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In wireless technology, interference of the signals is always major constrain to the researchers. The cognitive radio is also not an exceptional in this case. If the spectrum analysis at transmitter end is not done properly than there is a possibility to interference and hence the desired signal to noise ratio can't be obtained high.
Using different techniques we need to obtain high signal to noise ratio and that is the main aim of the project to develop interference cancellation technique for the cognitive radios. Beamforming is one of those effective methods to obtain high signal to noise ratio. Beamforming for reception in a network is easier than beamforming for transmission. The problems seem to be equivalent initially but they are quite different with each other and difficult as well. The design of downlink beamformers is less important of the individual links and more important of the system gain.
In todays narrowband cellular systems, each operator occupy a number of carrier frequencies which are allocated to the diverse cells according to some frequency arrangement. Different frequencies are used in neighbour cells to reduce the interference level from co-channel users in other cells. The beamformer designs for cognitive radio networks with improved SINR balancing technique are bit different from traditional wireless networks due to additional interference elements.
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High data rate services like web browsing and interactive multimedia application with the help of wireless communication is the current trend of technology. The traditional allocation for frequency spectrum is non-overlapping and hence the future wireless technologies are running short of this frequency spectrum. The solution of this constrain is to allocate frequency spectrum dynamically on the basis of technique cognitive radios.
The novel idea of cognitive radio was first coined by Joseph Mitola III and Gerald Q. Maguire, Jr. (1). Cognitive radio is capable enough to manage and execute itself without human interference. Cognitive radio provides intelligence to the transmitter and receiver to sense the availability of the spectrum automatically and accordingly change the related transmission parameters to optimise the use of available spectrum.
The main function of the cognitive radio is to sense the spectrum. The important requirement to achieve this is to detect the unused frequency band and make a use of it without interference with other users in same spectrum. The spectrum analysis and spectrum decision are major factors for spectrum management in cognitive radios to achieve high quality of service. Recently, cognitive radio has attracted research interests globally as this is the only possible way of solving shortage of frequency spectrum for future wireless technologies.
Each and every real-time problem has an unique and optimum solutions. Convex optimization is one of the methods to achieve these optimal solutions. Convex optimization is the very well known procedure to obtain the optimal solution especially for the field of engineering. Convex optimization, recently, became an essential tool to optimize the real time engineering problems. Convex optimization has an application in various fields such as estimation and signal processing, electronic circuit design, statistic and finance, communication and networks. With recent updates in computation to achieve optimization result, convex optimization is nearly same as linear programming. Convex optimization has ability to solve very large-practical engineering problems efficiently and reliably. Once an optimal solution is found with help of convex optimization then it is guaranteed that the solution is the best and there is no any other possible alternate.
Convex optimization which is a subdivision of mathematical optimization is related to the problem of minimizing convex functions. If the objective function is linear with the constrained space, the problem is linear programming problem and which can be easily solved by linear programming (2). If the objective function is concave then this is maximization problem. If the objective function is convex then this is a minimization problem. The important factor is the constrain set which is convex and this is very basic idea of convex optimization.
A convex optimization problem is one of the forms (2),
Subject to , = 1,..., m.
Where the function : are convex, i.e., satisfy
For all and all with = 1, The least-squares problem and linear programming problem are both special cases of the general convex optimization problem.
There is not any particular general formula to solve the convex optimization problems but there are different and efficient ways to solve them. Convex set is another basic but very important term to understand the theory of convex optimization.
A set C is convex if the line segment between any two points in C lies in C, i.e for any C and any with 0 â‰¤ â‰¤ 1, we have+ C. (2).
Figure 2.: Sample of convex and non convex set
Figure 1(a), the square includes its boundary (dark line) is convex. In figure 1(b), random shaped set is not convex as the line segment between two points in the set shown as dots is not contained within the set. Figure 1(c), even though the pentagon has surface but it contains some boundary points but not others and hence it is not convex.
