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Restoration of Beampattern Using Goal Attain Optimisation Abstract The desire beampattern of a sound source at the far field can be synthesised by controlling the relative amplitudes and phases of each array elements. However, when one or more elements in the array fail, the beampattern at the far field will be severely distorted. The Goal Attain optimisation algorithm is used here to adjust the relative gains of the remaining array elements so as to generate an optimised pattern which closely resembles the original beampattern. The quality of each optimised pattern is evaluated and compared based on a defined performance index. The effectiveness of the Goal Attain method is tested in both the Uniform Planar and Linear Array sonar systems. 1. Introduction The desire shape of the far field radiation pattern of a linear uniform array antenna system can be synthesised by controlling the relative amplitudes of the array elements. Phase shifters are the devices in an electronically scanned array that allows the antenna beam to be steered in the desired direction without physically re-positioning the antenna. Phased array antennas typically consist of individual elements that are essentially each little antenna on their own. Numbers of these elements are arrayed in patterns depending on the desired performance characteristics needed by the application, such as operating frequencies, antenna gain, sensitivity, and power requirements. When one or more array elements or its associated circuitries of an antenna system become defective, this will result in severe distortion of the original far-field pattern. The Goal Attain technique [1] can be use to restore automatically the original far field radiation pattern by adjusting the gains of the remaining working elements. 1 Least Mean Square optimisation is widely used in adaptive beamforming [2] where the adaptive array can automatically steer its beam toward the desired direction and null in the direction of the interference sound source. The Goal Attain technique is more relevant when dealing with faulty array elements since the Goal objective is chosen so as to restore maximally possible the correct beampattern to avoid degradation in performance of the array system. Other phased array antenna radiation pattern recovery methods such as Conjugate gradient-based algorithms [3] to minimize side lobes, enhancement of array system with simulated annealing techniques [4] and Genetic algorithms [5,6] also has been successfully employed. To use Goal Attain, two beam patterns are required, the original and the distorted pattern caused by faulty elements. The original beampattern is determined by numerically integrating the gains and phases of all the working elements in the sound source array. To synthesis the fail beam pattern [7], the phases and gains of the faulty array elements are null during the integration process. The effect of the faulty array elements on the beampattern is the overall reduction in the array gain factor. In addition, lobes symmetry is also affected in absence of non-coherent inter- element spacing between elements, Figure 1. The original beampattern is set as the target in Goal Attain. The Goal Attain then attempt to minimise the error between the faulty and the original pattern by adjusting the gain (weights) of the remaining working elements in the array. The availability of powerful DSPs and their efficient computation algorithms can be exploited to implement optimisation techniques in real time sonar array systems. Although the Goal Attainment optimisation technique is applied to linear and planar arrays in this paper, the procedure can be generalized for other array configurations. 2 2. Far Field BeamPattern Generation of Sound Source Array – Beamforming The linear and planar arrays considered in this work operates at a frequency of fc = 100KHz and the speed of sound in air of at 20º C is c = 344 m/sec; the inter-element distance ‘d’ avoiding grating is c 344 3.44mm 100,000 fc d 1.72mm 2 The linear array consists of M=16 and planar consists of MxN (8x8) elements. The resultant sound field at a particular far field point is determined by the vector addition of the sound field at that point radiated, by the individual sound sources. Conventional beamforming calculates the time delay of the signals arriving between array elements by shifting and summing of the incoming data by using numerical integration techniques. 