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Beamforming in Tight Specifications Environment

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1
Beamforming in Tight Specifications Environment using Generalized Minimum Mean Error (GMME) Algorithm
Bashir A. El-Jabu, Associate Professor Higher Industrial Institute Consultant for the General Post and Telecommunication Company (Tripoli, Libya) Jalal A. Srar, Dept. of Elec. Engr., University of 7th of October, (Misurata, Libya), jalalsaed @ yahoo.com Omar A. Abu-Ella1, Dept. of Elec. Engr., University of 7th of October, 2659 (Misurata, Libya), omar_abuella@yahoo.com
Abstract
—
Frequency spectrum is considered to be scarce and valuable resource. Techniques, which help in its utilization, are welcome. Among these techniques is the beamforming in which a set of weight vector is obtained for optimum error solution. Many algorithms are used in the optimization process. The least mean square error is often used because of its simplicity and reasonable performance. In tight specifications environments (low SINR, close angular separation between the desired signal and the interference) the authors noticed degradation in the LMS performance. Therefore, a higher order error function is proposed in this paper, a technique named generalized minimum mean error (GMME). Simulation tests showed that increasing the order of the error function leads to better performance with no significant increase in the computational costs.
1
2
T
ABLE OF
C
ONTENTS
1.
I
NTRODUCTION
......................................................1
2.
B
EAMFORMING
T
ECHNIQUES
...............................1
3.
B
EAMFORMING
U
SING
G
ENERAL
M
INIMUM
M
EAN
E
RROR
(GMME)
A
LGORITHM
......................1
4.
S
IMULATION
M
ODEL
.............................................2
5.
S
IMULATION
R
ESULTS
..........................................4
6.
C
ONCLUSION
.........................................................6
R
EFERENCES
............................................................
7
B
IOGRAPHY
..............................................................
7
1.
I
NTRODUCTION
Adaptive beamforming is one of the useful techniques used to utilize frequency spectrum in communications systems. Different algorithms are used. The most commonly used algorithm is the least mean-square (LMS) one. This paper is devoted to generalize the LMS algorithm (which has an error behavior of order 2) to take any order i.e. Generalized Minimum Mean Error (GMME), and apply this technique for antenna beamforming in tight specifications environment, to see its effect on performance and computational cost. Here, tight specifications environment mean one or more of the following conditions:
•
Higher noise levels. 1
1
1-4244-0525-4/07/$20.00 ©2007 IEEE.
2
IEEEAC paper #1681, Version 3, Updated December 18, 2006
•
Large number of interfering signals.
•
Higher interfering levels.
•
Close angular separations between the desired and some interferences signals. The mathematical derivation of the algorithm is carried out. The algorithm is investigated with respect to the number of interfering signals with different angular positions and several noise levels (SNRs) using simulation.
2.
B
EAMFORMING
T
ECHNIQUES
Adaptive beamforming can be classified into two major categories; non-blind adaptive algorithms, e.g. LMS, Recursive Least Square (RLS) and Sample Matrix Inversion (SMI), and blind adaptive algorithms e.g. Constant Modulus Algorithm (CMA) and Steepest Descent decision directed (SD-DD). Non-blind adaptive algorithms need statistical knowledge of the transmitted signal in order to converge to a weight solution. This is typically accomplished through the use of a pilot training sequence sent over the channel to the receiver to help identifying the desired user. On the other hand, blind adaptive algorithms do not need any training; hence, the term “blind” is used. They attempt to restore some type of characteristic of the transmitted signal in order to separate it from other users in the surrounding environment [1]-[5].
3.
B
EAMFORMING
U
SING
G
ENERAL
M
INIMUM
M
EAN
E
RROR
(GMME)
A
LGORITHM
The General Minimum Mean Error (GMME) criterion intends to find a weight vector that will minimize the mean of
m
th
order error between the combined signal (desired signal plus interfering and noise) and some desired (or reference) signal. This algorithm is based on the method of steepest descent as in LMS case. Changes in the weight vector are made along the direction of the estimated gradient vector. Accordingly,
[ ]
))(()()1(
k J k wk w
w
∇+=+
µ
(1)
2where
)(
k w
is the weight vector before adaptation,
)1(
+
k w
is the weight vector after adaptation,
µ
is a step-size parameter controlling the convergence and stability characteristics of the GMME algorithm,
)0(
>
µ
,
)(
J
w
∇
is the estimated gradient vector of the cost function with respect to
w
. One method for obtaining the estimated gradient of the function is to take the gradient of a single time sample of the
m
th order error,
m
k e E k J
)()(
=
which defines the GMME cost function, i.e.
