INTRODUCTION
Nowadays, wireless communications have been very popular and diverse applications
of them can be seen in various aspects of human life. Cellular wireless communications,
satellite communications, wireless radio of Police, radio taxies, Bluetooth,
Wireless Local Loop (WLL), pointtopoint microwave links, Wireless Local Area
Network (WLAN) and wireless wideband networks such as internet are some examples
of wireless communications. Growth in wireless technologies and user’s
demands, lead us to extend these systems in rural and urban areas, indoor and
outdoor environments, shortrange as well as longrange applications and then
optimizing them for long term scheduling. In this regard, wireless cellular
communications, wireless networks, wireless wideband communications and digital
radio transmitters takes precedence over other technologies due to their diversities,
flexibilities and number of users (Ahmadi, 2009; Alexiou
and Haardt, 2004).
In one hand, by optimum using of available frequency bandwidth and reusing
of radio spectrum and on the other hand, by decreasing the number of fixed stations
(to reduce costs), the required energy consumption will be decreased. In noisy
systems, by reducing internal noises, Signal to Noise Ratio (SNR) will be increased
and then the system can produce better Quality of Services (QoS). In systems
with interference, the main goal is to reduce interferences and increase Carrier
to Interference Ratio (CIR) in the way that consuming energy doesn’t increase.
Timespace techniques, such as, spacetime coding, low noise filters, channel
equalizers and high performance modulations, can decrease the effects of interferences
and noises and then reduce the power consumption. Furthermore, due to frequencyreuse,
cochannel interference will be increased and then the quality of system will
be decreased (Rappaport, 1999; Zhang
and HsiaoHwa, 2008).
Four methods are proposed to manage radio resources and decrease power consumption as follow:
• 
Increasing the performance of energy convertor devices, such
as, high performance radio amplifiers, mixers and modulators 
• 
Using low power electronic and communication devices, such
as, CMOSbased technologies and Radio Frequency Integrated Circuits (RFIC) 
• 
Reducing electrical energy consumption in such way that radio
systems transmit or receive in lower power. For example, some power control
techniques can reduce power consumption in each element of wireless systems 
• 
Applying antenna features to produce strong signals in transmitters
or receivers 
During these two decades, appropriate beamforming and concentrating radiated
power in specific directions are been so attractive. As so, in this research,
we focus on antenna beamforming. It is obvious that considering passive elements
as directional antennas or active antenna structures, such as, fixedbeam and
adaptive arrays, radio resources (power, time, frequency and space) can be used
for different communication systems as well as broadcasting wireless systems
(Kaiser et al., 2005).
NUMBER OF MOBILE SUBSCRIBERS AT A GLANCE
Currently, there are more than 5 billion mobile users in the world and this
number will reach to 5.5 billion users at the end of 2010. Figure
1 (extracted from www.computerweekly.com)
shows the number of mobile subscribers in the years between 2000 and 2010. As
shown, in 2000, there were 750 million users in the world and it is predicted
that this number growth to 4.6 billion in 2010. Figure 1 also
shows a 61% rise in the number of mobile users in time interval 2000 to 2010.
It is the highest growth between different communication technologies.
Figure 2 (extracted from www.itu.int)
shows the penetration rate of mobile technology between years 2000 and 2010.
It is predictable that the penetration rate in 2010 will be more than 80%. According
to Fig. 1 and 2 and the importance of wireless
communication systems in the human life, it is very obvious that power consumption
of these systems should be optimized, interference should be decreased and radio
resources should be managed.