2.1 Linear array The array factor of the beampattern is normalised with equal amplitudes w(m) = 1. The beamforming output from linear array is given as M G( ) w( m ) ei( 2mM 1 )* arg (1) m1 where arg d sin( ) 2 is scanning angle , = -90 :1:90 Υ is steering angle 3 w(m)=[1,1,…..,M], Uniform weight on each element d, inter-element distance between array elements M=16 array elements and m is the index on M The quantity |G()|2 is known as the beampattern, which is simply the power of the beamformer as a function of . 2.2 Planar array Planar array beampattern G* in two-dimensional is superposition of two beampatterns Sx in x-axis and Sy in y-axis G*( , ) Sx( , ) S y( , ) (2) where M i( 2m M 1 ) * arg x S x ( , ) wx ( m ) e m1 arg x d x cos( x ) sin( y ) 2 2 and =-90:1:90, are the scanning in x and y dimension and N i( 2n N 1 ) * arg y S y ( , ) w y ( n ) e n1 arg y d y sin( x ) sin( y ) 2 2 , , and wx(m) = wy(n) = w(m) are same as before. 4 2. Failed Beam Pattern Synthesis 3. To simulate the beampattern due to failed elements, equations (1) and (2) are used. A new weight vector wf(q) is used in equations (1) and (2) instead of w(m) . Null – zero is injected in w(m) at desired position to assimilate faulty element. wf(q) = [1,0,1,0,0,1,1…..Q= M-P] Array element # 2,4,,5……failed (3) where Q is the number of the remaining working elements and q is their index. P is the number of the faulty elements and p is the index . The beam pattern of the remaining elements excluding the contribution from the failed elements is evaluated as Q i( 2( q p )Q1 ) * arg G f ( ) w f ( q )e (4) q1 In equation (4) index q+ p cater for increased inter-element distance in multiple of d Figure 1. The purpose of wf(q) and (q+ p)*d is to null the contributions from the faulty elements (q+ p)*d = (1, 2, 3..)* d (5) 1 2 3 4 5 6 w=1 d w=0 w=1 w=0 w=0 w=1 2d 3d Figure 1. Non-uniform inter-element distance In the planar array system randomly selected row or column of array elements is failed similarly, Figure 2. The weights wx and wy are changed to wfx and wfy as in equ(3). 5 V U i ( 2( u p )U 1 ) * arg x i ( 2( v p ) V 1 ) * arg y G ( , ) w fx ( u ) e f w fy ( v ) e (6 ) u 1 v 1 where u, v and U, V are the indexes and numbers of the failed x-y planar array respectively, similar to q and Q in equation (5) for failed linear array. n m Figure 2. Failed elements of planar array are shown white 4. Optimization – Goal Attainment Multi objective optimization method of Gembicki [8] concerns the minimization of a set of objectives simultaneously. Mathematically for array gain optimisation the problem can be stated G f ( , ) W f * G* ( , ) (9) 0 ( lb ) W f 1( ub ) G* = {Sx* , Sy*} equation (2) is set as goal to be achieved by the objective in equation(6) associated with weights wfx and wfy . The problem formulation allows the objective to be under or over achieved. The degree of achievement of the goal is controlled by a vectors of weighting coefficients Wf = [wx , wy]. is the parameter which changes the size of feasible region of solution space (), figure3. Lower bound (lb) and upper bound (ub) is set to 0 and 1 respectively to normalized weights, Wf. 6 Sy Wf Gf Sy_o Go g2 g2 G* Sy* * Sg* x2 Sx_o Sx Figure 3. Two dimensional optimisation problem The sign of Wf determine the direction of search of the feasible function space for goal point G* = (Sx*, Sy* ). The +ve of Wf in equ.(6) makes the Gf(,) less than G*(,), while -ve makes Gf(,) greater than G*(,) and function converges closer to the solution. The starting weights wf (M). are carefully chosen so as not to produce local minima The starting optimization function |Gf (,)| then would be less than goal |G* (,)|, so W f ( 1 to 2 )* ( w f ) where w f [ wx ( m ) , w y ( n )] The magnitude of 1 to 2 is selected for the slackness in the solution and helps to increase speed of convergence. After optimisation the optimised weights are then wo ( w )o f 4. Optimization Objective Function and Pattern In case of planar array the optimization process evaluate the optimise weights wo =[wo_x , wo_y], compensating for the failed elements weights wf. The failed element beam pattern Gf( ) in eq.