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1. INTRODUCTION
Current and future demands in mobile communication for various
high speed multimedia data services entail a robust, high data rate
transmission system. Increasing numbers of users amid limited
spectrum motivate research on technology to expand the capacity and
increase spectral efficiency. At the same time, some detrimental
effects in randomly varying mobile communication environment like
multipath fading, co-channel interference and Doppler effects need to
be addressed. Adaptive beam forming are part of recent methods that
known to offer the solution for the abovementioned problems.
Adaptive modulation is a technique that varies some
transmission parameters to take advantage of favorable channel
conditions. Under bad channel conditions, a robust signal transmission
mode will be applied to ensure reliable data exchange. While, in good
channel, spectrally efficient mode that offer higher throughput is
applied. This mechanism ensures the most efficient mode is always
used based on certain criteria and constraints. The varying parameters
can be the symbol transmission rate, transmitted power level,
constellation size, BER, code rate or scheme, any combination of theseparameters [1]. Compared to the fixed system, which was designed
specifically for the worst case channel conditions, this adaptive
modulation offers higher spectral efficiency, higher throughput and
remarkable capacity enhancement without sacrificing BER or wasting
power [2].
Research on applications of adaptive antenna arrays have been
an interesting subject over past 40 years [3] contributing to the
invention of adaptive beam forming method. By taking advantage of
the fact that users collocated in frequency domain are typically
separated in spatial domain, the beam former is used to direct the
antenna beams toward the desired user while canceling signal from
other users [4]. The beam former electronically steer a phased array
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by weighting the amplitude and phase of signal at each array element
in response to changes in the propagation environment. Capacity
improvement is achieved by effective co-channel interference
cancellation and multipath fading mitigation.
Theoretically proven impressive performance, coupled with
enabling signal processing technologies has attracted researchers to
focus on better utilization of the methods discussed. This paper will
outline a few approaches of adaptive modulation and adaptive beam
forming techniques and highlight some of the recent works that
employ these techniques. Two important improvements on the CMA
performance are the dithered signed-error CMA (DSE-CMA) and the
pre-whitened CMA (PW-CMA). The DSE-CMA is an approach to reduce
the computational complexity of the CMA while retaining its robustness
and the PW-CMA aims at improving the convergence rate of the CMA in
case of channels exhibiting large frequency response deviations. In this
paper we review the two approaches and propose a new scheme
combining the virtues of the two. The combined scheme is
computationally simpler than the PW-CMA and provides better
convergence than the DSE-CMA. It is particularly suited for thesituations where ill-convergence needs to be treated with minimum
additional complexity and without loss of robustness.
In this work provides a review of Constant modulus algorithm
and direct inversion matrix methods used in mobile communication
systems. A comprehensive review includes some performance
comparisons, advantages and drawbacks of each method.
1.1. Problem outline
In the past, different algorithms are implemented in smart
antennas. Those algorithms tracks the signal received from the user.
The radiation pattern is adjusted to place nulls in the Direction of
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Interferers and Maxima in the direction of the desired user.so, that
algorithms has low computation complexity and poor convergence.
1.2.Objective
In order to avoid those problems two methods has to be
developed.They are conjugate gradient method and music algorithm.
This algorithms has improve the computation complexity and better
convergence.
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2. SMART ANTENNA FUNDAMENTALS
A smart antenna usually involves spatial processing and adaptive
filtering techniques. The field of application is very large, ranging from
signal to noise improvement to the user capacity enlargement of the
mobile network. A typical application will involve an adaptive algorithm
to create a beam to track a user or to eliminate noise sources and
therefore the smart antenna is also referred to as adaptive array or
adaptive beam former. This chapter discusses two algorithms, the
Least Mean Square algorithm and the Constant Modulus algorithm.
2.1. Smart antenna basics
The smart antenna is basically a set of receiving antennas in a certain
topology. The received signals are multiplied with a factor, adjusting
phase and amplitude. Summing up the weighted signals, results in the
Output signal. The concept of a transmitting smart antenna is rather
the same, by splitting up the signal between multiple antennas and
then multiplying these signals with a factor, which adjusts the phase
and amplitude. Figure 1 represents the concept of the smart antenna.
The signals and weight factors are complex.
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The following mathematical foundations on the smart antenna concept
can be found in [5]. If the wave front arrives at the array antenna as
shown in Figure 2, the wave front will be earlier on antenna element
k+1 than element k. The difference in length between the paths is
dsin. If the arriving signal is a harmonic signal or frequency, then the
signal arriving at antenna k+1 is leading in phase compared with
antenna k. The signal that arrives at antenna element zero is
considered to have a phase lead of zero. The signal that arrives at
antenna k, leads in phase with kdsin, where =2/ and is thewavelength.
The weight vector is defined by:
W=[w0,w1,w2,--wk-1 ]T ----------- (1)
Now the array factor is defined by:
[ W1 W2 W3 WK ]
X1
X2
X3
XK
Y
Figure 1, smart antenna concept for a receiving antenna
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F=wTv ----------------------- (2)
Where
V=[1 ej dsin e2j dsin---ej(k-1) dsin]------(3)
Here
K=no. of antennas
Wk=weight vector of antenna k
V=Array propagation vector
2.2. Adaptive beam forming
Popularly referred as smart antenna, adaptive beamforming is
one of antenna arrays application in mobile communication. With the
ability to adaptively directing the beam in specific directions it is
known to be an effective technique in canceling co-channel
interference. Some of the invaluable references that thoroughly
outlined all the methods and algorithms include [4] [26] [27] [28]
Adaptive beam forming can be done in many ways. Many
algorithms exist for many applications varying in complexity. Most of
the algorithms are concerned with the maximization of the signal to
noise ratio. A generic adaptive beam former is shown in Figure 3. The
weight vector w is calculated using the statistics of signal x (t) arrivingfrom the antenna array. An adaptive processor will minimize the error
e between a desired signal d(t) and the array output y(t).
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One of the simplest algorithms for adaptive processing is based
on the Least Mean Square (LMS) error. Although the complexity of the
algorithm is very low, its results are satisfying in many cases. The
algorithm is very stable and it needs few computations, which is
important for system implementation. The computational power of
many systems is limited and should be managed wisely.
The algorithm is based on knowledge of the arriving signal. The
knowledge of the received signal eliminates the need for beam
forming, but the reference can also be a vector that is partly known, orcorrelated with the received signal. For example, the training sequence
in the GSM standard, intended for channel equalization, could be used
for beam forming. The rest of the signal is unknown, and beam forming
using LMS can only be performed on the known training sequence.
