value
by solving the equation given by
 a is found
R Vn   a Vn
(2.24)
Where V n is Lx1 matrix comprising of unknown
variables.
Expanding equation (2.24) in matrix notation, one can
obtain
 R0, 0
R
 1, 0
 

 RL , 0
R0,1
R1,1

RL ,1
 R0, L  V1

 R1, L  V2

  

 RL , L  VL
(2.25)
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
V1


   V2
a

 



VL






Multiplying the matrices, a set of simultaneous
equations as defined in (2.26) are obtained as
R0, 0 V1  R0,1 V2  .............. R0, L V L   a V1
R1, 0 V1  R1,1 V2  .............. R1, L V L   a V2




 
R L , 0 V1  R L ,1 V2  .............. R L , L V L   a V L
(2.26)
Since there are L unknowns we have L
simultaneous equations which can be solved to obtain
V1 ,V2 ,......., VL
. These L values form Eigen vector
matrix.
III. MULTIPLE SIGNAL CLASSIFICATION
(MUSIC)
MUSIC [2] is an acronym which stands for Multiple
Signal Classification. It is a high resolution signal
parameter estimate and provides information about
AOA, incident signals, cross correlation, noise power of
the point sources. Multiple Signal Classification
method is widely used in signal processing applications
for estimating and tracking the frequency and emitter
location. MUSIC makes the assumption that the noise
in each channel is un-correlated making the noise
correlation matrix as the diagonal matrix. The incident
signals may be correlated creating a non diagonal signal
correlation matrix. However, under some high Signal
correlation conditions the MUSIC algorithm breaks
down and some other methods must be implemented to
correct this weakness.
One must know in advance the number of
incoming signals hence one must search the Eigen
values to determine the number of incoming signals. If
the number of signals is M, the number of signal Eigen
values and Eigen vectors is M and the number of noise
Eigen values and Eigen vectors are L-M (L is the
number of antenna array elements). Because MUSIC
exploits the noise eigenvector subspace, so it is
sometimes referred to as a subspace method.
The Eigen values and Eigen vectors for
correlation matrix R is found. M Eigen vectors
associated with the signals and L−M Eigen vectors
associated with the noise are separated. The Eigen
vectors associated with the smallest Eigen values are
chosen to calculate power spectrum. For uncorrelated
signals, the smallest Eigen values are equal to the
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International Journal of Engineering Trends and Technology (IJETT) – Volume 34 Number 7- April 2016
 a1 
a 
 2
s  
 
  
a M 
variance of the noise. The L× (L− M) dimensional
subspace spanned by the noise Eigen vectors is given
by
E N  e1 e2 e3 .......... ....... e L  M 
(3.1)
Where, ei is the ith Eigen Value.
The noise subspace Eigen vectors [5] are
orthogonal to the array steering vectors at the AOA
1 , 2 ,......... . M . Because of this orthogonality
consideration, one can evidence that the Euclidean
distance d
2
 a ( ) H E N E NH a ( )  0 for each and
every angle of arrival 1 ,  2 ,......... . M .Placing this
euclidean distance expression in denominator will
create acute peaks at the AOA. The MUSIC pseudo
spectrum is given by
PMUSIC 
1
Where, a1,a2, ……., aM are amplitude of M sources.
3. The Hermitian transpose „sH‟ of signal vector S is
given by

s H  a1* a 2*   a M*

4. The signal correlation matrix or the power matrix „P‟
is given by
a( ) H E N E NH a ( )
P  E[ ss H ]
(3.2)
Where, a( ) is steering vector for an angle

and E N is L x L-M matrix comprising of noise Eigen
vectors.
3.1 Simulation Methodology of MUSIC Algorithm
1. The steering vector „A‟ for an antenna array
comprising of L antenna elements is calculated by using
equation
1
1

1


 e i 2d sin 1
e i 2d sin  2  e i 2d sin  M 1 

A






 i 2d ( L 1) sin 1 i 2d ( L1) sin  2
i 2d ( L 1) sin  ( M 1) 
e
 e
e

2. The source amplitude matrix or signal amplitude
vector „s‟ is a column vector of order Mx1 is given by
Substituting the value of s and sH in above
equation one can obtain equation
 a1a1* a1a2*   a1aM* 
 a1 
 *
a 
*
* 
 a2 a1 a2 a2   a2 aM 
 2
p     * a1* a2*   aM*   


 


 


