International Journal of Engineering Trends and Technology (IJETT) - Volume4Issue4- April 2013
Performance Analysis of Conjugate Gradient
and Recursive Least Square Adaptive Filters
On Smart Antenna Systems
Naresh birudala1, M. Siva subramanyam2
1
Department of Electronics and Communication Engineering,
Sree Vidyanikethan Engineering College, TIRUPATI – 517 102, A. P., INDIA
2
Department of Electronics and Communication Engineering,
Sree Vidyanikethan Engineering College, TIRUPATI – 517 102, A. P., INDIA
Abstract: In this paper, Adaptive beam former design
using conjugate gradient method (CGM) and RLS
algorithm has been proposed. The performance of
CGM has fast convergence rate than steepest descent
[2], [12] and that it also has lower computational
complexity when compared with the classic recursive
least squares (RLS) algorithm. Simulation results
reveal that RLS and CGM algorithms have high
resolution for beam formation. However CGM has
good performance to minimize MSE and better
convergence as compared to RLS. Therefore, CGM is
found more efficient algorithm to implement in the
mobile communication environment to minimize MSE.
Keywords—Adaptive filtering algorithms, conjugate
gradient method and RLS.
I.INTRODUCTION
Smart antenna can be used to achieve different
benefits. By providing higher network capacity, it
increases revenues of network operators and gives
customers less probability of blocked or dropped calls.
Adaptive Beam forming [1] is a technique in which an
array of antennas is exploited to achieve maximum
reception in a specified direction by estimating the signal
arrival from a desired direction (in the presence of noise)
while signals of the same frequency from other directions
are rejected.
In many adaptive filtering algorithms based on
the conjugate gradient (CG) method of optimization have
been reported [3], [4]. In these works, several
modifications have been proposed to improve the
performance of the CG algorithm for various applications,
ISSN: 2231-5381
but usually, the analysis of the proposed algorithms has
not been shown. It is well known that the CG algorithm
has a faster convergence rate than steepest descent [2] and
that it also has lower computational complexity when
compared with the classic recursive least squares (RLS)
algorithm [3], but mostly, its analysis can only be found in
the optimization and matrix computation literature. Here,
we will describe, from the signal processing point of view,
two of the CG algorithm implementations and analyze
their performance in steady state. Some related
implementation ideas can also be found in [3] and [7]. In
addition, their performance under finite word-length
effects will be discussed. Due to the highly nonlinear
nature of the algorithms, a linear quantization model is
used in the analysis of the LMS [6], NLMS [8] and RLS
[5] algorithms, in general, cannot be applied.
In this paper is organized as follows. A brief
review on adaptive filters is given in section II, overview
on RLS technique in section III, and CGM technique in
section IV. The simulation results are presented in Section
V. Concluding remarks are made in Section VI.
II. ADAPTIVE FILTER
The principle of an adaptive filter is its timevarying, self- adjusting characteristics. An adaptive filter
usually takes on the form of an FIR filter structure, with an
adaptive algorithm that continually updates the filter
coefficients, such that an error signal is minimized
according to some criterion. The error signal is derived in
some way from the signal flow diagram of the application,
so that it is a measure of how close the filter is to the
optimum. Most adaptive algorithms can be regarded as
approximations to the Wiener filter, which is therefore
central to the understanding of adaptive filters
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International Journal of Engineering Trends and Technology (IJETT) - Volume4Issue4- April 2013
N1
y[n ] = ∑ c [n]x[n - k ]
k =0
Here, the ck[n] are time dependent filter coefficients (we
use the complex conjugated coefficients ck[n] so that the
derivation of the adoption algorithm is valid for complex
signals, too).
The block diagram of an adaptive filter is as
shown in fig.1.
process of weighting these complex weights w1…wN-1
adjusted their amplitudes and phases such that when added
together forms the desired beam
In recursive least-square (RLS) algorithm [6], the weights
are updated by the following equation
W(n-1)=W(n-2) + K(n)[d(n)-W(n-2)X(n)]
The RLS algorithm does not require any matrix
inversion computations as the inverse correlation matrix is
computed directly. It requires reference signal and
correlation matrix information.
