Get documents about "Analysis on Robust Adaptive Beamformers
Shared by: ijcsis
Categories
Tags
IJCSIS, call for paper, journal computer science, research, google scholar, IEEE, Scirus, download, ArXiV, library, information security, internet, peer review, scribd, docstoc, cornell university, archive, Journal of Computing, DOAJ, Open Access, March 2011, Volume 9, No. 3, Impact Factor, engineering, international, proQuest, computing, computer, technology
-
Stats
- views:
- 168
- posted:
- 4/9/2011
- language:
- English
- pages:
- 8
Document Sample
(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 3, March 2011
Analysis on Robust Adaptive Beamformers
T.S.JEYALI LASEETHA 1, DR.(MRS) R.SUKANESH2
1. Professor
Department of Electronics and Communication Engineering,
Anna University of Technology
Tirunelveli, Tamil Nadu
INDIA
email id: laseetha@gmail.com
2. Professor
Department of Electronics and Communication Engineering,
Madurai,Tamil Nadu
INDIA
Abstract: MVDR (minimum variance distortionless response) beamformer is the optimal beamformer which
utilizes the second order statistics of the actual data for obtaining the Covariance matrix from which the weight
vector of the antenna array is determined. In adaptive beamfomer which utilizes MVDR beamformer along
with SMI(sample matrix inversion), actual data is not available to calculate the covariance matrix. Instead,
covariance matrix is estimated from the available data. It includes finding the Matrix inversion. It may result in
bad conditioning. To avoid this, some amount of loading is introduced to the diagonal elements, which is called
diagonal loading. Diagonal loading can be inserted by adding a scaled version of identity matrix. Diagonal
loading imparts Robustness to the adaptive beamformer against signal mismatch due to low sample support and
helps to achieve desired sidelobe level and SINR improvement. Various methods in diagonal loading are
analyzed in this paper with different loading levels and a novel hybrid algorithm for MVDR-SMI beamformer
with colored adaptive diagonal loading is also proposed. The performance of the proposed methods is
compared with other methods such as Conventional, MVDR, MVDR-SMI, MVDR-SMI-Diagonal Loading,
MVDR-SMI-Colored –DL, MVDR-SMI-Adaptive DL by conducting simulations experiment. The proposed
method shows the improvement in directivity and SINR compared to other methods.
Key-words: Smart antennas, Adaptive beamforming, Uniform Linear Array, Minimum Variance Distortionless
Response Beamformer (MVDR), Sample-Matrix Inversion(SMI), Adaptive colored diagonal loading
1.Introduction
In Wireless Communications, smart technologies that results from the dynamic variation of an
are not only being applied at the antenna level, but element-space processing weight vector as opposed
also at the receiver for direction of arrival to a switched-beam or beam-space antennas, is
estimation, detection, diversity combining and controlled by an adaptive algorithm, which is the
equalization and at baseband processing software MVDR-Sample Matrix Inversion algorithm[2,7,10].
levels. The ultimate benefit of these techniques is to It minimizes cost function reduction of a link’s
increase cellular capacity and range. Adaptive SINR by ideally directing beams toward the signal-
beamforming reveals to be a complementary means of-interest (SOI) and nulls in the directions of
for signal-to-interference-plus-noise-ratio (SINR) interference. Many algorithms differ largely in
optimization [6,7,10]. In this paper, at antenna array complexity, correlation matrix, eigen value spread
elements level, the formation of a lobe structure, dependence, inherent gradient noise, limited
171 http://sites.google.com/site/ijcsis/
ISSN 1947-5500
(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 3, March 2011
dynamic range and limited number of samples used "{ { " { {IJJ{ I{ . It is arriving from an
[1], [2]. angle θ0 and is received by the ith sensor. The signal
S0(t) is a baseband signal having a deterministic
In optimum beamformers optimality can be amplitude and random uniformly distributed phase
achieved in theory if perfect knowledge of the and Fc is the carrier frequency. The symbol is
second order statistics of the interference is used to indicate that the signal is a pass band signal.