When the adaptive algorithm is not using this knowledge, but statistic
[ W1 W2 W3 WK ]
X1
X2
X3
XK
Y
Figure.2. Smart antenna concept for a beam forming
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information of the signal, it is called blind beam forming. There are
several algorithms for blind beam forming. For example the Constant
Modulus algorithm (CMA) uses the knowledge that the modulus of the
signal is constant. There are many modulation schemes where the
modulus is kept constant. CMA is one of the simplest blind beam
forming algorithms.
J. Litva and K. Y. Lo in Chapter 3 of [4] explained in detail the
basic concept of adaptive beamforming starting from the used of two
elements array to suppress interference. The fundamental method in
adaptive beamforming is to choose the weights of array elements in
order to optimize the beamformer response to fulfill certain criterion.
The criterion includes Minimum Mean-Square Error, Maximum Signal-
to-Interference Ratio and Minimum Variance was discussed in the
book. The choice of criteria is not critically important since they are
closely related to each other. The more important part is the adaptive
algorithms, which will determine the speed of convergence and
hardware complexity required. The algorithms include Least Mean
Squares algorithm (LMS), Direct Sample Covarince Matrix Inversion
(SMI), Recursive Least Squares Algorithms (RLS) and Neural Networks.The notion of partially adaptivity then explained which is the
alternative technique when the number of array elements becomes
very large until it is difficult to implement full adaptivity. Another
important component in adaptive beamforming is the reference
signals, also known as the prior knowledge of the signal of interest,
which is needed to decrease the complexity, improve accuracy and
achieve faster convergence. Two known types are temporal reference
and spatial reference.
Some benefits of using adaptive antennas in mobile
communication were listed. The performance improvement in terms of
BER and co-channel interference reduction was shown using a few
simulation results from some established literatures. Since spatial
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channel information available on the uplink and most of the research
done on it, the discussion was focused on this type of application. An
optimum criterion, which was explained in chapter 3 is directly
applicable here. Some adaptive algorithms that suitable in mobile
communication with their implementation issues then were briefly
discussed. This includes LMS algorithm, SMI technique, RLS algorithm
touching on the pro and cons of each of them. Other algorithms that
proposed to overcome shortcomings or improve the performance of
the three basic algorithms such as conjugate gradient method,
eigenanlysis algorithm, rotational invariance method, linear least
square error (LSSE) algorithm, and Hopfield neural network with
respective references are listed.
The estimation technique of spatial reference signals referred as
Angle of Arrival (AOA) of the desired signal was categorized into two
groups. The first group named as wavenumber estimation is based on
decomposition of a covariance matrix whose terms consist of
estimates of the correlation between the signals at the elements of an
array antenna. The techniques include Multiple signal classification
(MUSIC), modified forward-backward linear prediction (FBLP), PrincipalEigenvector Gram-Schmidt (PEGS), Estimation of Signal Parameters by
Rotational Invariance Techniques (ESPRIT). The second group is
parametric estimation cover a variety of maximum likelihood
estimation (MLE) techniques, which require a high computational
complexity. It is noted that the main drawback of AOA approaches is
requirement for array calibration and extra processing load. The
temporal reference may be a pilot signal that is correlated with the
wanted signal, or known PN code in CDMA. The alternative techniques
in case of unavailability of explicit reference signal called blind
adaptive beamforming were briefly described. They are Constant
Modulus Algorithm, Decision Directed Algorithm and Cyclostationary
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Algorithms. Finally, some implementation issues for downlink
application were discussed.
An earlier literature by B. D. V. Veen and K. M. Buckley [16]
introduced beamforming as a versatile form of spatial filtering. Started
with the basic concept, associated the explanation with FIR filtering.
Beamformer was classified into data independent and statistically
optimum beamformer. Independent of the received data, the first class
of beamformer chose a fixed antenna arrays weights. The later class
use statistical information of received data to select the weights.
Adaptive beamforming comes into picture for the fact that the data
statistics are often unknown and varying over time. Two basic adaptive
approaches, block adaptation and continuous adaptation were
discussed. In block adaptation, the statistics are estimated from
temporal block of array data while continuous adaptation the weights
are adjusted as the data is sampled. Two basics adaptive algorithms,
LMS and RLS also introduced. Partial adaptivity was highlighted.
Lal. C. Godara [17][18] contributed a thorough study on antenna
arrays application in mobile communication. Part I gave a briefoverview of mobile communications, antenna array terminology, the
usage of antenna arrays in mobile communication systems, the
advantages and improvements that it brings, design issues, and the
feasibility in implementation. Part II presented a detail depiction of
various beam-forming schemes, adaptive algorithms, DOA estimation
methods, and some issues on error sensitivities. Relevant details and
references were provided for further research on each topic.
The important comparison of different approach in beamforming
may be compared based on type of adaptive algorithm it used. Many
researches were done on each algorithm and some comparisons were
highlighted in [4] [18]. Simplicity of Least Mean Square (LMS)
algorithm makes it widely been used for tap coefficient adaptations of
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an adaptive processor in antenna array. However, this continuous
adaptation approach algorithm causes signal acquisition and tracking
problems due to its slow convergence in multipath fading channel. This
is not suitable for mobile communication and some other measures
need to be taken if this algorithm is to be used such as power control
or normalized LMS algorithm. Converging faster than LMS algorithm,
SMI has attracted to be applied in mobile communication. However,
implementation difficulties need to be considered since its complexity
requires advance hardware capability and the use of finite precision
arithmetic may cause numerical instability. RLS can be seen as the
solution for the slow convergence of LMS and high complexity of SMI.
This is provided that SNR is high and setting of a fading rate dependent
forgotten factor is correct [4]. Computer simulation results for mobile
communication application shown that RLS outperform LMS and SMI in
flat fading channels. Another algorithm, conjugate gradient method
was studied to mitigate multipath fading effect in mobile
communication and shown a better BER performance than RLS [28].
2.3. kalman based on LMS algorithm
The LMS algorithm can be considered to be the most commonadaptive algorithm for continues adaptation. It uses the steepest-
descent method and recursively computes and updates the weight
vector. Due to the steepest-descend the updated vector will propagate
to the vector which causes the least mean square error (MSE) between
the beamformer output and the reference signal. The following
derivation for the LMS algorithm is found in [1]. The MSE is defined by:
e2(t) =[d*(t)-wHx(t)]2-------(4)
where
d*(t) = complex conjugate of the desired signal.