Fig. 1: 
Number of mobile subscribers 

Fig. 2: 
Worldwide penetration of mobile subscribers 
ANTENNA DEVELOPMENTS IN WIRELSS COMMUNICATIONS
In the past, antenna had viewed as a single element, but, nowadays, antenna arrays have been one of the most important parts of current wireless communication systems. The evolution of antenna in wireless systems is as follow.
Omnidirectional antennas: Omnidirectional antennas are used for broadcasting
and local applications. For covering large zones the height and power of the
antenna should be increased (Kraus and Marhefka, 2006).
Directional antennas: In these antennas, signals are transmitted in
a specific direction. These antennas can be used instead of high power omnidirectional
antennas and we gain higher coverage and lower interference (Kraus
and Marhefka, 2006).
Cellular structure: In cellular mobile and multibeam satellite communications,
the coverage area is divided to several zones (cells) that each zone will be
covered by a basestation containing one omnidirectional or several directional
antennas. In this regard, by reusing radio resources (time, frequency and/or
code), both capacity and quality of service can be increased (Bellofiore
et al., 2002).
Cellular sectorization: In cellular systems that are cochannel interference
limited, directional antennas are very useful to decrease the received power
of interferers from sidelobes of antenna instead of increasing the power. This
means, each basestation equipped with three 120degree or six 60degree directional
antennas (Bellofiore et al., 2002; Etemad,
2008).
Space diversity: In order to improve the quality of received signals
in radio receivers and also overcoming multipath fading, space diversity is
used. In this technique, multiple antennas are located in such way that they
are too distant (about 510 wavelengths). Signals which received from multipath
are combined in a decision making box. Four methods are considered to combining
and finding the better signal, selection method that select the signal belongs
to higher quality path, scanning method that search for a new path if signal
reaches below a defined threshold, Equal Gain Combining (EGC) that considers
all paths with similar weights and Maximal Ratio Combining (MRC) that considers
paths with related SNRbased weights. Output of scanning and selection methods
just belongs to a path whereas output of two other methods is a combination
of the signals of all paths (Bhobe and Perini, 2001).
MIMO systems: In MultiInput MultiOutput (MIMO) systems that use multiple
antennas at both transmitter and receiver, have demonstrated the potential for
increased capacity and diversity gain in rich multipath environments same as
cellular mobile and wireless broadband communication systems. It is remarkable
that the maximum achievable diversity gain of a MIMO system with N_{T}
antenna elements at the transmitter and N_{R} antenna elements at the
receiver is N_{T}xN_{R}. Moreover, it has been proved that when
the fades connecting pairs of transmit and receive antenna elements are independently
and identically distributed (i.i.d.), the capacity of a Rayleigh distributed
flat fading channel increases almost linearly with the min {N_{T}, N_{R}}
(Godara, 2004, Nooralizadeh and Moghaddam,
2010; Moghaddam and Saremi, 2008).
Spacetime Coding: In multiple antenna systems by using spacetime coding,
such as, Alamouti coding, we can add high gain to the system instead of increasing
the power level of the transmitter (Badic et al., 2003).
Antenna arrays with fixed weights: There are several antenna arrays
(windows) with fixed weight, such as, Bartlett, Chebychev, Blackman, Hanning,
Hamming, Tukey, Natal, Gaussian and Kaiser. The main goal of these predefined
weighting windows is to obtain an appropriate gain for improving the transmitting
and/or receiving power (ElZooghby, 2005).
Phased array antennas: These antenna arrays have very huge applications
in recent years. In these antennas, for improving the transmitted or received
signal, processing is held in Intermediate Frequency (IF) or Radio Frequency
(RF) and the phase of each antenna element will be changed (Fakharzadeh
Jahromi, 2008).
Digital Beam Forming: In digital beamforming, instead of hardware changes,
major part of processing is done by digital processors in IF or baseband. This
type of processing, increase the flexibility, reduce the size of the system
and by using high speed processors and doing processes with software operations,
the energy consumption will be decreased. By appropriate bemforming at the transmitter,
receiver or both, we can reach to higher receiving power, better Signal to Noise
Plus Interference Ratio (SNIR) and lower power consumption. In the following
section, two major techniques for forming the radiation pattern of antenna arrays,
fixed beam and adaptive processing, are discussed with more details (Balanis
and Ioannides, 2007; Bellofiore et al., 2002;
Litva and Lo, 1996; Sarkar et al.,
2003; Sun et al., 2009).
ANTENNA ARRAY SIGNAL PROCESSING
With the direction of the incoming signals known or estimated via Direction of Arrival (DOA) estimation methods, the next step is to use spatial processing techniques to improve the reception performance of the receiving antenna array based on this information. Some of these spatial processing techniques are referred to as beamforming because they can form the array radiation pattern to meet the requirements dictated by the wireless system. Given a one dimensional (1D) Uniform Linear Array (ULA) of elements (Fig. 3) and an impinging wavefront from an arbitrary point source, the directional power pattern P(θ) can be expressed as:
where, α (x) is the amplitude distribution along the array, β is the phase constant and d (x, θ) is the relative distance of the impinging wavefront, with an angle of arrival θ, has to travel between points uniformly spaced a distance x apart along the length of the array.
The exponential term is the one that primarily scans the beam of the array in a given angular direction. The integral of Eq. 1 can be generalized for two and threedimensional configurations. Equation 1 is basically the Fourier transform of α (x) along the length of the array and is the basis for beamforming methods. The amplitude distribution α (x), necessary for a desired P(θ), is usually difficult to implement practically. Therefore, most of the times, realization of (1) is accomplished using discrete sources, represented by a summation over a finite number of elements. Thus, by controlling the relative phase between the elements, the beam can be scanned electronically with some possible changes in the overall shape of the array pattern.
This is the basic principle of array phasing and beam shaping. The main objective
of this spatial signal pattern shaping is to simultaneously place a beam maximum
toward the Signal of Interest (SOI) and ideally nulls toward directions of interfering
signal(s) or Signal Not of Interest (SNOI).