[ ]
( )
[ ]
mmw
k e E wk e E
)(2)(
∗∆
∂∂=∇
(2) where
( )
m
k e
)(
is the
m
th order error between the beamformer output
)(
k y
and the reference signal, i.e.
)()()(
k yk d k e
−=
(3) Therefore,
[ ]
{ }
[ ]
{ }
[ ]
⎭⎬⎫⎩⎨⎧
−∂∂=−∂∂=∂∂
∗
22**
)()(
)()()(
mmm
k yk d E
wk yk d E
wk e E w
(4) But,
[ ]
( )( )
[ ]
∗∗∗
−−∂∂=−∂∂
)()()()()()(
2
k yk d k yk d E
wk yk d E
w
( )( )
[ ]
∗∗
−−∂∂=
)()()()()()(
k xk wk d k xk wk d E
w
H H
[
]
)()()()(
)()()()()()()(
2
k wk xk xk w
k xk wk d k wk xk d k d E
w
H H H H
+−−∂∂=
∗∗
{ }
[ ]
{ }
[ ]
)()()()()(
k ek x E k d k yk x E
∗∗∗
−=−=
By taking the instantaneous value of the estimated one
[ ]
)()()()(
2
k ek xk yk d E
w
∗∗
−=−∂∂
(5) Substituting (5) in (4) leads to
{ }
2122
()()()()()2
mm
E e k E d k y k w wm E d k y k
∗ ∗−
∂ ∂
⎡ ⎤ ⎡ ⎤
= −
⎣ ⎦ ⎣ ⎦
∂ ∂
⎧ ⎫⎪ ⎪⎡ ⎤
⋅ −
⎨ ⎬⎣ ⎦⎪ ⎪⎩ ⎭
( )
( )
1221*2
()()()2()()()()2
mm
m x k e k e k m x k e k e k e k
−∗−∗
= −= −
( )
( )
122
)()()(
2
−∗
−=
mm
k ek ek x
m
(6) Substituting (6) into (2) gives
[ ]
( )
( )
( )
122
)()()()(
−∗
−=∇
mmmw
k ek ek mxk e E
(7) Let
pm
2
=
,...,2,1
=
p
, (7) becomes
( )
( )
( )
12
)()()(2)(
−∗
−=∇
p p pw
k ek ek pxk e E
(8) Inserting (8) into (1), the formula of adapted weights vector can be obtained as follows
( )
( )
)()()(2)()1(
1
k ek ek x pk wk w
p p
−∗
+=+
µ
(9) where
2p
is the algorithm order.
4.