IV. CONJUGATE GRADIENT METHOD
Fig.1. Block diagram of adaptive filter
III. RLS ADAPTIVE FILTER
The convergence speed of the LMS algorithm
depends on the Eigen values of the array correlation
matrix. The RLS algorithm does not require any matrix
inversion computations as the inverse correlation matrix is
computed directly. It requires reference signal and
correlation matrix information. An important feature of the
recursive least square algorithm is that its rate
convergence is typically an order of magnitude faster than
that of the simple least square.
Given an adaptive filter with an input x(n), an
impulse response w(n) and an output y(n) you will get a
mathematical relation for the transfer function of the
system
The problem with the steepest descent method
has been the sensitivity of the convergence rates to the
Eigen value spread of the correlation matrix. Greater
spreads result in slower convergences. The convergence
rate can be accelerated by use of the conjugate gradient
method (CGM) [5]. The goal of CGM is to iteratively
search for the optimum solution by choosing conjugate
(perpendicular) paths for each new iteration. The method
of CGM produces orthogonal search directions resulting in
the fastest convergence. The path taken at iteration n + 1 is
perpendicular to the path taken at the previous iteration n.
CGM is an iterative method, whose goal is to minimize
the quadratic cost function. The algorithm of CGM as
show below
K=0;xo=0;
While(||rk||2 >tolerance) and (k < max_iter)
K++
If k==1
P1=ro
Else
T
y(n) = w (n)x(n)
Βk=(rk-1.rk-1)/( rk-2.rk-2)
and
Pk=rk-1+βkpk-1
x(n) = [x(n), x(n-1), x(n-2)... x(n-(N-1))]
wT(n) = [w0(n), w1(n), w2(n) ... wN-1(n)]H are the time
domain coefficients
where H denotes the Hermitian (complex conjugate)
transpose. The weight vector W is a complex vector. The
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Endif
Sk=Ap k
Αk=(rk-1.rk-1)/( p k.sk)
Rk=rk-1-αksk
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Endwhile
X=xk
The main disadvantage of normalized RLS is its
high sensitivity to measurement noise. The normalized
RLS algorithm lead to amplification of the measurement
noise in low order filters especially when the reference
signal power is low.
CGM algorithm places adaptively the maxima in the
direction of desired user and nulls at the AOA of the
interferer for various values of N. Simulation results prove
that higher Value of antenna element gives better results.
0.9
N=5
N=6
N=7
0.8
0.7
V. EXPERIMENTAL RESULTS
M ean s quare error
The performance of the algorithm is evaluated
through radiation pattern and convergence analysis which
are particularly attractive measurement of the wireless
communications. In this paper we examined the signal
model corresponding to uniform linear equally spaced
array. When modeling an antenna array, we make the
following assumptions
0.6
0.5
0.4
0.3
0.2
0.1
0
0
2
4
6
8
1) The spacing between array elements is λ/2.
2) There is no mutual coupling between elements.
10
12
Iteration no.
14
16
18
20
Fig.3. Error signal for n=5, 6, 7 elements using CGM
algorithm
3) All incidents fields can be decomposed into a discrete
number of plane waves. That is there are finite numbers of
signals.
1
4) The bandwidth of the signal incident on the arrays is
small compared with the carrier frequency.
|A F |
0.8
d=0.5
d=0.25
d=0.125
0.6
0.4
N=5
N=6
N=7
1
0.2
|AF|
0.8
0
-90
0.6
-60
-30
0
AOA (deg)
30
60
90
0.4
Fig.4. Rectangular Beampttern for CGM Algorithm with
AOA desired user at 600 and Interference at 0 0
0.2
0
-90
-60
-30
0
AOA (deg)
30
60
90
Fig.2. Rectangular Beampttern for CGM Algorithm with
AOA desired user at 60 0 and Interference at 00
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The array factor plots for a spacing between the elements
of quarter wave length and one eight wave length
respectively. From these simulations it is evident that the
optimum spacing beam between the elements is half wave
length.
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International Journal of Engineering Trends and Technology (IJETT) - Volume4Issue4- April 2013
N=5
N=6
N=7
1
0.8
0.8
0.6
0.6
|AF n |
|AF n|
1
0.4
0.4
0.2
0.2
0
-90
-60
-30
0
AOA (deg)
30
60
90
Fig.5. Rectangular Beam pattern for RLS Algorithm with
AOA desired user at 60o and Interference at -20 for N=5,
6, 7 element
RLS algorithm places adaptively the maxima in the
direction of desired user and nulls at the AOA of the
interferer for various values of N. Simulation results prove
that higher Value of antenna element gives better results.