A least-squares is an optimization problem without constraints and an objective which is a sum of squares of terms of the form
For least-squares problems we are having efficient algorithms and software implementations as well to solve the problem with high level of accuracy and reliability. The least-squares problem can be solved within a time limit approximately proportional to . A computer can solve a least-squares problem with plenty of different variables and terms in just few seconds and those solutions duration will decrease exponentially in the future. In many cases we can solve even larger least-squares problems, by bringing some special structure in the coefficient matrix.
For extremely large problems, or for problems with exacting real-time computing requirements, solving a least-squares problem can be a real troublesome. But majority of cases, we can say that current methods are very efficient and extremely reliable. Recognizing an optimization problem as a least-squares problem is simple but we have to verify that the objective is a quadratic function and the associated quadratic form is positive semidefinite while the basic least-squares problem has a simple fixed form.
In weighted least-squares, the weighted least-squares cost (2).
where, are positive and minimized. The weight vectors are selected in such a way that it reflect differing levels of terms to get the output.
Linear Programming Method:
Linear programming is an another category of optimization problems in which the objective and constraint functions both are linear and form of this is as below (2).
where i = 1,2....m. and vectors c, âˆˆ and scalars âˆˆ R
There isn't any straight and simple mathematical formula for the solution of a linear program as what we have in least square method, but there are a various effective methods for solving these problems. No one can tell the exact number of arithmetic operations required to solve a linear program. The complexity for the linear programming is order of but with a constant that is less accurately characterized than for least-squares. This algorithm is reliable but not quite as reliable as least-squares methods. We can easily solve problems with hundreds of variables and constraints in a few seconds. If the problem has exploitable structure, this method can solve problems with tens of variables and hundreds of constraints.
It is still a challenge to solve extremely large with linear programs with least square method. In many other cases the original optimization problems don't have a standard linear program form, but can be transformed to an equivalent linear program and can easily then be solved.
Convex Optimization Method:
A convex optimization problem is one of the form as below (2).
subject to , i = 1, . . . ,m
where the functions ........ : â†’ R are convex which satisfy
For all and all with = 1, .
The least-squares problem and linear programming problem are both special cases of the normal convex optimization problem. There isn't any common mathematical formula to solve the convex optimization problems, but there are very effective and efficient methods to solve them. Like methods for solving linear programs, the interior-point methods are quite reliable to solve convex optimization. We can easily solve problems with hundreds of variables and thousands of constraints that too in few tens of seconds. By exploiting problem structure, we can solve far larger problems as well. But it is reasonable to expect that solving general convex optimization problems will become a future technology within a few years.
To use convex optimization is very similar as using least squares or linear programming. If we can formulate a problem as a convex optimization problem, then we can easily solve it just as we can solve a least squares problem efficiently. The challenge in using convex optimization is in recognizing and formulating the problem. Once this formulation is done, it is as easy to solve the problem as to solve least squares or linear programming.
CVX - Matlab Toolbox
CVX is a Matlab tool for solving Convex Optimization problems. CVX allow constraints and objectives to be specified using standard Matlab expression syntax. There are few software available to solve standard problems such as linear programs, quadratic programs, second-order cone programs, and semidefinite programs but compared to using these particular software for one or these types of problems, cvx can effectively simplify the specific problem. CVX can also formulate and solve constrained norm minimization, entropy maximization, determinant maximization, and many other problems.
CVX is implemented in Matlab and eventually turning Matlab into an optimization modeling language. Syntax for the programming are constructed using common Matlab operations and functions only and standard Matlab code can be easily assorted with these specifications. This permutation makes it simple to perform the calculations to build optimization problems and eventually to get the output obtained from their solution. CVX can also be used as an element of a larger system that uses convex optimization, such as a branch and bound method, or an engineering design framework. CVX provides exceptional modes to simplify the construction of problems from two specific problem classes.
