Uniform Linear Array Gain Optimization - MetaLab

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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( 2mM 1 )* arg             (1)
                                  m1

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
                                           m1
                             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
                                            n1
                             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 )Q1 ) * arg
                           G f (  )   w f ( q )e                                             (4)
                                       q1

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




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