X(t)=received signal from the antenna elements.
wH=output of the beam form antenna.
(.)H = Hermetian operator.
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The LMS algorithm converges to this optimum Wiener solution. The
basic iteration is based on the following simple recursive relation:
W(n+1)=w(n)+1/2(-(E(e2)))-------(5)
Or
W(n+1)=w(n)+x(n)e(n)------(6)
One of the issues on the use of the instantaneous error is concerned
with the gradient vector, which is not the true error gradient. The
gradient is stochastic and therefore the estimated vector will never be
the optimum solution. The steady state solution is noisy; it will
fluctuate around the optimum solution. By decreasing the precision
will improve but it will decrease the adaptation rate. An adaptive
could solve this issue by starting with a large and decrease the factor
when the vector converges.
An adaptive array is simulated in MATLAB by using the LMS algorithm.
When an array of 4 antennas is used, there is a maximum of 3 nulls
that can eliminate the interferer. Figure shows the convergence of the
array for 2 interferers as shown in results. The minimum error is a
result of the extra system noise that is added to all antennas. The
interference signals are Gaussian white noise, zero mean with a sigmaof 1. The extra system noise to all antennas is white noise with zero
mean and a sigma of 0.1. The received signals are MSK signals with an
oversampling of 4 and have amplitude of 1 in the simulations. The
true array output y(t) is converging to the desired signal d(t). After 40
samples the signal is at its minimum due to the system noise. The LMS
cannot filter the system noise, as it is not correlated for all four
antennas.The interferers are cancelled by placing nulls in the direction
of the interferers. The received signal arrives at an angle of 25 degrees
and the array response is 0 dB. The LMS algorithm clearly works
sufficient as the strong interferers are reduced.
2.4. Constant Modulus Algorithm
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The CM algorithm is used for blind equalization of signals that
have a constant modulus. The MSK signal, for example, is a signal that
has the property of a constant modulus. The algorithm that updates
the weight Coefficients are exactly the same as for the LMS algorithm.
2.5. Direct matrix inversion
The Direct matrix inversion is a computationally intensive
process. Various algorithms can be applied for direction of arrival
estimation and tracking problems, such as blind algorithms that use
the temporal constant modulus structure of the signal (without training
signal) or algorithms that use the spatial properties of received signals
or training signal method [Godara97]. The main Disadvantage of the
training signal method is the slower convergence rate. It can be
applied to 3G Communication systems because a pilot signal is
presented in the structure of the uplink CDMA frame of UMTS/ITM2000
physical channel. The reason for a dedicated pilot instead of a common
pilot is to support the use of adaptive antenna arrays.
2.6. Adaptive modulation
With the main objective to maximize the spectral efficiency,many approaches of adaptive modulation have been proposed in
literature. An early work includes in Chapter 13 of [15], where W.T.
Webb and L. Hanzo introduce variable rate QAM. The transmitter varies
the signal constellation size from 1bit/symbol corresponding to BPSK to
6bits/symbol star 64-QAM. In a good quality channel, the constellation
size is increased, and as the channel quality become worst, i.e. as the
receiver enters a deep fade, the constellation size is decreased to a
value, which provides an acceptable BER. Two choices of
implementation can be applied where to keep constant of one
parameter and varying the other parameter. Specifying a required BER
value leads to varying data throughput and vise versa. The chapter
also highlights two different types of switching criteria to control the
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modulation modes, Received Signal Strength Indicator (RSSI) system
and error detector switching system. RSSI system use SNR values
corresponding to the BER of interest as the switching thresholds while
the later system use the channel coder to monitor the channel quality.
Simulation results showed the performance improvement over fixed
modulation and comparisons of the two switching systems. It is
observed that RSSI typically offering a slightly higher number of
bits/sym at low SNRs for some BERs. This is due to the RSSI switching
systems ability to select a lower number of levels before any errors
occurred. RSSI is also more attractive in term of implementation
complexity because no additional BCH codec is needed.
Another literature [2] indicated the above switching system as
the channel state information (CSI) which specified the channel quality.
SNR based CSI corresponding to RSSI system was compared with error-
based CSI corresponding to error detector switching system. SNR
based CSI adapts with a faster rate, but relies on the computation of
adaptation or switching thresholds that may be inaccurate. Accuracy of
the threshold mechanism increases by taking into account higher order
statistics of SNR than just the mean.Studies on varying combination of parameters also attract a
great interest. In [16], A. J. Goldsmith and S.G. Chua proposed a
variable-rate and variable-power MQAM modulation scheme for fading
channels. The transmission rate and power is both optimized to
maximize spectral efficiency, while satisfying average power and BER
constraints. Spectral efficiency of the proposed technique was derived
and compared with Shannons capacity limit. A comparison in terms of
spectral efficiency between the proposed method and two fixed-rate
variable-power schemes also performed. One of the compared scheme,
using channel inversion method adapts the transmit power to maintain
a constant received SNR. The main drawback of this technique is it
suffers a large power penalty since it use most of it power to
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compensate for deep fades. The other method, called truncated
channel inversion maintains a constant received SNR unless the
channel fading falls below a given threshold point. It is acknowledged
this technique can achieve almost the same spectral efficiency as the
proposed method. However, the outage probability can be quite high.
A question raised on power adaptation of whether or not power
adaptation actually provides substantial performance gains over
constant power system. Goldsmith showed theoretically in [17] that
Shannon capacity can be achieved by varying both rate and power.
However, as stressed in [1], Shannon capacity assumes that BER is
arbitrarily small, coding schemes are random and of unbounded length
and complexity, and there is no delay constraint. Therefore, the
capacity results do not necessarily shown insight of practical system.
Moreover, [17] also highlights that varying both power and rate
achieve negligibly increase in channel capacity compared to varying
rate only. The results shown in [1] prove that constraining power or
rate to be constant causes only little lost in spectral efficiency. They
also concluded that spectral efficiency of adaptive modulation isrelatively insensitive to which degrees of freedom are adapted.
