# SIMULATIONS OF ADAPTIVE ALGORITHMS FOR SPATIAL BEAMFORMING

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3 Fig-2: classification of Adaptive algorithms Trained algorithm: Trained algorithms use training signal to adapt the weights of the array and minimize mean square error. The processor in the adaptive array has a prestored training signal and the array adapts its weights when the training signal is transmitted by the transmitter. This technique requires synchronization. These algorithms work very well, but the only cost paid is the excess transmission time or wastage of bandwidth. The trained algorithms are classified based on their adaptation criteria and they are least- mean squares method (LMS), sample matrix inversion (SMI) or least-squares method (LS) and recursive least-squares method (RLS). All these techniques minimize the squared error. Blind algorithm: Unlike training algorithm blind algorithm do not require training signals to adapt their weights. Therefore these algorithms save transmission bandwidth. Blind algorithms can be classified as property restoral algorithms, channel estimation algorithms, and dispread and respread algorithms. Property restoral algorithms restore certain properties of the desired signal and hence enhance the SNR. The property that is being restored may be the modulus or the spectral coherence. Blind property restoral algorithms can be classified as Constant Modulus (CM) algorithm, Spectral self- Coherence Restoral (SCORE) algorithms, and decision directed (DD) algorithms. Channel estimation techniques use the knowledge of the special code properties of the spread spectrum signals to obtain estimates of the channel parameters. These techniques first estimate the channel parameters and then use the channel estimates to form beams in the direction of the desired signals. A. Least Mean Square(LMS) Algorithm The LMS algorithm is the most widely used algorithm invented in 1960 by Stanford University professor Bernard Widrow and his first Ph.D. student, Ted Hoff. The main features that attracted the use of the LMS algorithm are low computational complexity, proof of convergence in stationary environment, unbiased convergence, and stable behaviour when implemented with finite-precision arithmetic. LMS algorithms are a class of adaptive filter used to mimic a desired filter by finding the filter coefficients

7 adaptive array output can be restored to a constant by measuring the variation in the signal s modulus and minimizing it by using the cost function. µ represents the rate of adaptation, controlled by the processing gain of the antenna array. If a large value µ of is taken then convergence becomes faster but makes the array system unstable/noisy. Conversely if a small value µ is taken then convergence becomes slow that is also not desirable. Therefore, value of is taken in between that satisfy the following conditions for good convergence and to avoid instability. The constant modulus cost function is a positive definite measure of how much the array output s envelope varies from the unity modulus used to minimize the result. The value of p = 1, q = 2 provides the deepest nulls of the four configurations and provides the best signal to interference noise ratio flow chart and equations of this algorithm is given below. Initialization: 1. Direction of arrival of noise and desired signal 2. Initial weight w(0) Computation: J(K) = [ y(k) p -1 q ] = cost function (J(k)) = x(k)(y(k) y(k)/ y(k) ) = gradient of cost function w(k+1) = w(k)-µx(k)(y(k)-y(k)/ y(k) ) = new weight e(k) = y(k) y(k)/ y(k) = error RESULTS In this paper the main focus is made on different adaptive algorithms for the formation of beam in the desired direction and eliminating the noise received. To do so Adaptive Algorithms code is written in MATLAB. The output of various parameters like phase, amplitude, rate of error change and the Beam formation of different algorithm is compared at different angle. The error value of the same code is obtained by simulating Adaptive Algorithm in Simulink. Here for the simplicity of the execution we have considered three different angles i.e., (1) angle of desired signal (2) two angles at which noise is generated. The complete procedure of executing the program is given below: 1. Input the angle of desired signal and noise angle values and select the Algorithm to be executed. 2. Beam formation at an angle of for all the algorithms is illustrated in Figures below. 3. In LMS Algorithm the phase of noise and the desired signal are in phase, the rate of convergence depends on the µ factor. The output of LMS Algorithm is as shown in Fig-3.

8 Fig-3: Output of LMS Algorithm 4. RLS Algorithm is the efficient algorithm as shown in Fig-4, where the rate of convergence is high and error is less. Fig-4: output of RLS Algorithm 5. SMI Algorithm involves only matrix operation, so the computation time and the error will be more. The Fig-5 illustrates the simulation results of SMI Algorithm. Fig-5: Results of SMI algorithm

9 6. CMA Algorithm is a Blind Adaptive Algorithm which is independent of training sequence. Hence the overhead will be less. The output of CMA Algorithm is as shown in Fig-6. Fig-6: Results of CMA Algorithm COMPARISON TABLE: Table-1: Comparison of Algorithms CONCLUSION Four Adaptive Algorithms are simulated for Spatial Beam forming both in MATLAB and Simulink; these tools have efficiency for signal processing. The results of simulated algorithms are compared with respect to various parameters like computation complexity, convergence rate, and magnitude of error. Accordingly the results showed

10 that RLS to be the best algorithm in spite of computational complexity followed by LMS, CMA, SMI based on their magnitude of error. By trading off these parameters one choose the better algorithms for the requirement. REFERENCES [1] B.Widrow, P.E.Mantely,L.J.Griffiths, and B.B.Goode, Adaptive Antenna Systems,Proc. IEEE, Vol.55,No.12, pp ,aug [2] A.Alexiou and M. Haardt, Smart Antenna technologies for future wireless systems: trends and challenges, Communications Magzine, IEEE, vol.42, pp , [3] Ch. Santhi Rani, Dr. P.V. Subbaia, Dr. K. Chennakesava Reddy, LMS and RLS algorithm for Smart Antennas in a CDMA Mobile Communication Environment. [4] M.Chryssomallis, Smart Antennas, IEEE Antennas Propagation Magzine Vol. 42, no. 3, June [5] P.J. Burt. Smart sensing within a pyramid vision. Proceedings of the IEEE, 76(8): , 1988.

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