available. It involves calculation of interference plus
noise correlation matrix . For real world X1(k) is the single observation or measurement of
scenarios, the adaptive methods are followed to this signal made at time instant k, at sensor 1, which
obtain optimality. In adaptive beamformer, the is given as
correlation matrix is estimated from the collected
I# { { I" " { { - {I# I$ I {{H# { { H$ { { H { {{ - J{ { (1)
data. In sample matrix Inversion technique a block
I" "{ {- (# I H { { - J{ {
of data is used to estimate the adaptive beamforming
(2)
weight vector. The estimate is not really a
substitute for true correlation matrix . Hence Hence the single observation or measurement
there is loss in performance. The SINR which is a made at the array of elements at the time instant k,
measure of performance of the beamformer called array snapshot is given as a vector
degrades as the sample support (the number of data)
is low. The lower band on sidelobe levels of the I{ { {I# { { I$ { { I% { { I { {{ (3)
beamformer when no interference sources were
found at an angle is also to be calculated. Training The general model of the steering vector[13] is
issues like the presence of desired signal in the given as
correlation matrix is also dealt with. The
{{ – { È { {{ % {{ % { È { {{
desired signal in the training set results in the #
cancellation and subsequent lose in performance. (4)
The paper is organized as follows. In Section 2, Also it is assumed that the desired signal,
Problem formulation and general model is interference signals and noise are mutually
presented. In Section 3 Adaptive beamforming with uncorrelated.
various beamforming methods are presented along
with the Novel Hybrid algorithm -Adaptive colored
diagonal loading. In Section 4 simulation
experiments are presented. Section 5 contains
Results and discussions. Section 6 presents the 3. Adaptive Beamforming
conclusions.
In optimum beamformer, a priori knowledge of true
statistics of the array data is used to determine the
correlation matrix which in turn is used to derive the
beamformer weight vector. Adaptive Beamforming
2. Problem Formulation And General is a technique in which an array of antennas is
Model exploited to achieve maximum reception in a
specified direction by estimating the signal arrival
An uniform linear array (ULA) of M elements or from a desired direction while signals of the same
sensors is considered. Let a desired signal S0 from a frequency from other directions are rejected. This is
point source from a known direction θ0 with achieved by varying the weights of each of the
steering vector ‘a0’ and L number of J(jammer or) sensors used in the array. Though the signals
interference signals from unknown emanating from different transmitters occupy the
directions{ # $ % {, specified by the same frequency channel, they still arrive from
steering vectors {I# I$ I% I { respectively different directions. This spatial separation is
impinge on the array. The white or sensor or thermal exploited to separate the desired signal from the
noise is considered as ‘n’. interfering signals. In adaptive beamforming the
optimum weights are iteratively computed using
A single carrier modulated signal "{ { is given by complex algorithms based upon different criteria.
For an adaptive beamformer, covariance or
correlation matrix must be estimated from unknown
172 http://sites.google.com/site/ijcsis/
ISSN 1947-5500
(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 3, March 2011
statistics of the array snapshots to get the optimum This beamforming method experiences the
array weights. The optimality criterion is to following drawbacks
maximize the signal-to-interference-plus-noise ratio
to increase the visibility of the desired signal at the 1) the computational complexity is more in the
array output. The determination of the presence of order of { $ { J { % { .
signals of interest is known as detection while the 2) In the case of large array, low sample
inference of their parameters likes, the angle of support i.e(M>>k), may result in
arrival θ0, is referred to as estimation. In this paper it singular matrix or ill-conditioned.