Realizing that previous work only deals with uncoded
modulation, Goldsmith and S.G. Chua [18] propose adaptive coded
modulation for fading channels. They applied coset codes since the
code design are separable from modulation design, which is well suited
to be combined with adaptive modulation system. Special cases of
coset codes, trellis and lattice codes were first applied to general class
adaptive modulation. Combination of trellis code with spectrally
efficient adaptive M-ary quadrature amplitude modulation (M-QAM)
introduced in [16] produce trellis-coded adaptive MQAM. Analytical and
simulation results shown that the new simple 4-state trellis-coded
adaptive MQAM achieves 3-dB effective coding gain relative to
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uncoded adaptive M-QAM and 3.6 dB for 8-state trellis. Compared with
traditional trellis codes and fixed-rate modulation, the new scheme
shown more than 20 dB power savings.
K. J. Hole, Henrik Holm [19] introduced a general variable-rate
constant-power Trellis Coded Modulation (TCM) scheme for frequency-
flat, slowly varying Nakagami Fading (NMF) channel. Their main
contribution is the development of a general technique to determine
the average spectral efficiency of the coding scheme for any set of 2L-
dimensional (2L-D) trellis codes. The paper concentrates on code sets
that can be generated by the same encoder and decoded by the same
decoder to avoid hardware complexity. It is assumed that the perfect
channel state estimation (CSI) is available at the decoder and reliable
feedback channel available between the encoder and decoder.
Considering channel estimation accuracy and feedback delay
problem, D. L. Goeckel [20] propose an adaptive trellis-coded
modulation schemes, which is proved to gain higher bandwidth
efficiency over their non-adaptive counterparts on time-varying
channels. The scheme was designed using only a single outdated
fading estimate when neither the Doppler frequency nor exact shapeof autocorrelation function of the channel fading process is known. This
issue concerning channel estimation errors and outdated feedback
have become one of critical issue in ensure the effectiveness of
adaptive modulation. Some works on this includes [21] [22]
Another way of categorizing adaptive modulation is based on the
adaptation algorithms used, including the constraints and the
objectives. Some typical constraints are upper bound BER, fixed
throughput and average transmitted power [23]. For some application
that required low delay such as speech and real-time video
communication, fixed throughput adaptive transmission is favored.
However, for data transmission systems, which can tolerate for higher
delay the variable throughput, maximum BER is usually utilized [24].
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Aiming to maximize the throughput, some works done on deriving the
optimum switching mode thresholds subject to the average BER
constraints [22] [25].
3. BENEFITS OF SMART ANTENNA
Multipath propagation, defined as the creation of multiple signal
paths between the transmitter and the receiver due to the reflection of
the transmitted signal by physical obstacles is one of the major
problems of mobile communications [6]. It is well known that the delay
spread and resulting intersymbol interference (ISI) due to multiple
signal paths arriving at the receiver at different times have a critical
impact on communication link quality. On the other hand, co-channel
interference is a major limiting factor on the capacity of wireless
systems, resulting from the reuse of the available network resources
(e.g., frequency, time) by a number of users. Smart antenna systems
can improve link quality by combating the effects of multipath
propagation or constructively exploiting the different paths, and
increase capacity by mitigating interference and allowing transmission
of different data streams from different antennas. More specifically,
the benefits of smart antennas can
be summarized as follows [6]:
3.1. Increased range/coverage:
The array or beam forming gain is the average increase in signalpower at the receiver due to a coherent combination of the signals
received at all antenna elements. It is proportional to the number of
receive antennas and also allows for lower battery life.Lower power
requirements and/or cost reduction: Optimizing transmission toward
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the wanted user (transmit beam forming gain) achieves lower power
consumption and amplifier costs.
3.2. Improved link quality/reliability:
Diversity gain is obtained by receiving independent replicas of
the signal through independently fading signal components. Based on
the fact that it is highly probable that at least one or more of these
signal components will not be in a deep fade, the availability of
multiple independent dimensions reduces the effective fluctuations of
the signal. Forms of diversity include temporal, frequency, code, and
spatial diversity obtained when sampling the spatial domain with smart
antennas. The maximum spatial diversity order of a non-frequency-
selective fading MIMO channel is equal to the product of the number of
receives and transmits antennas. Transmit diversity with multiple
transmit antennas can be exploited via special modulation and coding
schemes [6], whereas receive diversity relies on the combination of
independently fading signal dimensions.
3.3. Increased spectral efficiency:
Precise control of the transmitted and received power and
exploitation of the knowledge of training sequence and/or otherproperties of the received signal (e.g., constant envelope, finite
alphabet, cyclostationarity) allows for interference reduction/mitigation
and increased numbers of users sharing the same available resources
(e.g., time, frequency, codes) and/or reuse of these resources by users
served by the same base station/access point. The latter introduces a
new multiple access scheme that exploits the space domain, space-
division multiple access (SDMA). Moreover, increased data rates and
therefore increased spectral efficiency can be achieved by exploiting
the spatial multiplexing gain, that is, the possibility to simultaneously
transmit multiple data streams, exploiting the multiple independent
dimensions, the so called spatial signatures or MIMO channel
eigenmodes. It was shown [7] that in uncorrelated Raleigh fading the
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MIMO channel capacity limit grows linearly with min (M, N), where M
and N denote the number of transmit and receive antennas,
respectively.
Traditionally, smart antenna systems have been designed
focusing on maximization of one of the above-mentioned gains (beam
forming diversity, and multiplexing gains). Nevertheless the trade-offs
between these gains have been recently studied [8], and smart
antenna approaches have been proposed that combine the resulting
benefits [9].
4. SMART ANTENNA MODELS
In this work, we are interested more in adaptive array
antennas that can independently steer their main beam and nulls
to arbitrary directions. This process is generally called beam
forming. Their main difference from simple directional antennas
(and hence their smartness) is the following: Instead of just
directing the main beam towards the direction specified (e.g. by the
application), smart antennas can automatically adapt their
radiation pattern, in order to track the intended
receiver/transmitter and minimize transmission/reception gain (i.e.
create nulls) towards unintended receivers/transmitters .A large
number of alternative beamforming designs (e.g. digital,
microwave, aerial beamforming) and algorithms (e.g. Least Mean
Square, Constant Modulus Algorithm, etc.) have been proposed in
literature, a detailed tutorial of which can be found in [10]. Until
recently, adaptive array antennas had only been considered for
the use on base station in cellular systems, due to their large size, highcost, considerable power consumption, and complexity of design.
However, recently there have been proposed simple, analog, smart
antenna designs [11] [12] that are low cost and energy-efficient
enough to be used on wireless terminals. Theyre based on the
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concept of aerial beamforming and prototype antennas have been built
and tested [13].