Fig. 3: 
1D uniform linear array 
This process continuously changes to accommodate the incoming SOIs and SNOIs.
The signal processor of the array must automatically adjust the weight vector
W = [W_{1}, W_{2}...W_{M}]^{T} which corresponds
to the complex amplitude excitation along each antenna element.
It is usually convenient to represent the signal envelopes and the applied weights in their complex envelope form. This relationship is represented by Eq. 2:
where, ω_{c} is the angular frequency of operation and x (t) is the complex envelope of the received real signal r (t). The incoming signal is weighted by the array pattern and the output is represented by:
where, n indicates each of the array elements and W^{H}(t).x(t) is the complex envelope representation of y (t). Since, for any modern electronic system, signal processing is performed in discretetime, the weight vector W combines linearly the collected discrete samples to form a single signal output expressed as:
where, k denotes discrete time index of the received signal sample being considered. The concept of beamforming is applicable in both continuoustime and discretetime signals. Therefore, each element of the receiving antenna array possesses the necessary electronics to downconvert the received signal to baseband and for AnalogtoDigital (AD) conversion for digital beamforming.
To simplify the analysis, only baseband equivalent complex signal envelopes along with discretetime processing will be considered herein. Various adaptive algorithms have already been developed to calculate the optimal weight coefficients that satisfy several criteria or constraints. Once the beamforming weight vector W is calculated, the response of this spatial filter is represented by the antenna radiation pattern (beampattern) for all directions, which is expressed as:
where, 5, P (θ) represents the average power of the spatial filter output
when a single, unitypower signal arrives from angle θ. With proper control
of the magnitude and phase of W, the pattern will exhibit a main beam in the
direction of the desired signal and, ideally, nulls toward the direction of
the interfering signals (Balanis and Ioannides, 2007;
Bellofiore et al., 2002). In following subsections
three classes of beamformers, classical, fixedbeam and adaptive algorithms
are described.
(1) Classical beamformers: In classical beamforming, the beamforming weights are set to be equal to the array response vector of the desired signal. For each particular direction θ_{0}, the antenna pattern formed using the weight vector W_{b} has the maximum gain in this direction compared to any other possible weight vector of the same magnitude. This is accomplished because W_{b} adjusts the phases of the incoming signals arriving at each antenna element from a given direction θ_{0} so that they add inphase (or constructively). Because all the elements of the beamforming weight vector are basically phase shifts with unity magnitude, the system is commonly referred to as phased array. Mathematically, the desired response of the method can be justified by the CauchySchwartz inequality:
for all vectors W, with equality holding if and only if W is proportional to
α (θ_{0}). In the absence of array ambiguity, the effective
pattern in Eq. 5 possesses a global maximum at θ_{0}.
Even though the classical beamformer is the ideal choice to direct the maximum
of the beampattern toward the direction of a SOI, since the complex weight vector
W can be easily derived in closed form, it lacks the additional ability to place
nulls toward any present SNOIs, often required in pragmatic scenarios. This
is obvious when observing the expression in Eq. 5 where, besides
the look direction θ_{0}, control of the beampattern cannot be
achieved in the rest of the angular region of interest. Thus, to accommodate
all the requirements, a more advanced spatial processing technique is necessary
to be applied. As expected, the classical beamformer directs its maximum toward
the direction of the SOI but fails to form nulls toward the directions of the
SNOIs, since it does not have control of the beampattern, whereas the adaptive
beamforming algorithms achieve simultaneously to form a maximum toward the direction
of the SOI and place nulls in the directions of the SNOIs (Balanis
and Ioannides, 2007; Bhobe and Perini, 2001).
(2)Fixedbeam methods: Depending on how the beamforming weights are
chosen, beamformers can be classified as data independent or statistically optimum.
The weights in a data independent beamformer do not depend on the received array
data and are chosen to present a specified response for all signal and interference
scenarios. In practice, propagating waves are perturbed by the propagating medium
or the receive mechanism. In this case, the plane wave assumption may no longer
hold and weight vectors based on planewave delays between adjacent elements
will not combine coherently the waves of the desired signal (ElZooghby,
2005).
Matching of a randomly perturbed signal with arbitrary characteristics can
be realized only in a statistical sense by using a matrix weighting of input
data which adapts to the received signal characteristics. This is referred to
as statistically optimum beamforming. In this case, the weight vectors are chosen
based on the statistics of the received data. The weights are selected to optimize
the beamformer response so that the array output contains minimal contributions
due to noise and signals arriving from directions other than that of the desired
signal (Godara, 2004).
Any possible performance degradation may occur due to a deviation of the actual
operating conditions from the assumed ideal and can be minimized by the use
of complementary methods that introduce constraints. Due to the interest in
applying array signal processing techniques in cellular communications, where
mobile units can be located anywhere in the cell, statistically optimum beamformers
provide the ability to adapt to the statistics of different subscribers. There
exist different criteria for determining statisticallyoptimum beamformer weights
(Gross, 2005). Three of them are reviewed in this section.
The essential goal of the fixed beam methods is to locate the main lobe of the radiation pattern in the direction of the signal and set to zero the radiation pattern in the direction of interference signals. There exist three different criteria for determining statisticallyoptimum beamformer weights, that are Maximum Signal to Noise Plus Interference Ratio (MaxSNIR), Minimum Mean Square Error (MMSE) and Linearly Constrained Minimum Variance (LCMV).
(a) Maximum Signal to Noise Plus Interference Ratio (MaxSNIR): In the
case of more than one user in the communication system, it is often desired
to suppress the interfering signals, in addition to noise, using appropriate
signal processing techniques. There are some intuitive methods to accomplish
this, for example, the Multiple Side lobe Canceller (MSC). The basic idea of
the MSC is that the conventional beamforming weight vectors for each of the
signal sources are first calculated and the final beamforming vector is a linear
combination of them in a way that the desired signal is preserved whereas all
the interference components are eliminated. MSC has some limitations, however.
For instance, for a large number of interfering signals it cannot cancel all
of them adequately and can result in significant gain for the noise component.
The solution to these limitations is the maximum SNIR beamformer that maximizes
the output signal to noise and interference power ratio (Balanis
and Ioannides, 2007; Godara, 2004).
As depicted in Fig. 4, the output of the beamformer is given by:
where, all the components collected by the array at a single observation instant
are Nx1 complex vectors and are classified as s is the desired signal component
arriving from DOA
the interference component (assuming I such sources to be present) and n is
the noise component. In (7), we also separate the desired signal array response
weighted output, y_{s} = W^{H}.S and the interferenceplusnoise
total array response, y_{NI} = W^{H}.(n+i).