is assumed that the angle of arrival of the desired
signal is known
conventional beamforming
0
3.1 Estimation Of Correlation Matrix
-10
The correlation matrix can be estimated[6,7,8,9] -20
using different methods which would result in
different performance and behavior of the algorithm. -30
beam response in dB
In block adaptive Sample Matrix Inversion -40
technique, a block of snapshots are used to estimate -50
the ensemble average of and is written as[8]
-60
#
{ { { { {{ (# { { { { (5) -70
$
I" I" - -
-80
=H (6)
-90
where N is the number of snapshots used and k is
the time index, $ is the power of the desired signal
-100
-80 -60 -40 -20 0 20 40 60 80
angle inθ
and and are the jammer and noise correlation
matrices, respectively. The interference-plus-noise Fig 1 conventional beamforming showing the
correlation matrix is the sum of these two matrices beampattern
- (7)
Where $
H, and $ is the thermal noise Mvdr beamforming
power, I is the identity matrix. It is assumed that 0
thermal noise is spatially uncorrelated. -10
-20
-30
3.2 MVDR Beamformer
beam response in dB
-40
The MVDR beamformer whose pattern is shown in
Fig(2) is an adaptive high resolution beamformer -50
that minimizes the output power while maintaining -60
unity response in the desired direction.
Mathematically a weight vector ‘w’ is to be -70
calculated with the constrained optimization of -80
problem
-90
c J I I J c I" (8)
-100
-80 -60 -40 -20 0 20 40 60 80
Now the optimal weight vector may be written as angle inθ
#
{ {È { { #
{ { (9) Fig 2 MVDR-the optimum beamformer-
beampattern
173 http://sites.google.com/site/ijcsis/
ISSN 1947-5500
(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 3, March 2011
0
Mvdr-smi beamforming minimum loading level must be equal to noise
power. Diagonal loading increases the variance of
the artificial white noise by the amount $ . This
-10
-20
modification forces the beamformer to put more
-30
effort in suppressing white noise rather than
beam response in dB
-40 interference. When the SOI steering vector is
-50 mismatched, the SOI is attenuated as one type of
-60 interference as the beamformer puts less effort in
suppressing the interferences and noise[17].
However when $ is too large, the beamformer
-70
-80
-90
fails to suppress strong interference because it puts
most effort to suppress the white noise. Hence, there
-100
-80 -60 -40 -20 0 20 40 60 80 is a tradeoff between reducing signal cancellation
angle inθ
and effectively suppressing interference. For that
reason, it is not clear how to choose a good diagonal
loading factor $ in the traditional MVDR
Fig 3 MVDR-SMI beamformer with beam
response
beamformer.
3.3 Diagonal Loading (Dl) This conventional diagonal loading can be
thought of as a gradual morphing between two
To overcome the above mentioned drawback no. 2, different behavior, a fully adaptive MVDR solution
a small diagonal matrix is added to the covariance (H , no loading) and a conventional uniformly
matrix. This process is called diagonal loading[15] weighted beampattern (H ∞, infinite loading)[5].
or white noise stabilization which is useful to The conventional DL weight vector can be
provide robustness to adaptive array beamformers calculated as
against a variety of conditions such as direction-of-
arrival mismatch; element position, gain, and/or { - $
H{ #
{ { (10)
phase mismatch; and statistical mismatch due to
finite sample support[12,14]. Because of the where is the normalization constant
robustness that diagonal loading provides it is given by
{ { { - $
H{ #
{ {
always desirable to find ways to add diagonal
(11)
loading to beamforming algorithms and appropriate
amount of loading. But little analytical information and $ reduces the sensitivity of the beampattern to
is available in the technical literature regarding unknown uncertainities and interference sources at
diagonal loading [11]. To achieve a desired sidelobe the expenses of slight beam broadening[3]. The
level in MVDR-SMI beamformer sufficient sample choice of loading can be determined from L-Curve
support ‘k’ must be available. However due to non- approach[12] or adaptive diagonal loading. Beam
stationarity of the interference only low sample shape for Diogonal loading as shown in Fig(2) is
support is available to train the adaptive better when compared to previously stated methods
beamformer. We know that the beam response of an
optimal beamformer can be written in terms of its MVDR-diagonal loading
0
eigen values and eigen vectors. The eigen values are
random variables that vary according to the sample -10
support ‘k’. Hence the beam response suffers as the -20
eigen values vary. This results in higher sidelobe -30
beam response in dB
level in adaptive beam pattern. A means of reducing -40
the variation of the eigen values is to add a weighted -50
identity matrix to the sample correlation matrix.