5. CONJUGATE GRADIENT METHOD
The Conjugate Gradient method is an effective method for
symmetric positive definite systems. The method proceeds by
generating vector sequences of iterates, residuals corresponding to the
iterates, and search directions used in updating the iterates and
residuals.
The unpreconditioned conjugate gradient method constructs the thk
iterate kx as an element of },,{0100
rArspanxk+ so that )()(
xxAxx kTk
is minimized, where
x is the exact solution of Ax=b. This minimum is
guaranteed to exist in general only if A is symmetric positive definite.
The conjugate gradient iterates converge to the solution of Ax=b in no
more than n steps, where n is the size of the matrix.
In every iteration of the method, two inner products are performed in
order to compute update scalars that are defined to make the
sequences satisfy certain orthogonal conditions. On a symmetric
positive definite linear system these conditions imply that the distance
to the true solution is minimized in some norm.
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The iterates kx are updated in each iteration by a multiple k of the
search direction vector kp :
kkkk pxx += 1
Correspondingly the residuals kk Axbr = are updated as
kkk Aprr = 1
The choice kkkk ApprrTT
/11 = minimizes kk rArT 1
The search directions are updated using the residuals 11 += kkkk prp
where the choice 11/ = kkkkk rrrrTT
ensures that kr and 1kr are
orthogonal.
The following is parallel code fragment which performs the conjugate
gradient algorithm for solving Ax=b.
r_local = b_local
rho = Allreduce (r_local* r_local)
for k=1:itermax
if k=1
p_local=r_local
else
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beta=rho/oldrho
p_local = r_local + beta* p_local
end
p=Gather(p_local)
v_local=A_local*p
alpha = rho/ Allreduce(p_local*v_local)
x_local = x_local + alpha*p_local
r_local = r_local alpha*v_local
oldrho = rho
rho = Allreduce (r_local*r_local)
end
The algorithm is the same as that in serial computer. All matrices and
vectors, however, have distributed: various dot products are performed
by collecting partial results (using Gather) and Sum them up (using
Allreduce(SUM)).
Example 1
Assume A be a 57600x57600 sparse matrix,
=
TI
I
I
IT
A
,
=
51
1
1
15
T
CG method convergent in 8 iterations.
Result:
No of processor
used 1 2 4
Time used 0.95 1.13 1.12
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(1) In Network of work-stations, since the time used in both algorithm
are almost the same. The difference between them is when one
processor is used, the time recorded is just used in calculation. When
four processors are used, the time mainly used in message passing.
(2) In cluster, if we dont including the time used in showing x, the time
used in both algorithms are almost the same.
No of processor 1 2 4
Total time - Time showing
x1.17-0.65
= 0.52
4.52 -
3.86 =
0.66
3.48 -
2.68 =
0.8
The time used in message passing becomes longer if more processors
are used. Ex. When 2 and 4 processors are used, it used 0.17s and
0.37s to transfer message respectively.
Since this matrix is too fast to convergent, there is just 8 iteration
steps and the time used in calculation is not obvious in this program.
Example 2
In this example, I will let smaller number in diagonal, so it need more
iterates to convergent.
Assume A be a 57600x57600 sparse matrix,
=
TI
I
I
IT
A
,
=
41
1
1
14
T
Since A is still a symmetric and positive definite matrix, we can solve
the equations by conjugate gradient method.
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The solution convergent in 296 iterations.
Result:
No of processor
used 1 2 4 8
Time used 6.4 6.73 5.8 5.41
(1) In network, from the profile report, the time used in calculation is
shorted. But when 2 processors are used, we need to add up the time
used in calculation and message passing. We find that the total time
used is almost the same as that when one processor is used.When 4 and 8 processors are used, the speedup of parallel algorithm is
1034.18.5
4.6
4==S , 1830.1
41.5
4.68 ==S
The efficiency is
2759.04
1034.1
4==E , 1479.0
8
1830.18 ==E
(2) In cluster, comparing with example 7, we find that if it needs more
iteration to convergent, the time used in message passing increased.
No of processor 1 2 4 8
Time used in calculation 6.24 5.52 5.38 4.26
Time used in message
passing ~ 6.2 6.08 6.62
Although the time used in calculation decrease continually, the range
is too close. Ex. When 2 processors are used, the calculations time is
5.52s, it just only increased about 11%. In addition, the time used in
transferring message is around 6s. So there is no speedup.
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Advantage:
The cgmprovides good performance in a discontinuous traffic
when the number of interferers and their positions remain
constant during the duration of the block acquisition.
The main advantage of cgm is the good conversation.
The block diagram of direct matrix inversion algorithm as shown below
The above diagram is the smart antenna concept for a receiving
antenna. In this diagram, contains k anntenas and their correspondingweighted vectors are w1, w2, w3, ---, wk. y is the summation of the all
reference signal with vectors.
[ W1 W2 W3 WK ]
X1
X2
X3
XK
Y
Figure 3, smart antenna CGM concept
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The covariance matrix of the input vector X for a finite sample size is
defined as the maximum likelihood estimation of matrix R and can be
calculate as
R(N)=1/N*X.XH------------------(17)
Here
K=no.of antennas
X(t)=received signal from the antenna elements.
wH=output of the beam form antenna.
(.)H = Hermetian operator.
The optimum weight vector that correspond to the estimated matrix
Rk for any i-th channel is given by
W=R-1V------------------------------ (18)
Where
V=[1 ej dsin e2j dsin---ej(k-1) dsin]------(19)
Here
K=no. of antennas
W=weight vector
V=Array propagation vector.
The Direct Matrix Inversion Algorithm provides good performance in adiscontinuous traffic when the number of interferers and their positions
remain constant during the duration of the block acquisition. The DMI
algorithm employs direct inversion of the co-variance matrix R and
therefore it has faster convergence rate. The equation for co-variance
matrix R is given by
R=E[x(t).xH(t)] ----------------------------------(20)
The equation for correlation matrix r is given by equation
r=E[d(t).x(t)]----------------------------------(21)
The error e due to estimation can be computed by the equation
e=Rw-r-------------------------------------- (22)
or
e=x(n)-y(n)
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where
x(n)=reference signal
y(n)=received signal
Then to calculate the array factor by using following equation
Arrayfactor=w(n)*ejksin---------------------------(23)
Weight adaptation in the DMI algorithm can be achieved by using block
adaptation technique where the adaptation is carried over disjoint
intervals of time is the most common type. This block adaptation
technique is suitable for mobile communications where the signal
environment is highly time varying. The overlapping block adaptation
technique is computational intensive as adaptation intervals are not
disjoint but overlapping. This technique gives better performance but
the numbers of inversions required are more when compared to block
adaptation method. Another block adaptation technique is the block
adaptation technique with memory. This method utilizes the matrix
estimates computed in the previous blocks. This approach provides
faster convergence for spatial channels that are highly time correlated.