(4) 7 is chosen as the optimisation objective function so that during the optimisation process, the faulty array elements do not contribute in the optimised beam pattern. Goal Attain algorithm then iteratively evaluates the weights of all the working elements in the array. Each weight wo consists of the relative amplitude of individual working array element The weights are so adjusted that Gf() is optimise to Go() and approaches closest to the goal G*(). The optimise objective function is describe here with linear and planar array. The optimised beam patterns would then be described from figure 3 as M i( 2( m p ) M 1 ) * arg G ( ) w ( m ) e (7) o o m 1 and for planar array Go ( , ) S x _ o ( , ) S y _ o ( , ) (8) 6. Performance Measure After the optimised pattern Go() has been determined, it is compared with the failed beampattern to evaluate if any improvement in performance has occurred. Gain performance indicator η (SNR) is set to the ratio of the beamwidth, defined as the gain of the main beam above -3db, to the side lobes as GBW SNR ( f ,o ) 20 log 10 3db (10) Go( ) GBW 3db 8 Figure 4 depicts the above expression. The rationale for η is that the main beam is pointing towards the desired direction and the side lobes give rise to undesirable noise entering the receiver from other undesirable directions. -3db Figure 4. 3db beamwidth used in equ. (9) 7. Results Sixteen element uniform linear array optimization results were evaluated at different steering angles with different positions of failed elements. Matlab’s optimization tool box [9] ‘fgoalattain’ commands was used for implementation of algorithm in section 5. Figures 5-6 illustrate the beampattern results. The reference pattern |G*()| is obtained equ. (1) and (2), failed patterns |Gf()| from equ. (4) and equ.(6), optimized beampattern |Go()| from equ.(7) and equ.(8). The selected failed elements in equ. (3) and (4) are concatenate as ‘0.000’ in the weight column at the left of Figure 5 . . 9 Figure 5. Goal optimisation for failed elements numbers 2, 9 and 14. Element separation d=1.72mm The η of the three-radiation patterns |G*()| , |Gf()| and |Go()| in Figures 5-6 were evaluated using equation (10). The η at different steering angles for the uniform linear array is shown in Figure 7 8. Conclusion The performance index η (SNR) measure shown in Figures 5 and 6 demonstrates that the Goal Attain algorithm provide good results when it is used to restore beam pattern due to element failures. The poor angular resolution of Go(,) can be improved if a smaller degree increment is used in the scanning instead of the 1° selected to reduce the computation burden. The planar array optimisation in Figure 6 demonstrates the failed elements in the column or row vector shown in Figure 2. The planar array optimization can be further investigated for discrete scalar failed planar array elements. 10 Optimized Weights: w(n)(y) = [0.91 1.00 0.73 0.16 0.34 0.08 0.00 0.08] w(m)(x) =[1.00 0.89 0.39 0.00 0.01 0.10 0.41 0.01] Figure 6. Failed element vector three or x-axis and vector seven on x-axis, see figure2. Scanning angles theta and psi range –90 to 90 degrees. SNR = η, Angular resolution 5 degrees, amplitude is normalized. Figure 7 shows the optimisation results of uniform linear array (ULA) when steered off center. The best results are obtained within 40 of the scanning angle. The effect of optimisation on steering angle similar to ULA in Figure 7 can also be studied for planar array. The optimised pattern can be chosen for both signal transmission and reception. 11 Figure 7. η (SNR) vs. scanning angle plot of uniform linear failed elements array in fig. 5. References 1) Engineering Optimization by S. S. Rao, Wiley & Sons, Inc. NewYork, USA, 1996 2) Signal Processing for Intelligent Sensor Systems by David C. Swanson, Marcel Dekker, Inc.Newyork 2000. 3) Peters, T. J., “Conjugate gradient-based algorithms to minimize the sidelobes level of planar arrays with element failures,” IEEE trans. Antennas Propagat., Vol. 39, 1497-1504, 1991 4) Rodriguez, J. A. and F. Ares, “Otpimization of the performance of Arrays with failed elements using the simulated annealing techniques,” J. Electromagn. Waves Appl., Vol 12, 1625-1637, 1998 5) Yeo, B. and Y. Lu, “Array failure correction with genetic algorithm,” IEEE trans. Antennas Propagat., Vol. 47, 823-828, 1999 12 6) Rodriguez, J. A. and F. Ares, “Finding defective elements in planar arrays using genetic algorithms,” Progress In Electromagnetic Research, PIER 29, 25-37, 2000 7) Antenna Theory Analysis and Design by C. A. Balanis –2ed, Wiley & Sons, Inc. NewYork, USA, 1997. 8) W. Gembicki, “Vector Optimization for Control with Performance and Parameter Sensitivity Indices”, Ph.D. Dissertation, Case Western Reserve Univ., Cleveland, Ohio, 1974. 9) Optimization ToolBox for Use with MATLAB – User Guide, MathWorks, Inc. Natick, Mass. USA, 1996 13

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