-60
The result of diagonal loading of the correlation
matrix is to add the loading level to all the eigen -70
values. This in turn produces the bias in these eigen -80
values in order to reduce their variation which in -90
term produces side bias in the adaptive weights that -100
-80 -60 -40 -20 0 20 40 60 80
reduces the output SINR. Recommended loading angle in θ
levels of $ ≤ $ $
where $ is the noise Fig 4 MVDR-Diagonal Loading
$
power and is the diagonal loading level. The
174 http://sites.google.com/site/ijcsis/
ISSN 1947-5500
(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 3, March 2011
{ - $
H{ #
{ { (14)
3.4 Colored Diagonal Loading(Cdl) where [4]
In the presence of colored noise, DL can be applied 0
MVDR-adaptive diagonal loading
which is termed as colored diagonal loading (CDL)
and the morphing process may result in a -10
beampattern of our choosing. The colored diagonal -20
loading is similar to but the diagonal -30
loading level of $ = ∞ , end point, can be altered
beam response in dB
-40
by the term[5] -50
{ - $
{ #
{ { (12)
-60
-70
where is the covariance matrix that captures -80
the desired quiescent structure. It may be -90
determined directly 1)based on apriori information – -100
-80 -60 -40 -20 0 20 40 60 80
where , need not be diagonal or 2) desired angle in θ
weight vector –where must be diagonal. It is
given as Fig 6 MVDR- Adaptive Diagonal Loading
beampattern
#
I {? I C { {{ (13)
3.6 Adaptive coloured White Noise
where is the desired quiescent weight vector. Stabilization (ACDL)
The colored diagoinal loading shows no As already discussed, white noise stabilization is
improvement in pattern shape as shown in Fig.5 nothing but diagonal loading in which the adaptive
MVDR -colored-Diagonal loading
colored loading technique is imbedded to get a
0 novel hybrid method proposed as
#
-10
-20 ? - C { { (15)
-30
beam response in dB
-40
-50 MVDR-adaptive colored diagonal loading- the proposed algorithm
0
-60
-10
-70
-20
-80
-30
beam response in dB
-90
-40
-100
-80 -60 -40 -20 0 20 40 60 80
-50
angle in θ
-60
Fig 5 MVDR-Colored Diagonal Loading -70
-80
-90
-100
3.5 Adaptive Diagonal Loading (ADL) -80 -60 -40 -20 0 20 40 60 80
angle in θ
In this method the loading level is calculated
assuming the apriori information about the SNR is Fig 7 MVDR- Adaptive Colored Diagonal
available. The SNR can be estimated from link Loading beampattern
budget or using some SNR estimated algorithm.Gu
and Wolf proposed a variable loading MVDR.(VL-
MVDR) in which the loading level is chosen
as( $ {[16]
175 http://sites.google.com/site/ijcsis/
ISSN 1947-5500
(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 3, March 2011
have a compact size. Hence a maximum of 16
elements are chosen for further analysis
4. Simulations and Experiments
Table1. Effect of Changing number of antenna
For the proposed hybrid algorithm ,a 16 element elements
Uniform Linear Array is considered with SNR of 20 Desired
Beam forming method
dB for the desired signal coming from θs = 0° and signal Jammer1 Jammer2 Jammer3
INR of 70 dB for three jammer signals coming from θ=0° θ = 20° θ=-20° θ =-70°
the directions θi = -20°, 20° and 70°. The element Beam
spacing is d = 0.5 λ. Beam Beam Beam response
The various methods of beamforming are obtained response response response Power(in
to compare them with the performance of the Mvdr- Power Power Power dB)