Fig. 4: 
Antenna array structure with MaxSNIR weighting 
Consequently, the weighted array signal output power is
where, R_{ss} is the autocovariance matrix of the signal vectors S and the weighted noise plus interference output power is
where, R_{NI} is the autocovariance matrix of the vectors i+n. Therefore, the weighted output SNIR can be expressed as:
By appropriate factorization of R_{NI} and manipulation of the SNIR expression, the maximization problem can be recognized as an eigendecomposition problem. The expression for W that maximizes the SNIR is found to be:
This is the statistical optimum solution in maximizing the output SNIR in a noise plus interference environment, but it requires a computationally intensive inversion of R_{NI}, which may be problematic when the number of elements in the antenna array is large.
(b) Minimum Mean Square Error (MMSE): If sufficient knowledge of the
desired signal is available, a reference signal d can be generated. A block
diagram of an antenna array system using reference signals is shown in Fig.
5. These reference signals are used to determine the optimal weight vector.
This is done by minimizing the mean square error of the reference signals and
the outputs of the Melement antenna array.

Fig. 5: 
Antenna array structure with MMSE or LCMV techniques 
The concept of reference signal use in antenna array system was first introduced
by Widrow in where he described several pilotsignal generation techniques (Balanis
and Ioannides, 2007; Moghaddam and Saremi, 2010).
For beamforming considerations, the reference signal is usually obtained by a periodic transmission of a training sequence, which is a priori known at the receiver and is referred to as temporal reference. Note that information about the direction of the desired signal is usually referred to spatial reference. The temporal reference is of vital importance in a fading environment due to lack of angle of arrival information. The array reference signal need not necessarily be an exact replica of the desired signal, even though this is what occurs in most of the cases. In general, it can be unknown but needs to be correlated with the desired signal and uncorrelated with any possible interference.
As depicted in Fig. 5, at each observation instance k, the error e (k) between the reference signal d (k) and the weighted array output y (k) is given by:
Mathematically, the MMSE criterion can be expressed as:
In order to get a meaningful result, the objective function needs to have explicit dependency on the conjugate of the weight vector. This simply usually translates into changing transposition to conjugate transposition (or Hermitian). Therefore, we have
To minimize the objective function, we set Eq. 14 to zero. Considering additionally the expectation value of the minimum of e (k)^{2}, it yields
where, R_{xx} = E{X.X^{H}} is the signal autocovariance matrix and r_{xd} = E{X.d^{•}} is the reference signal covariance vector. Thus, the optimal MMSE weight solution is given by Eq. 16.
and is usually referred to as the WienerHopf solution. One disadvantage using this method is the generation of an accurate reference signal based on limited knowledge at the receiver.
(c) Linearly Constrained Minimum Variance (LCMV): In the MMSE criterion,
the Wiener filter minimizes the MSE with no constraints imposed on the solution
(i.e., the weights). However, it may be desirable, or even mandatory, to design
a filter that minimizes a mean square criterion subject to a specific constraint.
The LCMV constrains the response of the beamformer so that signals from the
direction of interest are passed through the array with a specific gain and
phase. However it requires knowledge, or prior estimation, of the desired signal
array response α (θ_{0}) with DOA θ_{0}. Its
weights are chosen to minimize the expected value of the output power/variance
subject to the response constraints (Balanis and Ioannides,
2007, Godara, 2004). That is
Cε C^{NxK} has K linearly independent constraints and gεC^{Kx1} is the constraint response vector.
The constraints have an effect of preserving the desired signal while minimizing contributions to the array output due to interfering signals and noise arriving from undesired directions. The solution to this constrained optimization problem requires the use of the Lagrange multiplier vector bεC^{K}. Letting F (W) = W^{H}.R_{xx}.W be the cost function and G (W) = C^{H}. Wg* = 0 be the constraint function, the following expression is formed:
F (W) has its minimum value at a point W subject to the constraint G (W) = C^{H} Wg^{*} = 0, i.e., when H (W) is minimum. Therefore, to find the minimum point in Eq. 18, we differentiate with respect to W and set it equal to zero, which yields:
where, the existence of [C ^{H}.R_{xx}^{1}.C] follows from the fact that R_{xx} is positive definite and C is fullrank. Therefore, the LCMV estimate of the weight vector is
As a special case, a requirement would be to force the beam pattern to be constant in the boresight direction; concisely, this can be stated mathematically as:
where, g is a complex scalar which constrains the output response to α (θ_{0}). In this case, the LCMV weight estimate is
For the special case when g = 1 (i.e., the gain constant is unity), the optimum solution of Eq. 23 is termed as the Minimum Variance Distortionless Response (MVDR) or Minimum Variance (MV) beamformer. The advantage of using LCMV criteria is general constraint approach that permits extensive control over the adapted response of the beamformer. It is a flexible technique that does not require knowledge of the desired signal autocovariance matrix R_{xx}, the noise plus interference autocovariance matrix R_{NI}, or any reference signal d (k). A certain level of beamforming performance can be attained through the design of the beamformer, allowed by the constraint matrix. However, the LCMV is computationally complex.
(d) Simulation results of fixedbeam methods: By comparing three wellknown methods for fixedbeam forming the antenna array pattern, considering 8element uniform linear antenna array, the following simulation results are obtained. These results can be extended to different element numbers or other array geometries.
As depicted in Fig. 6, when the angle of interference signal is close to the angle of the main one, the amplitude of the main signal (maximum gain of Array Factor (AF) that pointed to the source signal) will be decreased. The performance of MaxSIR and MMSE methods are similar and they have higher gain than MVDR method.
Figure 7 shows that the SNIR in MMSE algorithm is higher than other methods. After that, MVDR has high SNIR. In MaxSIR, because the angle of the main signal is not considered in calculations, it has lower SNIR than others.
As illustrated in Fig. 8, Bit Error Rate (BER) for MMSE method is lower than other methods. After that, before the angle of 5 degrees, MaxSIR has lower BER. In other words, in low angular differences, MMSE has the best performance among all other methods. Despite the MaxSIR algorithm, BER of other methods tend to zero for high angular differences, i.e., the accuracy of weighting algorithm for MMSE and MVDR methods are increased in high angular differences.
Figure 9 shows the variations of Normalized Mean Square Error (NMSE) in terms of different signal to noise ratios. As depicted in this figure, NMSE in MMSE method is lower than others. MaxSIR is the second one. It means that, in low SNRs, the best algorithm in terms of NMSE, is MMSE. In high SNRs, the performance of all methods is similar.

