# Adaptive Digital Beam Forming using LMS Algorithm

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3 The quadrature baseband signal can be represented as s(t) is the complex base band signal ( ) ( )is the real part ( ) ( )is the imaginary part ( ) These complex base band signals are multiplied by the complex weights ( ) Adaptive Digital Beam Forming using LMS Algorithm ( ) ( ) Complex weight for the antenna element Relative amplitude of the weight Phase shift of the weight For each antenna element the complex multiplication is done: ( ) *, ( )-, ( )-+ As shown in fig 6. III. LMS ALGORITHM Least mean square (LMS) algorithm was introduced by B.Widrow and Hoff in 1960 [4], which is an iterative method based on minimizing Fig.6: the complex mean square weight error[11]. multiplication This algorithm is derived from the wiener filter. LMS algorithm is based on gradient descent method, which makes consecutive corrections to the weight vector in the direction of negative gradient vector which finally leads to the minimum mean square error. In figure, the output of individual sensors are first multiplied with corresponding weights and then they are combined so that antenna arrays radiation pattern is optimized to have maximum gain in the direction of desired signal and nulls in the direction of interference. The weights are computed by LMS algorithm. Consider a uniform linear array with Fig.7: N- LMS isotropic Adaptive elements. Beam Let Forming the output Network of antenna array be x(t) is given by s(t) Desired signal arriving at angle. ( ) interfering signals at angle. n(t) noise. ( )& ( ) steering vector. Let, all signals are represented by their sample values. From steepest descent method, the weight equation is represented as ( ) ( ), ( * ( )+)- Stepsize (controls the convergence characteristics of LMS algorithm) ( ) Mean square error between beamformer output y(n) and the reference signal. ( ), ( ) ( )- ( * ( )+) Gradient vector ( * ( )+) ( ) 65 Page

4 Here r and R values are simplified by using the instantaneous values of covariance matrices and is given by Therefore the weights update equation is, - ( ) Where, ( ) ( ) The LMS algorithm is initiated with some initial weights, to normally converge and stay stable value should be. IV. Experimental Analysis In this paper, the adaptive digital beam forming algorithm was developed by means of simulation software called Matlab. The simulation is performed with a linear array antenna and the number of element taken are 4, d =, k = are fixed. The LMS algorithm contains three steps in each recursion: 1. Computation of the processed signal with the current set of weights. 2. Generation of the error between the processed signal and the desired signal. 3. Adjustment of the weights with the new error information by the gradient method. The input at each element of array antenna consists of sum ofinput signal which is taken as complex MSK signal in some desired direction, introduced two interrupts in the direction and introduced random noise signal n. the above signals are shown in fig 7. The proposed work is divided in following steps 1. Initially simple LMS algorithm by keeping initial weights zeroes and with fixed step-size. Minimum error is calculated and the corresponding weights are fixed and are used as initial weights in case 2 and case In second step, LMS algorithm is developed by setting initial weights derived from case 1 and step-size is varied in proper steps. Here sample by sample optimization is done to get the desired signal. 3. Finally, Performed LMS algorithm for entire samples to get the direction of the desired signal by varying step-size and by setting initial weights. The following are the observations By knowing the minimum error from entire error graph, one can fix the weights where minimum erro Fig.8: Input signals R is produced because error minimum means the output of the beam former is equals to the desired signal. Therefore we have selected the weights corresponding to the minimum error. 66 Page

5 Fig.10: different values of weights Min(e) W1 W2 W3 W i i i i i i i i i i i i i i i i i i i i i i i i Table 1: list of optimized weights with minimized error main lobe at 67 Page

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