0
-90
-30
0
AOA (deg)
30
60
90
The array factor plots for a spacing between the elements
of quarter wave length and one eight wave length
respectively. From these simulations it is evident that the
optimum spacing beam between the elements is half wave
length.
VI. CONCLUSION
N=5
N=6
N=7
0.9
0.8
0.7
Mean s quare error
-60
Fig.7. Rectangular Beampttern for RLS Algorithm with
AOA desired user at 40 0 and Interference at 250
1
0.6
0.5
0.4
0.3
0.2
0.1
0
d=0.5
d=0.25
d=0.125
0
5
10
15
20
25
30
Iteration no.
35
40
45
50
Fig.6. Error signal for n=5,6,7 elements using RLS
algorithm
In this paper, the non-blind adaptive beam
forming algorithms such as RLS and CGM have been
analyzed on a smart antenna system. It was noticed that
increasing the number of elements of the antenna array
ensures better performance. Conventionally, the LMS
adaptive algorithm has been used to update the combining
weights of adaptive antenna array. In an environment
yielding an array correlation matrix with large Eigen
values spread the algorithm converges with a slow speed.
This problem is solved with the RLS and CGM algorithm
by replacing the gradient step size with a gain matrix. It
was noticed that increasing the number of elements of the
antenna array ensures better performance. Also conclude
that the optimum spacing beam between the elements is
half wave length.
REFERENCES
[1] Salvatore Bellofiore, Jeffrey Foutz, Constantine A.
Balanis, and Andreas S. Spanias, “Smart-Antenna System
for Mobile Communication Networks Part 2:
Beamforming and Network Throughput” IEEE Antenna's
and Propagation Magazine, Vol. 44, NO. 4, August 2002.
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International Journal of Engineering Trends and Technology (IJETT) - Volume4Issue4- April 2013
[2] O. Axelsson, Iterative Solution Methods. New York:
Cambridge Univ.Press, 1994.
[3] G.K. Boray andM.D. Srinath, “Conjugate gradient
techniques for adaptive filtering,” IEEE Trans. Circuits
Syst. I, vol. 39, pp. 1–10, Jan. 1992.
[4] T. Bose and M. Q. Chen, “Conjugate gradient method
in adaptive bilinear filtering,” IEEE Trans. Signal
Processing, vol. 43, pp. 1503–1508, Jan. 1995.
[5] G. E. Bottomley and S. T. Alexander, “A novel
approach for stabilizing recursive least squares filters,”
IEEE Trans. Signal Processing, vol. 39, pp. 1770–1779,
Aug. 1991.
M.Siva subramanyam received the
Bachelor degree, B.Tech (ECE) and Master degree M.Tech
(DECS) from the University of JNTU. He is currently
working as Assistant Professor, Department of ECE in Sree
Vidyanikethan Engineering College, Tirupathi.
[6] C. Caraiscos and B. Liu, “A roundoff error analysis of
the LMS adaptive algorithm,” IEEE Trans. Acoust.,
Speech, Signal Processing, vol. ASSP-32, pp. 34–41, Feb.
1984.
[7] P. S. Chang and A. N. Willson, Jr., “Adaptive filtering
using modified conjugate gradient,” in Proc. 38th Midwest
Symp. Circuits Syst., Rio de Janeiro, Brazil, Aug. 1995,
pp. 243–246.
[8] , “A roundoff error analysis of the normalized LMS
algorithm,” in Proc. 29th Asilomar Conf. Signals, Syst.,
Comput., Pacific Grove, CA, Oct. 1995, pp. 1337–1341.
AUTHORS PROFILE
Naresh Birudala was born in
Andhra Pradesh, India in 1987. He received the bachelor
degree, B.Tech (ECE) from the University of JNTU,
Hyderabad. He is currently pursuing Master degree,
M.Tech (CMS) in sree Vidyanikethan Engineering
College, Tirupathi.
ISSN: 2231-5381
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