In Semidefinite programming mode, CVX applies a matrix interpretation to the inequality operator, so that linear matrix inequalities and SDPs can be expressed in a more natural form. In Geomatric Programming mode, CVX accepts all of the special functions and combination rules of geometric programming, including monomials, posynomials, and transforms such problems into convex form so that they can be solved efficiently (3)
Disciplined convex programming is meant to support the formulation and construction of optimization problems that the user wants from the beginning of the convex. Disciplined convex programming imposes a set of rules, which are called the DCP ruleset. Problems which hold to the ruleset can be rapidly and automatically verified as convex and converted to solvable form and problems that breach the ruleset are rejected, even if the problem is convex. It is not like that such problems cannot be solved using DCP, they just need to be re-written in such a way that they don't violate the DCP ruleset.
CVX is not a tool to check whether your problem is convex or not. Prior knowledge about convex optimization is much needed to use CVX effectively. On contrary, if CVX accepts your problem that means that your problem is surely a convex problem. In combination with a course on convex optimization, CVX can be very helpful in learning some basic convex analysis.
CVX can't solve very large problems, so if the problem is very large then CVX is doubtful to work on problem well. For such problems CVX can play an important role. Before initialising to develop a specialized large-scale method, CVX can be used to solve simplified versions of the same problem.
Each and every CVX models must be proceed by the command cvx_begin and terminated with the command cvx_end. On top of that, cvx_begin sdp and cvx_begin gp call upon special semidefinite and geometric programming modes, these modifiers may be combined when appropriate such as cvx_begin sdp quiet invokes SDP mode and quiet the solver output. All variable declarations, objective functions, and constraints must be written in between these two commands.
Beamforming is a powerful and productive approach to transmit and receive signals in a spatially selective way in the presence of noise. Beamforming is a continuously developing sphere that has number of research and practical applications to radar, sonar, microphone array speech/audio processing, communications, biomedicine, radio astronomy and other areas (4). In the last decade, there has been renewed interest in beamforming carried by applications in wireless communications, where multi-antenna techniques have emerged as one of the key technologies to acquire the growth of the number of users and eventually increasing demands for high data-rate services.
Recently, there has been significant progress in the field of beamforming technique with convex optimization approach. Motivated by the fact that the traditional adaptive beamforming techniques such as minimum variance distortionless response(MVDR) beamforming lack robustness against even small mismatches in the desired signal steering vector (4) (5) (6).Transmit beamforming is a relatively new and dynamically developing research field. Classical beamforming is matched to a single steering vector of interest and its aim is to ensure that the inner product of the beamforming weight vector and the steering vector of interest is large, while the inner product of the beamforming weight vector and all other steering vectors is small. This rule applies to both receive beamforming and transmit beamforming towards an uni-receiver.
Beamforming technique changes the directionality of the array with the help of interference. When transmitting the signal (7), a phase and amplitude of the signal are controlled by beamformer as each transmitter in terms of create constructive and destructive interference pattern in signal wavefront. Beamforming techniques can be divided into two sub categories:
Initially using information about the position of the sensors in space and the wave direction, conventional beamforming technique is using a fixed set of weightings and phase to merge the signals from the sensor. Adaptive beamforming is further divide into two sub categories which are desired signal maximization mode and Interference signal minimization or cancellation mode. Adaptive beamformer combines initial information with properties of signal which is received at receiver array in order to remove unnecessary noise which is captured by array from different direction and there for improves the signal to interference and noise ratio (SINR) (8).
In least squares method, no constraints are imposed on the solution. Sometimes use of such method may give unsatisfactory results, in such case we have to consider constraints into account. In adaptive beamforming techniques which involves spatial processing in which we try to minimise the variance of the beamformer's output while distortionless response is maintain along the direction of a target signal of interest. In the temporal counterpart to this problem, we may be required to minimise the average power of a spectrum estimator, while a distortionless response us maintained at a particular frequency. In such a case resulting is referred to minimum variance distortionless response (MVDR) estimator (9).