S
IMULATION
M
ODEL
a. Continuous Signal Model
The adaptive beamforming system is shown in Fig. 1. The desired signal is represented by
)2(
)(
d c
t f jd d
e At s
φ π
+
=
(10) where
d
A
is its amplitude,
c
f
is its operating carrier frequency,
d
φ
is its phase, and the interfering signal is represented by
)2(
)(
I c
t f j I
e At I
φ π
+
=
(11) where
I
A
and
I
φ
are its amplitude and phase. Therefore, the array output signal (or beamformer input signal) can be written as follows
)()()()()()(
t nat I at st x
I d d
++=
θ θ
(12) where
)(
t n
denotes an
1
×
vector of normal Gaussian noise signal with zero mean and standard deviation equal to
3
1
W
2
W
M
W
∑
)(
k y
+
desired k d
=
)(
)(
k e
)1(
φ ω
+
=
t j
e I
)(
φ ω
+
=
t j
eSignal Desired
i
θ
d
Fig.1 Adaptive beamforming system
n
σ
, and
)(
d
a
θ
and
)(
I
a
θ
are the steering or (array propagation) vectors given as follows
⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡
=
−−−
d d
d M jd jd
eea
θ θ
λ π λ π
θ
sin)1(
sin
22
1)(
(13)
⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡
=
−−−
I I
d M jd j I
eea
θ θ
λ π λ π
θ
sin)1(
sin
22
1)(
(14) such that
d
θ
is the direction of arrival (DOA) of the desired signal, and
I
θ
is the interfering signal DOA. The simple model in (12) can be generalized to q signals (including the desired one) to take the form )()()()(
1
t nt sat x
iqii
+=
∑
=
θ
(15) where )(
i
a
θ
is the array propagation vector of the
i
th signal given by
⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡
=
−−−
ii
d M jd ji
eea
θ θ
λ π λ π
θ
sin)1(
sin
22
1)(
(16) with
qi
i
...,,2,1,
=
θ
represents the AOAs of the incident signals on the array. In a matrix form, (12) becomes )()()()(
t nt s At x
+Θ=
(17) where )(
Θ
A
is an
q M
×
matrix of the steering vectors given by ])(...,)(),([)(
21
q
aaa A
θ θ θ
=Θ
(18) and
⎥⎥⎥⎦⎤⎢⎢⎢⎣⎡
=
)()()(
1
t st st s
q
(19)
b. Discrete Signal Model
In (17), if the input data vector is sampled
K
times, at
t
1
,…,
t
K
, the sampled data may be expressed as
N S A X
+Θ=
)( (20) where
X
and
N
are the
K
×
matrices containing
K
snapshots of the input data vector and noise vector, respectively, i.e. )](...,),2(),1([)](...,),(),([
21
K x x xt xt xt x X
K
==
(21) )](...,),2(),1([)](...,),(),([
21
K nnnt nt nt n N
K
==
(22) and
S
is the
K q
×
matrix containing
K
snapshots of incident signals, i.e. )](...,),2(),1([
)](...,),(),([
21
K s s s
t st st sS
K
==
(23) The time index
t
is dropped for simplicity.
c. Initialization the Weights Vector
We can initialize the weights vector by equating it to the first column of the
M
×
identity matrix. (the first element is 1 and all other elements are zero), supposing that the elements are working as omnidirectional antenna in the beginning of simulation.
d. Iterative Algorithm
To construct the adaptive beamforming algorithm we will replace the notation
K
for sample time by iteration notation
l
as follows
)()(
k X l X
=
(24) where
K k
...,3,2,1
=
,
)(
k X
is the
k
th sample of the input data and the reference signal is )()()()(
1
k S l d or k S l d
d
==
(25) where )(
k S
d
is the
k
th sample of the desired signal. At the
l
th iteration the output signal of the beamformer is )()()(
l X l wl y
H
=
(26)
4the error is )()()(
l yl d l e
−=
(27) and the weights are
( )
( )
1
)()()(2)()1(
−∗
+=+
p p
l el el X pl wl w
µ
(28) To decrease the required number of operations in each iteration we can rewrite (28) as follows
( )
( )
)()()(2)()1(
1
l X l el e pl wl w
p p
−∗
+=+
µ
(29)
Here,
µ
the adaptation step size, may be used to normalize the algorithm by using the inequality
max
21
λ µ
≤
, where
max
λ
denotes the maximum eigenvalue of the input data covariance matrix and is given by [6]
[ ]
{ }
)()(eigmax
max
l X l X
H
=
λ
(30)
e. Simulation Environment
The system considered to evaluate the performance of the GMME algorithm is with the following assumptions:
•
Uniform linear array with 4, 8, and 16 elements, and 2/
λ
spacing between any consecutive elements.
•
Phase delay between signals (desired and interference) is
o
0
=
φ
or
o
90
=
φ
.
•
Noise is a random Gaussian distributed signal with zero mean and standard deviation
n
σ
.
5.
S
IMULATION
R
ESULTS
The simulation is performed to obtain the following performance indicators:
•
Beam Pattern Characteristics.
•
Error behavior.
•
SINR against Angular Separation.
