Study the Behavioral Change in Adaptive Beamforming of Smart Antenna Array Using LMS and RLS Algorithms

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3 Where =array correlation matrix. = the signal correlation vector. In general, we do not know the signal statics and thus must resort to estimating the array correlation matrix ( ) and the signal correlation vector ( ) over a range of snapshots of for each instant in time. The instantaneous estimates of these values are given as and (k) If we substitute the instantaneous correlation approximation, we have the LMS solutions. = (6) Where the error function is given by as e(k)= d (k)- (k) (7) Where, e(k) = error signal, d (k) = reference signal, (k) = (k)+ (k)+ (k) (8) (k) = desired signal vector. (k) = interfering signal vector. (k) = zero mean Gaussian noise for each channel. to the instant of time when the algorithm is initiated. The resulting rate of convergence is therefore typically an order of magnitude faster than the simple LMS algorithm. The RLS algorithm also converges much more quickly than the LMS algorithm. VI. RESULTS AND DISCUSSION We consider here a linear array of 21 elements with interelement spacing of 0.5, total number of data samples taken is 100. All the elements are uniformly excited. Smart antenna array is optimized using two distinct adaptive algorithms LMS and RLS. Case 1: Optimization using LMS algorithms is shown below, V. OVERVIEW OF RLS ALGORITHM Unlike the LMS algorithm which uses the method of steepestdecent to update the weight vector, the Recursive Least Square (RLS) algorithm uses the method of least square to adjust the weight vector [3],[8]. In the method of least squares, we choose the weight vector w(k), so as to minimize a cost function that consists of the sum of squared errors over a time window. In the method of steepest decent, on the other hand, we choose the weight vector to minimize the ensemble average of the squared errors. In the exponentially weighted RLS algorithm, at time k, the weight vector is chosen to minimize the cost function (9) Where e(i) is the error signal, and is a positive constant close to, but less than one, which determines how quickly the previous data are de-emphasized. In a stationary environment, however, should be equal to 1, since all the data past and present should have equal weight. The RLS algorithm is obtained from minimizing equation by expanding the magnitude squared and applying the matrix inversion lemma. The RLS algorithm can be describes by the following equations (10) (11) The initial value of p(k) can be set to, (12) Where i is the m*m identity matrix, and is a small positive constant called the regularization parameter, which is assigned with a small value for high SNR and a large value for low SNR. An important feature of the RLS algorithm is that it utilizes information contained the input data, extending back Fig.4: Array factor plot using LMS algorithm where N=21, interelement spacing=0.5, desired user AOA=-20 deg, interferers AOA=20 deg. In case of fig.5, 6 we observed that after optimizing the adaptive or smart antenna array using least mean square algorithm the convergence of normalize weight and mean square error is obtained after the 50 th number of iteration. So using LMS algorithm in adaptive beamforming of a smart antenna array to achieve the optimum solution it is taken more time than the RLS algorithm. Fig.4 represents the radiation pattern of the linear array using LMS. Fig.5: Normalize weight vector plot w.r.t iteration number obtained using LMS for desired user AOA=-20 deg. interferers AOA=20 deg. ISSN: Page 312

4 Fig.6: Mean square error plot obtained using LMS algorithms for AOA=-20 deg. interferers AOA=20 deg. Fig.8: Normalize weight vector plot w.r.t iteration number obtained using RLS for desired user AOA=-20 deg. interferers AOA=20 deg. Where in fig.8, 9, we observed that after optimizing the adaptive antenna array using recursive least square algorithm the convergence of normalize weight and mean square error is obtained after the 30 th number of iteration. But using RLS algorithm in adaptive beamforming of a smart antenna array to achieve the optimum solution it is taken lesser time than the LMS algorithm. Fig.7 represents the radiation pattern of the linear array using RLS algorithm with desired signal direction at -20 degree and undesired signal direction at 20 degree, with interelement spacing 0.5. All the element of the array is uniformly excited to achieve the optimum solution fast. Case 2: Optimization using RLS algorithm is shown below, Fig.9: Mean square error plot obtained using RLS algorithms for AOA=-20 deg. interferers AOA=20 deg. Fig.7: Array factor plot using LMS algorithm where N=21, interelement spacing=0.5, desired user AOA=-20 deg, interferers AOA=20 deg. VII. CONCLUSIONS In this paper a smart or adaptive array system is optimized using different adaptive beamforming algorithms such as LMS & RLS. The convergence speed of LMS algorithm depends on the eigan values of the correlation matrix. In an environment yielding an array correlation matrix with large eigan values spread it converges slowly in a dynamic channel environment. This problem is solved by the RLS algorithm. In both cases the reference signal is needed. Simulation results revealed that RLS algorithm involves more computations than the LMS algorithm; it provides better response towards co channel interference and safe side to the main lobe. It is also revealed that the convergence rate of RLS is faster than that of the LMS which can be visualized from the simulated results where we can see that when the adaptive or smart antenna array is optimized using LMS the convergence of normalize weight and mean square error is obtained after the 50 th number of iteration, but in case of RLS convergence of normalize ISSN: Page 313

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