Conventional Beamforming Technique
Array signal processing techniques are used in various engineering applications such as radar, sonar, speech enhancement and wireless communication. In radar, array of antennae is placed at different point in space to respond to incident electromagnetic waves and the requirement is to detect the source responsible for radiating the electromagnetic wave by estimating angle of arrival of the waves.
The requirement in sonar is also similar to that of the radar, but the sensors consist of hydrophones designed to respond to incident acoustic waves. Similar techniques are also used to listen to the voice of a desired speaker in the presence of background noise.
Assume there are M antenna elements are placed uniformly. The spacing d between the adjacent elements in the array is assumed to be where is the wavelength of the transmitted signal. Consider there are U uncorrelated sources and the angle of arrival of the ith source is .
Figure 4.: Array Antenna
The actual angle of incidence of a plane wave is related to the equivalent electrical angle as following,
d = spacing between the adjacent sensors
= wavelength of the incident waves.
Figure 4.: Different Phase Angle
Spatial delay =
Corresponding time delay = =
Corresponding phase change is,
= for = d/2
Assuming that the source of interest is and treating all other sources are interferences, the requirement is to mitigate interference using the beamforming techniques. The beamformer would attempt to steer a beam towards the direction of the source of interest and would place nulls along the direction of interferers. This will increase the signal to interference ratio of the signal at the receiver. We would use MVDR based beamformers and adaptive beamformers.
The contribution of the ith source to the received signal at the array of antennae can be written as
where is an Mx1 steering vector. Therefore, the received signal at the array of antenna at time instant n can be written as
In figure 1, the beamformer output is the weighted sum of the signal received by the each antenna,
The MVDR beamformer that steers a beam towards the direction of the source of interest and place nulls along the directions of interferers can be designed as
where R is the autocorrelation matrix of the received signal vector at the array of antenna
and is the angle of arrival of the desired source.
The estimate of the variance of the signal impinging on the array along the direction corresponding to can be written as (9) (10),
Example: Assume there are two sources and each of them are radiating signal at an angle 0.6 and -1 and assuming there are three antennas generates received signal if size 3 x5000 as the superposition of signal 1 and 2 as follows,
[Y1]= arraysignal(0.6,M,N) and [Y2]= arraysignal(-1.0,M,N)
We will get the magnitude response using equation below,
for various values of in the range .
Figure 4. : Magnitude Response
Figure 4.: Phase Response
For the beamformer output z(n) = 1,2...N. The constellation diagram of the output and received signal is as below,
Figure 4.: Constellation Diagram
If we increase the number source to three by adding two more interfere to the part above and if the arrival angle of the third source is 0.2 and forth arrival angle is 1.3, the received signal,
The magnitude response of the received signal is as below,
Figure 4. : Magnitude Response
Figure 4.: Phase Response
The constellation diagram of received signal and output of the beamformer is as below
Figure 4.: Constellation Diagram
If we increase even more number of sources with an arrival angle of 1.3 than received signal and output signal would be,
And the magnitude response is as below,
Figure 4. : Magnitude Response
Figure 4.: Phase Response
The constellation diagram of received signal and output of the beamformer for particular number of sources is as below
Figure 4.: Constellation Diagram
Array Signal Function
% Produces signal vector of size MxN at an array of M antennas for a signal impinging at angle theta.
x = sqrt(1/2)* (sign(randn(N,1)) + j*sign(randn(N,1)) );
s = exp(-j*theta*(0:1:M-1)');
Y = s*transpose(x);
noisevar = 0.01;
N = 5000;
R = (Y*Y')/N;
w = (inv(R)*s)/(s'*inv(R)*s);
% Spatial spectrum/ Array response
m2 = conj(w(1))+ conj(w(2))*exp(-j*angle2) + conj(w(3))*exp(-2*j*angle2);
plot(angle2,20*log10(abs(m2)),'r'); grid on;
% The beamformer output
z = w'*Y;
Adaptive Beamforming Technique:
If the transmitted signal of the desired source is known, the beamformer can be adaptively updated using the complex LMS algorithm by treating the known signal as the desired response d(n):
where is the error signal which is the difference between the desired response d(n) and the beamformer output z(n)
is the beamformer weight vector and is the array input vector of size Mx1.