This technique works better when the signal environment is stationary.When an array of 4 antennas is used, there is a maximum of 3 nulls
that can eliminate the interferer. Figure shows the convergence of the
array for 2 interferers as shown in results. The minimum error is a
result of the extra system noise that is added to all antennas. The
interference signals are Gaussian white noise, zero mean with a sigma
of 1. The extra system noise to all antennas is white noise with zero
mean and a sigma of 0.1. The received signals are MSK signals with an
over sampling of 4 and have amplitude of 1 in the simulations. The
true array output y(t) is converging to the desired signal d(t). After 40
samples the signal is at its minimum due to the system noise. The LMS
cannot filter the system noise, as it is not correlated for all four
antennas.The interferers are cancelled by placing nulls in the direction
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of the interferers. The received signal arrives at an angle of 25 degrees
and the array response is 0 dB. The LMS algorithm clearly works
sufficient as the strong interferers are reduced.
5.1. FLOWCHART
The flowchart of direct matrix inversion as shown below. By using
flowchart to implement the mat lab code easily.
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star
t
Assign k,
W=1/k*[1 e-jsin e-j2sin -----e-j(k-1) sin]
Noise=sin+j cos
nan =signal_n1 * e-j(k) sin
x(n)=noise+n1+n
2+x
Yan =w*x(an)
Error= x(an)- Yan
W=w+.error.x(an)
Arrayfactor=w* e-j(k) sin
En
d
Fig 4. Direct matrix inversion algorithm flowchart
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5.2. Algorithm of conjugate gradient algorithm
Define the of k, %k=no. of antennas,=angle
W=1/k*[1 e-jsin
e-j2sin
-----e-j(k-1) sin
] %weighted vector
Noise=sin+j cos % system noise for every antenna
X=noise
For an=1:k
nan =noise * e-j(k) sin %received signal from noise source an(i.e antenna an)
end
x(n)=noise+n1+n2+x %total signal(i.e all antenna signals)
For an=1:k
Yan =w*x(an) % received signal
Error= x(an)- Yan
W=w+.error.x(an)
End
Arrayfactor=w* e-j(k) sin
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6. RLS ALGORITHM
As was mentioned in the previous section, the SMI technique has
several drawbacks. Even though the SMI method is faster than the
LMS algorithm, the computational burden and potentialsingularities can cause problems. However, we can recursively
calculate the required correlation matrix and the required
correlation vector. Recall that in Eqs. (8.60) and (8.61) the estimate
of the correlation matrix and vector was taken as the sum of the
terms divided by the block length K. When we calculate the weights
in Eq. (8.66), the division by K is cancelled by the product xx
(k)r(k). Thus, we can rewrite the correlation matrix and thecorrelation vector omitting K as
where k is the block length and last time sample k and Rxx (k),r (k)
is
the correlation estimates ending at time sample k. Both summations
(Eqs. (8.67) and (8.68)) use rectangular windows, thus they equallyconsider all previous time samples. Since the signal sources can
change or slowly move with time, we might want to deemphasize the
earliest data samples and emphasize the most recent ones. This can be
accomplished by modifying Eqs. (8.67) and (8.68) such that we forget
the earliest time samples. This is called a weighted estimate.
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where is the forgetting factor.
The forgetting factor is also sometimes referred to as the exponential
weighting factor [37]. is a positive constant such that 0
1. When = 1, we restore the ordinary least squares algorithm.
= 1 also indicates infinite memory. Let us break up the summation in
Eqs. (8.69) and (8.70) into two terms: the summation for values up
to i = k1 and last term for i = k.
Example 6.1 For an M = 4-element array, d = /2, one signal
arrives at 45, and S(k) = cos(2(k 1)/( K 1)). Calculate the
array correlation for a block of length K = 200 using the standard
SMI algorithm and the recursion algorithm with = 1. Plot the
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trace of the SMI correlation matrix for K data
points and the trace of the recursion correlation matrix vs. block
length k where 1 < k < K. Solution Using MATLAB, we can
construct the array steering vector for the angle of arrival of 45.
After multiplying the steering vector by the signal S(k) we can then
find the correlation matrix to start the recursion relationship in Eq.
(8.71). After K iterations, we can superimpose the traces of both
correlation matrices. It can be seen that the recursion formula
oscillates for different block lengths and that it matches the SMI
solution when k = K. The recursion formula always gives a
correlation matrix estimate for any block length k but only
matches SMI when the forgetting factor is 1. The advantage of the
recursion approach is that one need not calculate the correlation
for an entire block of length K. Rather, each update only requires
one a block of length 1 and the previous correlation matrix. Not
only can we recursively calculate the most recent correlation
estimates, we can also use Eq. (8.71) to derive a recursion
relationship for the inverse of the correlation matrix. The next
steps follow the derivation in [37]. We can invoke the ShermanMorrison-Woodbury (SMW) theorem [38] to find the inverse of Eq.
(8.71). Repeating the SMW theorem Example 6.2:Use the RLS
method to solve for the array weights and plot the resulting
pattern. Let the array be an M = 8-element array with
spacingd= .5withareceivedsignalarrivingattheangle0 =
30,aninterferer at 1 = 60. Use MATLAB to write an RLS
routine to solve for the desired weights. Use Eqs. (8.71), (8.78),
and (8.81). Assume that the desired received signal vector is
defined by xs(k) =a0s(k) where s(k) =
cos(2*pi*t(k)/T);withT=1ms.LettherebeK=50timesamplessuc
hthatt=(0:K1)*T/().Assume that the interfering signal vector
is defined byxi(k) = a1i(k) where i(k) = sin(pi*t(k)/T);. Let the
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desired signal d(k) = s(k). In order to keep the correlation matrix
inverse from becoming singular, add noise to the system with
variance n =.01.Beginwiththeassumptionthatallarrayweights are
zero such that w(1) = [0000 0 000]T . Set the forgetting factor
Advantages The advantage of the RLS algorithm over SMI is that it is no
longer necessary to invert a large correlation matrix. The
recursive equations allow for easy updates of the inverse of
the correlation matrix.