Fig. 6: 
Maximum array factor gain vs. angular difference 

Fig. 7: 
Signal to noise plus interference ratio vs. angular difference 

Fig. 8: 
Bit error rate vs. angular difference 

Fig. 9: 
Normalized MSE in different SNRs 
(3) Adaptive processing: As previously shown in fixedbeam methods,
statistically optimum weight vectors for beamforming can be calculated by the
Wiener solution. However, knowledge of the asymptotic Second Order Statistics
(SOS) of the signal and the interferenceplusnoise was assumed. These statistics
are usually not known but with the assumption of ergodicity, where the time
average equals the ensemble average, they can be estimated from the available
data. For timevarying signal environments, such as wireless cellular communication
systems, statistics change with time as the source and interferers move around
the cell.

Fig. 10: 
Adaptive antenna array structure 
For the timevarying signal propagation environment, a recursive update of
the weight vector is needed to track a moving mobile so that the spatial filtering
beam will adaptively steer to the target mobile’s timevarying DOA, thus
resulting in optimal transmission/reception of the desired signal. To solve
the problem of timevarying statistics, weight vectors are typically determined
by adaptive algorithms which adapt to the changing environment (Fuhl
and Bonek, 1998, Haykin, 1996).
In adaptive beamforming, according to the system condition, antenna array and its radiation pattern adjusted dynamically. Thus, in this system, there is a processing unit. Types of antennas (sensors) and forwarding information to the processor depend on the application. For example, communication system that uses information of different signals to process the main signal and separating it from others is one of these applications.
Figure 10 shows a generic adaptive antenna array system consisting of an Melement antenna array with a real time adaptive array signal processor containing an update control algorithm. The data samples collected by the antenna array are fed into the signal processing unit which computes the weight vector according to a specific control algorithm. Steadystate and transientstate are the two classifications of the requirement of an adaptive antenna array. These two classifications depend on whether the array weights have reached their steadystate values in a stationary environment or are being adjusted in response to alterations in the signal environment. If we consider that the reference signal for the adaptive algorithm is obtained by temporal reference, a priori known at the receiver during the actual data transmission, we can either continue to update the weights adaptively via a decision directed feedback or use those obtained at the end of the training period.
Several adaptive algorithms can be used such that the weight vector adapts
to the timevarying environment at each sample. As depicted in Fig.
11, there are two major types of adaptive weighting algorithms, i.e., trainingbased
methods and blind methods. In trainingbased methods, such as, Least Mean Squares
(LMS) and Recursive Least Squares (RLS), one reference signal is required. In
contrast, in blind methods, such as, Constant Modulus (CM), Least Squares (LS),
Decision Directed (DD) and Conjugate Gradient (CG), the only thing that is required
is the DOA of the main signal (source) and other information should be obtained
from received signal.