Eventually beamformer attempts to steer a beam towards the desired source and would place nulls along the direction of interferers.
In some applications, beamformers can be designed without a need for training signal (a desired response). One method is to employ the constant modulus algorithm (CMA) which is aimed at minimising the following cost function
where the dispersion constant is defined as .
The adaptive rule for the CMA algorithm is therefore
An adaptive beamformer technique performs adaptive spatial signal processing with an array of antennas in order to transmission or reception of the signal from different direction without any mechanically steer of the array. The main difference between conventional and adaptive beamformering technique is the ability of the former to adjust its performance to suit differences in the different environment. Adaptive beamformer is potential enough to reduce the sensitivity to certain directions of arrival so that counteract jamming by random transmission.
Signal To Noise Interference (SINR) Improvement:
Cognitive radios are able to improve spectrum utilization by observing availability of spectrum band and opportunistically utilizing the unused spectrum bands (1). In the SNIR balancing technique, the secondary users (SU) access the spectrum occupied by the primary users (PU) without creating harmful interference to the PU.
The power allocation and beamforming approach for multiple-user systems have been highly interested research topic over last decade for practical applications in communication system. The beamformer designs for cognitive radio networks with improved SINR balancing technique are bit different from traditional wireless networks due to additional interference elements.
In (8), a beamforming approach has been proposed to maximize the ratio between the received SU signal power and the interference power leakage to the PUs. In (8), a robust beamforming technique has been developed to limit the probability of the interference leakage to PUs. In (11), robust CR network beamformers have been developed that meet specific target SU. To avoid infeasibility issues, In[X], SINR balancing technique using max-min fairness approach with additional PU interference constraints has been developed. In earlier signal-to-noise ratio (SNR) balancing techniques have been developed in the context of broadcast CRN beamforming (12). However, this approach of (12) assumes that the same data stream is transmitted to all users and, therefore, it determines a single beamforming weight vector without taking into account cross-talk interference. Hence, CR networks downlink beamforming problem (13) significantly differs from that of (12).
The technique proposed in (14) is a direct extension of the SINR balancing technique of (15), that is, the uplink and downlink power allocation algorithms have been rectified in (14) to include the PU interference constraint. But main drawback of this approach is that if more power is available at the SNBS, it cannot be used to enhance the SU SINRs as this will also increase the interference leakage to PUs. In this project we will develop a new approach for downlink CRN beamforming technique that avoids such a drawback. The main purpose is to use the downlink power allocation values obtained in the previous iterations to optimally determine the attenuation required along the direction of the PUs when beamformers are designed in virtual uplink mode (13).
Optimal Downlink Beamforming Using Convex Optimization:
In several ways, Beamforming for reception in a network is easier than beamforming for transmission. The problems seem to be equivalent initially but they are quite different with each other and difficult as well. As the radio channel is reciprocal, a uplink beamformer should also work well in the downlink beamforming. However, whereas a beamformer at the receiver side will only affect the signal quality for one specific user, a signal transmitted from an antenna array will be received not only by the desired user but also by all other users in the surroundings. (16) Thus, the design of downlink beamformers is less important of the individual links and more important of the system gain, since all the different beamformers in a system will affect the system performance equally. The channel knowledge is another major difference. There are also some blind methods that don't require any training symbols but take advantage of knowledge about the transmitted signal modulation. On the contrary, estimation of the channel is only possible at the mobile, which might require extra feed-back link to the base station. In todays narrowband cellular systems, each operator occupy a number of carrier frequencies which are allocated to the diverse cells according to some frequency arrangement. Different frequencies are used in neighboring cells to reduce the interference level from co-channel users in other cells. When antenna arrays are introduced in the system to increase the capacity of the user, the idea is to use the directivity of the antenna to reduce the co-channel interference.