Adaptive colored Diagonal Loading. (in dB) (in dB) (in dB)
conventional 0 -20 -20 -26.5
comparison of various diagonal loading methods
0
MVDR 0 -91 -66 -91
-10
-20 MVDR-SMI 0 -58 -61 -72
-30
DL 0 -72.5 -72.5 -85
beam response in dB
-40
-50 CDL -6 -50 -57 -66.5
-60
-70
ADL 0 -72.5 -72.5 -85
MVDR-diagonal loding
-80
MVDR-coloured DL
Mvdr-adaptive DL
ACDL 0 -52 -56.5 -62
-90
Mvdr-adapt-col-DL
-100
-80 -60 -40 -20 0 20 40 60 80
angle in θ
Table2: Beam response of the signals - desired
Fig 8 Beampattern of various diagonal loading and jammers – using various methods
methods
3dB beamwidth
elements
5. Results and Discussion
MVDR
MVDR
Diagon
conven
ve DL
Adapti
Adapti
Colore
- SMI
No.of
tional
d DL
ve
al
5.1 Number of elements
For the ULA which is considered for simulation 4 26.2 19.5 17.1 17 17 17 16
work , the beampatterns were analyzed by changing
the number of elements as 4, 8, 12,16, 24, 50 and 8 12.8 15.4 13.3 14.8 14.8 14.8 25.5
100. As the number of elements increases, the 7
beampattern shows higher resolution i.e the 3 dB
beamwidth becomes much narrower from to 26° to 12 8.4 8.7 6.9 8.4 8.5 8.7 8.5
1° for conventional beamformer and 17° to 1° for
adaptive diagonal loading beamformer. The detailed 16 6.25 6.4 6.4 6.4 6.4 6.6 13.2
results are tabulated in Table1. Finer or sharper
beams are obtained when more number of elements 20 5.1 5.2 6 5.3 5.3 5.2 6.8
are used. Sharper the beam, the beamformer is not
24 4.4 4.5 4.5 4.3 4.3 4.3 4.3
susceptible to jammers. But the number of side
lobes also increased. The 3-dB beamwidth of the 50 2 2 2 2 2 2 2
different beamformers are tabulated in Table 2. A
trade off can be obtained to reduce the cost and to 100 1 1 1 1 1 1 1
176 http://sites.google.com/site/ijcsis/
ISSN 1947-5500
(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 3, March 2011
5.2 Noise effect 6. Conclusion
An ULA with 16 elements is considered for A new Hybrid Robust adaptive beamforming
analyzing the effect of noise on the peaks of the algorithm is proposed with Adaptive Colored
signal power. Signal to noise ratio (SNR) was varied diagonal loading based on data dependent approach.
in steps of 10 dB starting from 10 dB till 60 dB. As This method is computationally efficient.
SNR increases the peaks becomes sharper. It also Simulation results show that the proposed method
showed that the interference sources were provide robustness against steering vector errors and
suppressed to a maximum extent, so that it will not random array position perturbations by comparing it
be a disturbance while extracting the signal even in with various diagonal loading methods
the presence of strong interferers
References
5.3 Training issues with the number of array [1] Viktor V.Zaharov, Marvi Teixeira, “SMI-
MVDR Beamformer Implementations for Large
snapshots
Antenna Array and Small Sample Size”, IEEE
Increasing the number of array snapshots lead to
Transactions on Circuits and Systems- I Regular
complexity and computational cost but the
Papers Vol.55 No.10, November 2008.
performance of the beamformer increases. It is a
trade off between the cost and the performance [2]. Biao Jiang, Ye Zhu, “A New Robust Quadratic
Constraint Beamforming against Array Steering
Vector Errors”, International Conference on
5.4 Element Spacing Communications Circiuts and Systems 2004, Vol 2,
The spacing between the elements for an 16 element 29-29 June 2004.