Fig. 11: 
Adaptive array processing algorithms 
In the following, Sample Matrix Inversion (SMI), LMS and RLS algorithms of
trainingbased category and CM, LS and DD algorithms of blind category is reviewed
with more details. In addition, some simulation results of LMS and CM algorithms
are illustrated (Godara, 2004; Gross,
2005).
Sample Matrix Inversion (SMI) algorithm: If the desired and interference signals are known a priori, (16) provides the most direct and fastest solution to compute the optimal weights. However, the signals are not known exactly since the signal environment undergoes frequent changes. Thus, the signal processing unit must continually update the weight vector to meet the new requirements imposed by the varying conditions. This need to update the weight vector, without a priori information, leads to estimating the covariance matrix, R_{xx} and the crosscorrelation vector, r_{xd}, in a finite observation interval. Note that this is a blockadaptive approach where the statistics are estimated using temporal blocks of the array data. The adaptivity is achieved via a sliding window, say of length L symbols. The estimates R_{xxest} and r_{xdest} can be evaluated as:
where, N_{1} and N_{2} are, respectively, the lower and upper limits of the observation interval such that N_{2} = N_{1} +L1. Thus, the estimate for the weight vector is given by
The advantage of the method is that it converges faster than any adaptive method
and the rate of convergence does not depend on the power level of the signals.
However, two major problems are associated with the matrix inversion. First,
the increased computational complexity cannot be easily solved through the use
of integrated circuits and second, the use of finiteprecision arithmetic and
the necessity of inverting a large matrix may result in numerical instability
(Li and Stoica, 2006).
Least Mean Square (LMS) algorithm: The LMS algorithm is probably the
most widely used adaptive processing algorithm, being employed in several communication
systems. It has gained popularity due to its low computational complexity and
proven robustness. It incorporates new observations and iteratively minimizes
linearly the meansquare error. The LMS algorithm changes the weight vector
W along the direction of the estimated gradient based on the negative steepest
descent method. By the quadratic characteristics of the mean squareerror function
E {e (k)^{2}} that has only one minimum, the steepest descent
is guaranteed to converge. At adaptation index K, given a MSE function E {e
(k)^{2}} = E {d (k)W^{H}. x (k)^{2}}
the LMS algorithm updates the weight vector according to
where, the rate of change of the objective function e (k)^{2} has been derived earlier in (14) and μ is a scalar constant which controls the rate of convergence and stability of the algorithm. In order to guarantee stability in the meansquared sense, the step size μ should be restricted in the interval
where, λ_{max} is the maximum eigenvalue of R_{xx} Alternatively, in terms of the total power of the x
where
is the total input power. Therefore, a condition for satisfactory Wiener solution convergence of the mean of the LMS weight vector is
where, M is the number of elements in the array. A significant drawback from
the use of the LMS algorithm is its slow convergence for colored noise input
signals. The LMS algorithm is a member of a family of stochastic gradient algorithms
since the instantaneous estimate of the gradient vector is a random vector that
depends on the input data vector x (k). It requires about 2 M complex multiplications
per iteration, where M is the number of weights (elements) used in the adaptive
array. The convergence characteristics of the LMS depend directly on the Eigenstructure
of R_{xx}. Its convergence can be slow if the Eigenvalues are widely
spread. When the covariance matrix Eigenvalues differ substantially, the algorithm
convergence time can be exceedingly long and highly data dependent. Therefore,
depending on the Eigenvalue spread, the LMS algorithm may not have sufficient
iteration time for the weight vector to converge to the statistically optimum
solution and adaptation in real time to the timevarying environment will not
be able to be performed. In addition, employing the LMS algorithm, it is assumed
that sufficient knowledge of the desired signal is known (Enosawa
et al., 2006; Moghaddam et al., 2010a,
b).
Recursive Least Squares (RLS) algorithm: Unlike the LMS algorithm which uses the method of steepest descent to update the weight vector, the RLS adaptive algorithm approximates the Wiener solution directly using the method of LS to adjust the weight vector, without imposing the additional burden of approximating an optimization procedure. In the method of least squares, the weight vector W (k) is chosen so as to minimize a cost function that consists of the sum of error squares over a time window, i.e., the LS solution is minimized recursively. In the method of steepestdescent, on the other hand, the weight vector is chosen to minimize the ensemble average of the error squares. The recursions for the most common version of the RLS algorithm are a result of the Weighted Least Squares (WLS) objective function
where, the error signal e (i) has been defined earlier and 0<λ≤1 is an exponential scaling factor which determines how quickly the previous data are deemphasized and is referred to as the forgetting factor. Usually, λ is chosen close to, but less than, unity. However, in a stationary environment λ should be equal to 1, since all data past and present should have equal weight. Differentiating the objective function e (k)^{2} with respect to W* and solving for the minimum yields
Furthermore, defining the quantities
and
the solution is obtained as:
The recursive implementations are a result of the formulations
and
The R^{1} (k) can be obtained recursively in terms of R^{1} (k1), thus avoiding direct inversion of R (k) at each time instant k.
An important feature of the RLS algorithm is that it utilizes information contained
in the input data, extending back to the time instance the algorithm was initiated.
The resulting rate of convergence is therefore typically an order of magnitude
faster than the simple LMS algorithm. This improvement in performance, however,
is achieved at the expense of a large increase in computational complexity.
Other drawbacks associated with its implementation are potential divergence
behavior in a finiteprecision environment and stability problems that usually
result in loss of symmetry and positive definiteness of the matrix R^{1}
(k) (Balanis and Ioannides, 2007; Chen
et al., 2004; Santi Rani et al., 2009;
Skolnik, 2002).
Constant Modulus (CM) algorithm: This algorithm is first proposed by
Godard and it uses the constant envelope feature that is existed in some techniques
that modulate information in phase or frequency of the signal such as, Mary
Frequency Shift Keying (MFSK) and Mary Phase Shift Keying (MPSK) modulations.
By calculating this envelope, adaptive beamforming algorithm can be managed.
CM algorithm uses a cost function, named as diffraction function of order p
and after minimization, the optimum weights can be obtained. The Godard’s
cost function is shown in Eq. 36.
where, p and q are equal to 1 or 2. Godard showed that if R_{p} is
defined as in Eq. 38, the slope of the cost function will
be zero.
where s (k) is the memoryless estimation of y (k) and then the estimation error is:
If we assume that p = 1, the cost function has the form as in Eq.
40.
By rewriting the error signal in Eq. 38 and 41
can be derived.
Updating equation of weights is:
It has been shown that the fastest convergence is obtained by using p = 1.
This method has some problems. One of them is that the algorithm simply locks
on the strongest signal with constant envelope, even if this signal is interference.
In multiuser environments, by changing the initial condition of array, i.e.,
array weighting before the starting time, we can lock on different signals,
if signals have the same power. Another problem of this algorithm is higher
convergence time in comparison with other algorithms that use MMSE criteria
directly (Ghadian and Moghaddam, 2010; Yuvapoositanon
and Chambers, 2002).
Decision Directed (DD) algorithm: In this algorithm, a reference signal
is generated based on the outputs of a threshold decision device. The beamformer
output y (k) is demodulated to obtain the signal q (k). The decision device
then compares q (k) to the known alphabet of the transmitted data sequence and
makes a decision in favor of the closest value to q (k) denoted by r (k). The
reference signal is obtained by modulating r (k) then the cost function for
the beamformer is established. The DD algorithm convergence depends on the ability
of the receiver to lock onto the desired signal. Since, it may not always be
able to do that, the convergence is not guaranteed (Moghaddam
and Saremi, 2010; Moghaddam and Saremi, 2008).
Least Squares (LS) algorithm: Using the standard array model, we can write the received signal at the array output as:
where, X (k) = [x_{1} (k), x_{2} (k),..., x_{M} (k)]^{T}, S (k) = [s_{1} (k), s_{2} (k),..., s_{M} (k)]^{T} is the signal vector and N(k) = [n_{1} (k), n_{2} (k),..., n_{M} (k)]^{T} is the noise vector.
The LS algorithm minimizes the Maximum Likelihood (ML) criterion as Eq.
45 to find proper A that equals to weighting vector (Balanis
and Ioannides, 2007, Shirvani Moghaddam and Saremi, 2010).
Simulation results of least mean squares algorithm: This section shows some simulation results of an antenna array considering LMS algorithm in different environmental conditions such as noise, interference and number of array elements.