Antenna arrays brings new possibilities in the design of mobile communications systems. It is well known that the system capacity is more limited in the downlink than in the uplink (17). But still literature on beamforming for transmission is very less compared to the beamforming for a receiving antenna array. By having knowledge of the channel to nearby co-channel users, it is possible to actively hold back the signal to the interfered users. In (18), Rashid-Farrokhi et al. formulate the beamforming design as a constrained optimization problem and present an algorithm that is shown to find the global optimum.
In this project we are considering the same optimization problem, namely to minimize the total transmitted power while maintaining assured quality of service(QoS) for all users, but using different solution using convex optimization. It has a many advantages. Firstly, the
solution can be efficiently calculated using standard algorithms for semidefinite optimization
(19) (20). On the other hand, it is very easy to introduce other modifications, for example adding extra constraints on the dynamic range. In most cases this method gives a result in the form of a standard fixed beamformer, in the remaining cases the solution can be implemented using a time-varying beamformer - a space time code (20). In (21), Visotsky and Madhow consider the algorithm in (18) for the special case of rank one channels and claim that it does not give the optimal solution unless an extra scaling of the problem is introduced. However, this results confirm that the algorithm in (18) does indeed give the correct solution also without any rescaling (22).
When the antenna array is used in uplink mode, the channel can be estimated directly from the received data, whereas in the downlink, the transmitting beamformer must be based on information collected in the uplink (22). There are few schemes have been proposed for the transformation from uplink to downlink. In a Time Division Duplex system with short time slots, the downlink channel is virtually unique to the uplink channel, while in a Frequency Division Duplex system, the channel fades identically at the two duplex frequencies. The optimal solution must be solved in cooperation because every transmitted signal will affect all individual receivers. The optimal solution provides a valuable point of reference for assessment of other algorithms and for use in system simulations and to observe system capacity.
We can define "Optimal beamforming" in many different ways but main choice is completely based on the operators' perspective. The system should provide its QoS to as many users as possible with lower cost. In our case we will need to express the QoS requirement in form of a lower threshold on the received Signal to Interference Noise Ratio (SINR) at each mobile node.
With this constraint we need to minimize total transmitted power at all base stations. This will result in an optimization problem which is formulated as below,
min Total transmitted power
Let's express this term in mathematical way that the signal received at mobile i is
Signal received at mobile station i
Channel vector from base station k to mobile i, including the channel gain
Beamforming vector for transmission from base station k to mobile i
Signal intended for mobile i
Noise for mobile i
This equation follows the received SINR which is,
so for the above equation, the optimum downlink beamforming problem can be written as below,
subject to ,
We can write above equation as below as well,
subject to ,
This is a quadratic optimization problem with quadratic non-convex constraints.
Generally the original constraint set in not convex but we will prove that the problem still can be efficiently solved by convex optimization. To solve this problem, let's introduce the matrices and we can use the rule
We can re-write this problem as below,
Here means W is positive semidefinite. Relaxing the rank of gives a semidefinite optimization problem with a solution which is always within the lower bound for the original problem. This technique is called a Langrangian relaxation.
There are four transmitting which are carrying two different singal and for two different user 1 and user 2 respectively. Let's assume that threshold limit for SINR is 100 and noise for the channel is 0.01.
Figure 4.: Downlink Beamformers for user 1 & 2
So the transmission power for signal 1 can be written as below with the assumption that data signa are uncorrelated and have normalized power ] = 1 ,
and same for the signal 2,
So total transmission power will be,
Let's calculate Signal to Interference and Noise Ratio (SINR) for User 1,
We can write SINR equation for User 2 same as above,
As discuss earlier, our main aim is to minimize transmission power with two constrain of SINR for both user 1 and 2. We put this in mathematical form
CVX in Matlab can perform this operation very effectively and gives the optimal solution. In this particular problem we select a random channel so every time it gives a different value but for real time channel it will work perfectly well.