The RLS algorithm also converges much more quickly than the
LMS algorithm.
Interference is low Less noise
7. LMS ALGORITHM
The LMS algorithm can be considered to be the most common
adaptive algorithm for continues adaptation. It uses the steepest-
descent method and recursively computes and updates the weight
vector. Due to the steepest-descend the updated vector will propagate
to the vector which causes the least mean square error (MSE) between
the beamformer output and the reference signal. The following
derivation for the LMS algorithm is found in [1]. The MSE is defined by:
e2(t) =[d*(t)-wHx(t)]2-------(4)
where
d*(t) = complex conjugate of the desired signal.
X(t)=received signal from the antenna elements.
wH
=output of the beam form antenna.(.)H = Hermetian operator.
The LMS algorithm converges to this optimum Wiener solution. The
basic iteration is based on the following simple recursive relation:
W(n+1)=w(n)+1/2(-(E(e2)))-------(5)
Or
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W(n+1)=w(n)+x(n)e(n)------(6)
One of the issues on the use of the instantaneous error is concerned
with the gradient vector, which is not the true error gradient. The
gradient is stochastic and therefore the estimated vector will never be
the optimum solution. The steady state solution is noisy; it will
fluctuate around the optimum solution. By decreasing the precision
will improve but it will decrease the adaptation rate. An adaptive
could solve this issue by starting with a large and decrease the factor
when the vector converges.
An adaptive array is simulated in MATLAB by using the LMS algorithm.
When an array of 4 antennas is used, there is a maximum of 3 nulls
that can eliminate the interferer. Figure shows the convergence of the
array for 2 interferers as shown in results. The minimum error is a
result of the extra system noise that is added to all antennas. The
interference signals are Gaussian white noise, zero mean with a sigma
of 1. The extra system noise to all antennas is white noise with zero
mean and a sigma of 0.1. The received signals are MSK signals with an
oversampling of 4 and have amplitude of 1 in the simulations. The
true array output y(t) is converging to the desired signal d(t). After 40samples the signal is at its minimum due to the system noise. The LMS
cannot filter the system noise, as it is not correlated for all four
antennas.The interferers are cancelled by placing nulls in the direction
of the interferers. The received signal arrives at an angle of 25 degrees
and the array response is 0 dB. The LMS algorithm clearly works
sufficient as the strong interferers are reduced.
Disadvantages
The disadvantage of the LMS algorithm is difficulty for easy
updates of the inverse of the correlation matrix.
The LMS algorithm converges much slower.
Interference is high
high noise
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7. RESULTS
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0- 2
0
2
p
h
a
s
e
(ra
d
)
d e s i re d s i g n a l 1 0 d e g r e e s i n t e r fe r e r s - 1 0 a n d - 4 0 d e g r e e s
p h a s e ( d )p h a s e ( y )
Fig.6 .Phase response of LMS algorithm when N=5
0 5 0 1 0 0 1 5 0 2 0 0 2 5 00 . 5
1
1 . 5
2
a
m
pl
itu
d
e
| d |
| y |
.
| |Fig.7. Amplitude response of LMS algorithm when N=5
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li
0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0 2 0 00
0 . 5
1
s a m p le ( i n d e x )
am
plitude
| e r r o r |
Fig.8. Error response of LMS algorithm When N=5
-80 -60 -40 -20 0 20 40 60 80-30
-25
-20
-15
-10
-5
0
5
10amplitude response antenne, desired signal: 10 degrees, interferers: -10 and -40 degrees
(dB)
angle(degrees)
Fig.9.Normalize array factor of LMS algorithm when N=5
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0 5 0 1 0 0 1 5 0 2 0 0 2 5 0-2
0
2
phase(rad)
d e s ire d s ig n a l 1 0 d e g re e s in te rfe re rs -1 0 a n d -4 0 d e g re e s
p h a s e (d )
p h a s e (y )
li
l i
li
Fig .10.Phase response of LMS algorithm when N=8
0 50 100 150 200 250-2
0
2
phase(ra
desiredsignal 10degrees interferers -10and-40degrees
phase(d)
phase(y)
0 50 100 150 200 2500.5
1
1.5
amplitude
|d|
|y|
0 20 40 60 80 100 120 140 160 180 2000
0.5
1
sample(index)
amplitude
|error|
Fig.11. Amplitude response of LMS algorithm when N=8
0 50 100 150 200 250-2
0
2
phase(rad desiredsignal 10degrees interferers -10and-40degrees
phase(d)
phase(y)
0 50 100 150 200 2500.5
1
1.5
amplitude
|d|
|y|
0 20 40 60 80 100 120 140 160 180 2000
0.5
1
sample(index)
amplitude
|error|
Fig.12. Normalize array factor of LMS algorithm when N=8
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-80 -60 -40 -20 0 20 40 60 80-30
-25
-20
-15
-10
-5
0
5
10amplitude response antenne, desired signal: 10 degrees, interferers: -10 and -40 degrees
(dB)
angle(degrees)
Fig.13. Normalize array factor of LMS algorithm when N=8
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0-2
0
2
phase(rad)
d e s ire d s ig n a l 1 0 d e g re e s in te rfe re rs -1 0 a n d -4 0 d e g re e s
p h a s e (d )
p h a s e (y )
li
| |
| |
i
| |
Fig .14.Phase response of LMS algorithm when N=20
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0 50 100 150 200 250-2
0
2
phase(rad) desiredsignal 10degrees interferers -10 and -40degrees
phase(d)
phase(y)
0 50 100 150 200 2500.8
1
1.2
amplitude
|d|
|y|
0 20 40 60 80 100 120 140 160 180 2000
0.5
1
sample(index)
amplitude
|error|
Fig.15. Amplitude response of LMS algorithm when N=20
0 50 100 150 200 250-2
0
2
phase(ra
desiredsignal 10degrees interferers -10and-40degrees
phase(d)
phase(y)
0 50 100 150 200 2500.8
1
1.2
amplitude
|d|
|y|
0 20 40 60 80 100 120 140 160 180 2000
0.5
1
sample(index)
amplitude
|error|
Fig.16. Normalize array factor of LMS algorithm when N=20
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-80 -60 -40 -20 0 20 40 60 80-30
-25
-20
-15
-10
-5
0
5
10amplitude response antenne, desired signal: 10 degrees, interferers: -10 and -40 degrees
(dB)
angle(degrees)
Fig.17. Normalize array factor of LMS algorithm when N=20
-80 -60 -40 -20 0 20 40 60 80-30
-25
-20
-15
-10
-5
0
5
10amplitude response antenne, desired signal: 10 degrees, interferers: -10 and -40 degrees
(dB
)
angle(degrees)
N=5
N=8
N=20
Fig.18. comparison the Normalize array factor of LMS algorithm
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0 50 100 150 200 250-2
0
2
phase(ra
desiredsignal 25degrees interferers 0and-40degrees
phase(d)
phase(y)
0 50 100 150 200 2500.5
1
1.5
amplitude
|d|
|y|
0 20 40 60 80 100 120 140 160 180 2000
0.5
1
sample(index)
amplitude
|error|
Fig .19.Phase response of direct matrix inversion algorithm when N=5
l
0 5 0 1 0 0 1 5 0 2 0 0 2 5 0
0 . 5
1
1 . 5
a
m
p
litu
d
e
| d |
| y |
.