Fig. 12: 
BER vs. SNR for different number of ULA elements in a noisy
channel 

Fig. 13: 
Normalized MSE vs. SNR for different number of ULA elements
in a noisy channel 
As depicted in Fig. 12 and 13, increasing
the signal to noise ratio and also the number of array elements are the reason
for decreasing BER and NMSE. Higher number of array elements introduces more
computational complexity but it offers lower performance criteria (BER and NMSE)
rather than lower number of array elements. As it shows, in a noisy channel,
BER will be equal to zero in SNR = 8 dB and SNR = 0 dB for M = 2 and M = 16,
respectively. On the other hand, NMSE will be equal to zero in SNR = 15 dB and
SNR = 10 dB for M = 2 and M = 16, respectively.
Figure 14 and 15 show the simulation
results of a receiver equipped with a uniform linear array in the case of noisy
channel with one interference signal that its power equals to source signal.
It means Signal to Interference Ratio (SIR) equals to 0 dB. Due to adding an
interference, BER and NMSE in the Fig. 14 and 15
are higher than those belong to Fig. 12 and 13.
To illustrate the effect of the power of interference signal, simulations are
repeated for an 8element ULA in SIR = 0, 1, 3 and 10 dB. In SIR = 0 dB the
power of source and interference signals are the same and in SIR = 10 dB, source
signal is 10 times stronger than interference signal. Figure
16 and 17 show BER and NMSE of a LMSbased 8element
ULA adaptive antenna array system for different SIRs.
These simulation results show that the antenna array equipped with LMS algorithm
introduces good performance under different conditions. In noise dominant environment,
increasing the number of array elements has a sensible impact on system performance.

Fig. 14: 
BER vs. SNR for different number of ULA elements in a noisy
channel with one interferer (SIR = 0dB) 

Fig. 15: 
Normalized MSE vs. SNR for different number of ULA elements
in a noisy channel with one interferer (SIR = 0dB) 

Fig. 16: 
BER vs. SNR for different SIRs in a noisy channel with one
interferer and 8element ULA 