•
SIR against Number of Interfering Signals. The results can be summarized as follows:
a. Beam Pattern Characteristics:
In this case, the patterns are plotted for 8 antenna elements, with desired DOA equal to 0
o
, and with two interfering DOAs from 15
o
, 30
o
, and
p
=1 (LMS), 2 (fourth order-LMF), and 4 (8
th
order). The results are as displayed in Fig.2 and Fig.3, from which one can observe that 8
th
order GMME is the best of the three, where its maximum gain in Fig.2 Beam pattern of LMS, LMF and 8th Order GMME with SNR=0 dB Fig.3 Beam pattern of LMS, LMF and 8
th
Order GMME with SNR=-10 dB the desired DOA rise above the maximum gains of LMF and LMS beam patterns with approximately 4 dB in case of -10 dB SNR environment, and it has the smallest value in interferences DOAs. Also we can observe that accuracy of DOAs selectivity is improved by increasing the order of beamforming algorithm. All of these higher order advantages appear clearly at lower SNR, i.e. the higher order of GMME algorithm is more suitable for this tight specifications environment.
b. Error behavior:
In this case, the standard deviation of the error between the desired signal and the output signal is measured two times, one with one interfering signal and the other with the same previous 6 interfering DOAs for 8 and 16 antenna elements, with desired DOA equal to 0
o
. The curves are shown in Fig. 4 to Fig. 7, and from these we conclude that
-80 -60 -40 -20 0 20 40 60 80-30-25-20-15-10-50Angle in Degrees
N o r m a l i z e d G a i n ( d B )
8th OrderLMSLMF
-80 -60 -40 -20 0 20 40 60 80-30-25-20-15-10-50Angle in Degrees
N o r m a l i z e d G a i n ( d B )
8th OrderLMSLMF
5
•
At low signal to interference plus noise (SINR) levels around -6 dB, LMS is the best algorithm since it gives the lowest standard deviation (STD) of error.
•
As the noise level increased i.e. SINR decrease less than -6 dB, LMF will be the dominant up to a certain point at which the 8th Order GMME will be the dominant; the order of dominant algorithm increased in the same manner if the SINR is decreased.
•
Increasing the algorithm order will minimize the ripples in the STD of error curve. In other wards, the curve is smoother using higher orders algorithms.
•
As the number of elements decreases, the performance decreases (more error as expected) the higher order performance is better.
•
LMS curves starts with small values of STD and increases rapidly with decreasing SINR, but as the order increases the curves increases more slowly. Generally the minimum error value for the 8
th
Order of GMME algorithm is smaller than the error value for LMS and LMF algorithms.
c. SINR Performance against Angular Separation:
In this case, SINR is measured with one interfering signal which has DOA of over 180
o
(-90
o
to 90
o
) for four elements array, when the desired signal arrives from the bore-side once and end-fire another time. These measurements are plotted in Fig.8 and Fig. 9. We can observe that SINR for 8
th
Order GMME is the highest one for the most DOAs of interfering signal. However, in case of the bore-side, the SINR of LMS becomes greater than SINR of LMF and 8
th
Order GMME when the interfering direction is beyond 60
o
i.e. wide angular separation between desired and interfering DOAs. This means that higher order algorithm is more efficient in case of close angular separation between desired and interfering DOAs i.e. in tight specifications environment. Fig.4 Standard deviation of the error versus SINR for 16 antenna elements using six interfering signal Fig.5 Standard deviation of the error versus SINR for 16 antenna elements using a single interfering signals Fig.6 Standard deviation of the error versus SINR for 8 antenna lements using a single interfering signal
-24-22-20-18-16-14-12-10-8-6
00.050.10.150.20.25SINR (dB)
s t a n d a r d d e v i a t i o n o f t h e e r r o r
8 OrderLMSLMF
Fig.7 Standard deviation of the error versus SINR for 8 antenna elements using six interfering signals
-24-22-20-18-16-14-12-10-8-6
00.050.10.150.20.25SINR (dB)
s t a n d a r d d e v i a t i o n o f t h e e r r o r
8th OrderLMSLMF
-30-25-20-15-10-50
0.10.120.140.160.180.20.220.24SINR (dB)
s t a n d a r d d e v i a t i o n o f t h e e r r o r
8th OrderLMSLMF
-30-25-20-15-10-50
0.080.10.120.140.160.180.20.220.24SINR (dB)
s t a n d a r d d e v i a t i o n o f t h e e r r o r
8th OrderLMSLMF

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