l
l
| |
Fig.20 .Amplitude response of direct matrix inversion algorithm when N=5
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.
li
0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0 2 0 00
0 . 5
1
s a m p le ( in d e x )
a
m
p
litu
d
e
|e r r o r |
Fig .21.error response of direct matrix inversion algorithm when N=5
-80 -60 -40 -20 0 20 40 60 80-20
-15
-10
-5
0
5
amplitude response antenne pattern
(dB)
angle(degrees)
Fig.22. Normalize array factor of Direct matrix inversion algorithm when N=5
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0 5 0 1 0 0 1 5 0 2 0 0 2 5 0- 2
0
2
phase(rad)
d e s i r e d s ig n a l 2 5 d e g re e s in t e r fe re rs 0 a n d -4 0 d e g re e s
p h a s e (d )
p h a s e (y )
li
| |
| |
l i
li
| |
Fig.23 .Phase response of direct matrix inversion algorithm when N=8
0 5 0 1 0 0 1 5 0 2 0 0 2 5 00 . 5
1
1 . 5
a
m
p
litu
d
e
| d |
| y |
.
| |Fig .24.Amplitude response of direct matrix inversion algorithm when N=8
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0 5 0 1 0 0 1 5 0 2 0 0 2 5 00 . 5
1
1 . 5
a
m
p
litu
d
e
| d |
| y |
.
l
l
| |
Fig .25.Amplitude response of direct matrix inversion algorithm when N=8
-80 -60 -40 -20 0 20 40 60 80-20
-15
-10
-5
0
5
amplitude response antenne pattern
(
dB)
angle(degrees)
Fig.26. Normalize array factor of Direct matrix inversion algorithm when N=8
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0 5 0 1 0 0 1 5 0 2 0 0 2 5 0- 2 0
0
2 0
4 0
p
h
a
s
e
(ra
d
)
d e s ir e d s ig n a l 2 5 d e g r e e s in t e r fe r e r s 0 a n d - 4 0 d e g r e e s
p h a s e ( d )
p h a s e ( y )
li
| |
| |
.
l i
li
| |
Fig.27 .Phase response of direct matrix inversion algorithm when N=20
i i l i
0 5 0 1 0 0 1 5 0 2 0 0 2 5 00
5
1 0x 1 0
1 7
am
p
litude
| d |
| y |
l i
li
| |Fig .28.Amplitude response of direct matrix inversion algorithm when N=20
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0 50 100 150 200 250-20
0
20
40
phase(rad)
desired signal 25 degrees interferers 0 and -40 degrees
phase(d)
phase(y)
0 50 100 150 200 2500
5
10x 10
17
amplitude
|d|
|y|
0 20 40 60 80 100 120 140 160 180 2000
0.5
1
sample(index)
amplitude
|error|
Fig .29.error response of direct matrix inversion algorithm when N=20
-80 -60 -40 -20 0 20 40 60 80-20
-15
-10
-5
0
5
amplitude response antenne pattern
(dB)
angle(degrees)
Fig.30. Normalize array factor of Direct matrix inversion algorithm when N=20
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-80 -60 -40 -20 0 20 40 60 80-20
-15
-10
-5
0
5
amplitude response antenne pattern
(dB)
angle(degrees)
N=5
N=8
N=20
Fig.31. comparison the Normalize array factor of direct matrix inversion algorithm
-80 -60 -40 -20 0 20 40 60 80-30
-25
-20
-15
-10
-5
0
5
10amplitude response antenne, desired signal: 10 degrees, interferers: -10 and -40 degrees
(dB)
angle(degrees)
DMI
CMD
Fig.32. Comparison plots between DMI and CMA algorithms when N=5
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-80 -60 -40 -20 0 20 40 60 80-20
-15
-10
-5
0
5
amplitude response antenne pattern
(dB)
angle(degrees)
CMD
DMI
Fig.33. Comparison plots between DMI and CMA algorithms when N=8
-80 -60 -40 -20 0 20 40 60 80-20
-15
-10
-5
0
5
amplitude response antenne pattern
(dB)
angle(degrees)
CMD
DMI
Fig.34. Comparison plots between DMI and CMA algorithms when N=20
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-80 -60 -40 -20 0 20 40 60 80-30
-25
-20
-15
-10
-5
0
5
10amplitude response antenne pattern
(dB)
l
Fig.35. Normalize array factor of CM algorithm when N=5
-80 -60 -40 -20 0 20 40 60 80-30
-25
-20
-15
-10
-5
0
5
10
amplitude response antenne pattern
(dB)
angle(degrees)
Fig.36. Normalize array factor of CM algorithm when N=8
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-200 -150 -100 -50 0 50 100 150 200435
440
445
450
455
460
465
470
475
480
485amplitude response antenne pattern
(dB)
angle(degrees)
Fig.37. Normalize array factor of CM algorithm when N=20
8. CONCLUSION
In this work,direct matrix inversion and constant modulus
algorithm are used to update the combining weights of adaptiveantenna array. However, its fast convergence presents an acquisition
compare to LMS algorithm.These algorithms are good computation
complexity.Smart antennas technology suggested in this present work
offers a significantly improved solution to reduce interference levels
and improve the system capacity. With this novel approach, each
users signal is transmitted and received by the base station only in
the direction of that particular user. This drastically reduces the overall
interference in the system. Further through adaptive beam forming,
the base station can form narrower beams towards the desired user
and nulls towards interfering users, considerably improving the signal-
to-interference-plus noise ratio.
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Recommended