Fig. 17: 
Normalized MSE vs. SNR for different SIRs in a noisy channel
with one interferer and 8element ULA 
Also in channel with interference signals, system is able to reject the interference
signals. Besides, simulations show that the performance of the system will be
changed in terms of different SIRs.
Simulation results of constant modulus algorithm: In this section, performance of constant modulus algorithm, in 8element uniform linear array, different SNRs, two channels (pure noisy and noisy channel with one interferer) are evaluated based on BER, NMSE and polar radiation pattern. Simulations are carried out under stationary scenarios. To get each result, simulations are repeated 1000 times. In all simulations, the SOIDOA (source signal) is 40°. Also, Binary Phase Shift Keying (BPSK) modulation is employed.
As depicted in Fig. 18, after 1000 snapshots, main lobe of antenna array pattern is adjusted to 40°C.
Figure 19 and 20 show BER and normalized
MSE of a receiver equipped with digital beamforming system based on CMA. This
beamforming system includes 8element ULA. As depicted in these figures, BER
and NMSE are increased whereas the number of iterations (snapshots) is increased.
It is obvious that after 1000 iterations, BER converge to 0.015 and NMSE converge
to 0.01. In this interferencefree simulation, the step size was set equal to
0.001.
After 1000 snapshots, BER and NMSE will be constant. In this situation, for
different signal to noise ratios, antenna array beamforming considering CM algorithm
is repeated and output BER and NMSE are plotted in Fig. 21
and 22. As expected, increasing SNR force BER and NMSE to
be decreased.
Figure 23 shows antenna array radiation pattern of a receiver
equipped with 8element ULA applying CM algorithm after 1000 snapshots. As shown,
the main lobe is pointed to SOIDOA (40°) and the first null is in the direction
of interferer (20°). As depicted in Fig. 24 and 25,
in the case of noisy channel with one interference, also increasing the SNR
is the reason for decreasing BER and NMSE.
In this research, the effect of adaptive step size on the results is also investigated.
One can deduce from simulation results that increasing the power and number
of interference signals is the reason to decrease the CMA step size and to increase
the number of array elements.

Fig. 18: 
Polar radiation pattern of 8element ULA in a noisy channel
(SNR = 10dB, SOIDOA=40°) 

Fig. 19: 
BER vs. snapshots for 8element ULA in a noisy channel (SNR
= 10dB, SOIDOA = 40°) 

Fig. 20: 
NMSE vs. snapshots for 8element ULA in a noisy channel (SNR
= 10dB, SOIDOA= 40°) 

Fig. 21: 
BER vs. SNR for 8element ULA in a noisy channel (SOIDOA=40°) 

Fig. 22: 
NMSE vs. SNR for 8element ULA in a noisy channel (SOIDOA=40°)


Fig. 23: 
Polar radiation pattern of 8element ULA in a noisy channel
with one interferer (SNR = 10dB, SIR = 0dB, SOIDOA = 40°, SNOIDOA
= 20°) 

Fig. 24: 
BER vs. SNR for 8element ULA in a noisy channel with one
interferer (SIR = 0dB, SOIDOA = 40°, SNOIDOA=20°) 

Fig. 25: 
NMSE vs. SNR for 8element ULA in a noisy channel with one
interferer (SIR = 0dB, SOIDOA=40°, SNOIDOA=20°) 
In addition, some new modified versions of CMA are introduced in Ghadian
and Moghaddam (2010).
Comparing adaptive processing algorithms: Table 1
compares three trainingbased and two blind methods and show advantages as well
as disadvantages. As we know, trainingbased methods have a real convergence
point but they require a training sequence or reference signal. In contrast,
blind methods may be diverged and their performance depends on channel conditions,
however, they don’t have a large amount of information as trainingbased
methods (Li and Stoica, 2006, Liberti
and Rappaport, 1999).
Table 1: 
Comparison of adaptive beamforming techniques 

CONCLUSIONS
Today, wireless communication systems have progressed in the way that their effects on various aspects of human life are very obvious. Smart Antenna systems have received much attention in the last few years because they can increase system capacity (very important in urban and densely populated areas) by dynamically tuning out interference while focusing on the intended user along with impressive advances in the field of digital signal processing.
In this study, antennas are divided into 5 categories, i.e., omnidirectional, directional, different windows, phased array and digital beamforming (DBF) methods. The Fixed beamforming approaches, mentioned in which included MMSE, MaxSNIR and LCMV methods were assumed to apply to fixed arrival angle emitters. If the arrival angles don’t change with time, the optimum array weights won’t need to be adjusted. However, if the desired arrival angles change with time, it is necessary to devise an optimization scheme that operates onthefly so as to keep recalculating the optimum array weights. The receiver signal processing algorithm then must allow for the continuous adaptation to an everchanging electromagnetic environment. The adaptive algorithm takes the Fixed beam forming process one step further and allows for the calculation of continuously updated weights. The adaptation process must satisfy a specified optimization criterion. Several examples of popular adaptive algorithms include trainingbased, LMS, SMI and RLS and blind ones such as CM and DD algorithms. We discussed and explained each of these techniques. According to simulation results of fixedbeam as well as adaptive processing, it is obvious that by appropriate beamforming methods in transmission or reception, better radio signals with lower energy consumption, higher SNIR and lower BER and NMSE will be achieved.
In recent years, researchers focused on DBF in wireless cellular systems, satellite
networks and wideband systems such as WiMAX (Etemad, 2008,
Hoymann and Wolz, 2006) and according to capabilities
of these techniques and baseband processing, it will be a great technology in
the future. As a summery, some research subjects on digital antenna array signal
processing are as follow:
• 
New LMSbased algorithms such as, Variable Step Size LMS (VSSLMS)
and Normalized LMS (NLMS) (Wang et al., 2003) 
• 
Combined algorithms such as, RLSCMA, LMSRLS, LSRLS, SMICMA,
SMILMS, LSCMA, BartlettCMA (Bouacha et al., 2008;
Djigan, 2007; Nooralizadeh and
Shirvani Moghaddam, 2009; Nooralizadeh et al.,
2009; Moghaddam and Saremi, 2008, 2010) 