ULA was varied as λ/4, λ/2, 3λ/4 and λ
which in turn vary the effective aperture length of [3]Y.X.Zou,S.C.Chan ,et.al, “ Recursive Robust
the array. Among the four choices λ/2 showed the Variable Loading MVDR Beamforming in
best performance for the particular frequency used Implusive Noise Environment”, IEEE Asia Pacific
for simulation. When the distance between the Conference on Circuits and Systems 2008, DOI:
elements is increased beyond λ/2, it resulted in /10.1109/APCCAS.2008.4746190.
spatial aliasing i.e a lot of spurious peaks were
obtained which correspond to different frequencies. [4] Pekka Lilja, Harri Saarnisaari, “Robust Adaptive
Below λ/2 the resolution of the beams was not Beamforming in Software Defined Radio with
satisfactory. Adaptive Diagonal Loading”, IEEE Military
Communications Conference 2005, DOI:
Number of elements = 16, loading level= 100 10.1109/MILCOM.2005.1606058.
0
-5
[5]John D. Hiemstra, “Colored Diagonal Loading”,
SINR-CDL
-10
SINR-ADL Proceedings of the 2002 IEEE Radar
SINR-ACDL
conference,DOI: 10.1109/NRC.2002.999682.
-15
-20 [6]H.L.Van Trees, “Detection, Estimation , and
SINR in dB
-25 Modulation Theory”, Part IV, Optimum Array
-30 Processing, Wiley , NY,2002.
-35
[7]Dimitris G.Manolakis, Vinay.K.Ingle, “Statistical
-40
and Adaptive Signal Processing”, Artech House,
-45
2005
-50
0 50 100 150 200 250 300
number of snapshots k [8]Simon Haykin, “Adaptive Filter Theory”,
Prentice Hall of India, 1996.
Fig 9 Training issues with the number of
snapshots [9]Frank Gross, Smart Antennas for Wireless
Communications with Matlab, McGraw-Hill, 2005
177 http://sites.google.com/site/ijcsis/
ISSN 1947-5500
(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 3, March 2011
[10]Lal.C.Godara, “Applications of Antenna Arrays Communications, Circuits and Systems, 2004, PP
to Mobile communications, Part I: Performance 765-768
Improvement, Feasibility, and system
considerations”, Proceedings of IEEE Vol.85, No.7 [14]Jian Li and Petre Stoica, “Robust Adaptive
July 1997 Beamforming”,John Wiley & Sons Publications,
2006.
[11] Louis B.Fertig, “Statistical Performance of the
MVDR Beamformer in the presence of Diagonal [15]Jian Li and Petre Stoice&Zwang, “On Robust
Loading” Proc of IEEE Sensor Array and Capon Beamforming and Diagonal loading”IEEE
Multichannel Signal Processing Workshop, PP 77- Trans signal proc vol51, pp, 1702-1715 July 2003.
81, 2000
[16]J.Gu and P.J.Wolf,“Robust Adaptive
[12] John D. Hiemstra, “Robust Implementations of Beamforming using variable loading” In proc IEEE
the Multi stage Wiener Filter”, Ph.D Dessertation , workshop on senior Array and Multichannel signal
April 4, 2003. processing 2006.
[13] Biao Jiang, Ye Zhu, “A New Robust Quadratic [17]Chung-Yang Chen, “Quadratically constrained
Constraint Beamforming against Array Steering beamforming robust against Direction-Of-Arrival-
Vector Errors”, Proc of International Conference on Mismatch” IEEE Transactions on signal
processing,Vol 55,No.8,Aug 2007.
178 http://sites.google.com/site/ijcsis/
ISSN 1947-5500
Related docs
Other docs by ijcsis
Comparative Analysis between Split and HierarchyMap Treemap Algorithms for Visualizing Hierarchical Data
Views: 15 | Downloads: 0
Non-Preemptive Multi-Constrain Scheduling for Multiprocessor with Hopfield Neural Network
Views: 5 | Downloads: 0
Reliable Multipath Routing Protocol (RMRP) For Mobile Ad Hoc Networks Using Adaptive Video Compression
Views: 10 | Downloads: 1
Single CCTA-Based Four Input Single Output Voltage-Mode Universal Biquad Filter
Views: 36 | Downloads: 0
A Cloud Computing Architecture for E-Learning Platform, Supporting Multimedia Content
Views: 42 | Downloads: 0