 
 

a a * a a *   a a * 
aM 
M M
 M 1 M 2


5. The signal vector „S‟ is MxM diagonal matrix
comprising of only diagonal elements of matrix „P‟
given by
a1a1*

 0
S  

 
 0

0
a2 a2*


0
 
 


 
0 

0 
 

 
* 
aM aM

6. The Signal subspace is a LxL matrix given by
Rs  ASA H
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International Journal of Engineering Trends and Technology (IJETT) – Volume 34 Number 7- April 2016
7. The Noise subspace is LxL matrix given by
1 0  0  2
0 1  0  
 0
Rn   2 I   2 
     

 
0 0  1  0
Where,

2
is variance of noise and
matrix
2

0
0

 0

 

  2 

0

Since there are L unknowns we have L
simultaneous equations which can be solved to obtain
I is identity
R  R S  Rn
RS is signal subspace and Rn is noise subspace.
9. Find the Eigen values of array correlation matrix by
performing Eigen Value Decomposition
The Eigen values of LxL Array Correlation
matrix R is found by solving the characteristic
equation given by
values
solution
10. The noise Eigen values are selected from all Eigen
values
PMUSIC 
1
a ( ) E N E NH a ( )
H
E N is the noise Eigen vectors
This power spectrum is computed by keeping
E N constant and varying „θ‟ in the range –90°<
θ+0.001< 90°.
R  I  0
The
. These L values form Eigen vector
matrix.
11. The power is calculated by using
8. The array correlation matrix is given by
Where,
V1 ,V2 ,......., VL
to
equation
1 , 2 ,......, L .
gives
L
Eigen
3.2 FLOW CHART of MUSIC Algorithm
The Eigen Vector for specific Eigen value
by solving the equation given by
 a is found
R Vn   a Vn
Where V n is Lx1 matrix comprising of unknown
variables.
Expanding equation in matrix notation, one can obtain
 R0, 0
R
 1, 0
 

 RL , 0
R0,1
R1,1

RL ,1
 R0, L  V1

 R1, L  V2

  

 RL , L  VL

V1


   V2
a

 



VL






Multiplying the matrices, a set of simultaneous
equations as defined in are obtained
R0, 0 V1  R0,1 V2  .............. R0, L V L   a V1
R1, 0 V1  R1,1 V2  .............. R1, L V L   a V2




 
R L , 0 V1  R L ,1 V2  .............. R L , L V L   a V L
Figure 3.1: FLOW CHART of MUSIC Algorithm
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International Journal of Engineering Trends and Technology (IJETT) – Volume 34 Number 7- April 2016
IV. SIMULATION
We used the MATLAB software to implement the
MUSIC spectrum. The parameters used are:
No of antenna elements L
No of mobile users to be detected M
Direction from where the em wave strikes the
antenna θ
C. Multiple Source Simulation for less Antenna
Elements
L=8, M=2, θ =10° and 45°
A. Single Source Simulation for less Antenna
Elements
L=8, M=1, θ=45°
Fig 4.3: Multiple uers simulation for less antenna
elements
Fig 4.1: Single user simulation for less antenna
elements
B. Single Source Simulation for more Antenna
Elements
D. Multiple Source Simulation for more Antenna
Elements
L=100, M=2, θ=10° and 60°
L=100, M=1, θ=60°
Fig 4.2: Single user simulation for more
Fig 4.4: Multiple user simulation for more antenna
antenna elements
elements
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International Journal of Engineering Trends and Technology (IJETT) – Volume 34 Number 7- April 2016
E. Simulation for closely spaced multiple mobile
1.
users
L=100, M=4, θ=10°, 15°, 20°, and 25°
2.
3.
The performance of the MUSIC algorithm
improves substantially if the no of antenna
elements are increased.
For widely spaced mobile users with less
or more antenna elements the resolution of
MUSIC is high and bias is low
For closely spaced mobile users with less
or more antenna elements the resolution of
MUSIC is high and bias is low.
REFERENCES
[1] Gross, F.B., 2005. Smart Antennas for Wireless Communication
with MATLAB. McGraw-Hill.
[2] Godara, Smart antennas, CRC Press Boca Raton, FL, 2004
Fig 4.5: MUSIC simulation for closely spaced users
F.
Simulation for widely spaced multiple mobile
users
L=100, M=4, θ =10° , 30°, 50° , and 70°
[3] Lal. C. Godara.“Limitations and capabilities of directions of
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Fig 4.6: MUSIC simulation for widely spaced users
V. CONCLUSION
Precise AOA estimation can be obtained by varying
the parameters of the antenna. The results of the present
study can be summarized as:
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