For the above specific example CVX gives an optimal solution for transmission power is 0.6212 and weight vector for user 1 and user 2 are and respectively ;
= [0.2092 - 0.4638i -0.3331 + 0.2023i -0.0122 + 0.1170i 0.2032]
= [-0.0539 + 0.0888i -0.1170 - 0.1789i 0.2509 - 0.1047i 0.1582]
If we repeat the same program 1000 times we can have basic idea of optimal transmission power for random channel. Histogram for the particular experiment is as below
Figure 4.: Histogram
General coding for downlink beamforming technique using CVX.
N = 4;
r = 100;
s = 0.01;
l = 0.1
h1 = (randn(N,1)+j*randn(N,1))/sqrt(2);
h2 = (randn(N,1)+j*randn(N,1))/sqrt(2);
R1 = h1*h1';
R2 = h2*h2';
variable W1(N,N) hermitian complex;
variable W2(N,N) hermitian complex;
trace(R1*W1)-r*trace(R1*W2) >= r*s,
trace(R2*W2)-r*trace(R2*W1) >= r*s,
W1 >= 0;
W2 >= 0;
[v1,D1] = eig(W1)
w1 = sqrt(D1(4,4))*v1(:,4)
[v2,D2] = eig(W2)
w2 = sqrt(D2(4,4))*v2(:,4)
SINR1 = trace(R1*W1)/(trace(R1*W2)+s)
Now repeat the same program 1000 times and get the result out of it,
Nt = 4;
gamma1 = 100;
sigma = 0.01;
l = 0.1;
h1(:,i) = (randn(Nt,1)+j*randn(Nt,1))/sqrt(2);
h2(:,i) = (randn(Nt,1)+j*randn(Nt,1))/sqrt(2);
R1 = h1(:,i)*h1(:,i)';
R2 = h2(:,i)*h2(:,i)';
variable W1(Nt,Nt) hermitian complex;
variable W2(Nt,Nt) hermitian complex;
SINR1 = trace(R1*W1)-gamma1*trace(R1*W2);
SINR2 = trace(R2*W2)-gamma1*trace(R2*W1);
SINR1 >= gamma1*sigma,
SINR2 >= gamma1*sigma,
W1 >= 0;
W2 >= 0;
In general, this sort of problem cannot be solved in reasonable time but using convex optimization which has an inbuilt structure that makes it possible to find the global optimum effectively and efficiently. Note that for the optimal , all constraints will hold with equality. Assume for example that the first inequality is strict then can be scaled down to give equality in the first constraint. This will give a smaller value of the cost function but will not break any of the other constraints, which contradicts that the solution was optimal (22) . In mixed services system, different users may require different QoS. This is easily handled using different for the different individual user groups. Since all the beamformers in the system are drawn in all the constraints, we have obvious hint that the solution must be calculated centrally for the whole system. Also, the channel between every group of base station and mobile has to be recognized. That's why the main application may be as a benchmark in the evaluation of more realistic strategies.
Cognitive radio can able to utilize spectrum effectively by monitoring the availability of spectrum holes and unused spectrum band (1) (13). The power distribution and beamforming problems for multi-user have been widely studied over the decade and on the basis of this study more recent work on SINR balancing for designing beamformers (13) (23) (24). Conventional and adaptive beamforming techniques are used in sensor arrays for directional transmission and reception. Adaptive beamforming is used to detect the desired signal at the output of sensor array with the help of spatial filtering and interference cancellation. To develop algorithm to improve SINR for cancellation of interference in cognitive radio networks can be achieved with the help of beamforming techniques.
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