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1 ATI Material Material Do Not Duplicate ATI Material

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3 Global Optimization 2 While LMS methods are computationally fast, quantization of the phase will result in errors. Also, it is necessary to have receiver hardware at each element of the phased array as well as an elaborate calibration technique. Global search methods can place very deep nulls in the desired directions, while maintaining the characteristics of the antenna main beam. Since the solution space is predefined by the quantized amplitude and phase coefficients of the particular antenna system, these global methods do not require continuous amplitude and phase shifts. ATI Material Additionally, these methods deal with the coherent output power of the antenna array and therefore do not require receiver hardware at each element in the antenna array. ATI Material Material Do Not Duplicate

4 Optimization Methods 3 Local Conjugate Gradient Methods Quasi-Newton Methods Simplex Methods ATI Material Global Random Walk Particle Swarm Genetic Algorithms Material Do Not Duplicate ATI Material

5 Optimization Methods 4 Conjugate Gradient Random Walk Genetic Algorithm Global Optimization Poor Fair Good Discontinuous Functions Poor Good Good Non-differentiable Functions ATI Material Poor Good Good Convergence Rate Good Poor Fair Material Do Not Duplicate ATI Material

6 Genetic Algorithms 5 Genetic Algorithms (GA) are robust, stochastic-based search methods, modeled on the concepts of natural selection. The strong survive to pass on their genes, while the weak are eliminated from the population. Examples Design of layered material for broadband microwave absorbers. Extraction of natural resonance modes of radar targets from backscattered response data. ATI Material Economics, Ecology, Social Systems, Machine Learning, Chemistry, Physics, etc. ATI Material Material Do Not Duplicate

7 Terminology 6 Population set of trial solutions. Generation successively created populations. Parent member of the current generation. Child member of the next generation. Chromosome coded form of a trial solution. Fitness a chromosomes measure of goodness. ATI Material ATI Material Material Do Not Duplicate

8 Chromosome Coding 7 GAs operate on a coding of the parameters, instead of the parameters themselves. In binary coding, the parameters are each represented by a finitelength binary string. Chromosomes are the combination of all the encoded parameters. (A string of ones and zeros) Binary coding yields very simple binary operators. ATI Material R1 L1 C1 R2 L2 C ATI Material Material Do Not Duplicate

9 Genetic Algorithm 8 No No Initialize Population Selection of Parents CrossOver and Mutation Temp Population Full? Replace Population Termination Criteria Met? End Yes ATI Material Yes Evaluate Fitness Evaluate Fitness Material Do Not Duplicate ATI Material

10 Initialize Population 9 Random Fill The initial population is created by filling chromosomes with random numbers. A Priori Chromosomes in the initial population are created with information about the solution. ATI Material ATI Material Material Do Not Duplicate

11 Parent Selection 10 Proportionate selection Probability of selecting an individual is a function of the individual s relative fitness. ATI Material Material Do Not Duplicate ATI Material

12 Parent Selection 11 Tournament selection N individuals are selected at random, the individual with the highest fitness in the sub population is selected. Population N randomly selected chromosomes ATI Material Parent = Chromosome with best Fitness Material Do Not Duplicate ATI Material

13 Crossover and Mutation 12 The crossover and mutation operations accept the parent chromosomes and generate the children. Many variations of crossover have been developed, with single-point crossover being the simplest. In mutation, an element in the chromosome is randomly selected and changed. ATI Material ATI Material Material Do Not Duplicate

14 Crossover and Mutation Single Point Crossover Mutation 13 Parent 1 a1 a2 a3 a4 a5 b1 b2 b3 b4 b5 Parent 2 Child 1 a1 a2 b3 b4 b5 b1 b2 a3 a4 a5 Child 2 a1 a2 a3 a4 a5 a6 a7 ATI Material a1 a2 A3 a4 a5 a6 a7 Material Do Not Duplicate ATI Material

15 Population Replacement 14 Generational The GA produces an entirely new generation of children, which then replaces the parent generation. Steady-State Only a portion of the current generation is replaced by children. ATI Material ATI Material Material Do Not Duplicate

16 Fitness Function 15 The only connection between the physical problem and the GA. The value returned by the fitness function is proportional to the goodness of a trial solution. ATI Material ATI Material Material Do Not Duplicate

17 16 GA Optimization Guidelines Population Size: Typically Large populations enable faster convergence by providing more genetic diversity. Smaller populations yield faster execution, especially for complicated fitness functions. Probability of Crossover: Typically Crossover is the primary way a GA searches for new, better solutions. A probability of 0.7 has been found to be optimal for a wide variety of problems. Probability of Mutation: Typically The probability of mutation should generally be low. Mutation introduces new genetic material into the search, but tends to push the population s average fitness away from the optimal value. ATI Material Replacement Strategy: Generational vs. Steady-State Steady-state generally converges faster. Lower values of replacement percentage usually converge faster. ATI Material Material Do Not Duplicate

18 Particle Swarm 17 Originated in studies of bird flocking and fish schooling. The potential solutions (Particles) fly through the solution space subject to both deterministic and stochastic rules. Particles are pulled toward the local and global best solution with linear attraction forces. ATI Material ATI Material Material Do Not Duplicate

Harmonious Flight 18 The ability of animal groups such as this flock of starlings to shift shape as one, even when they have no leader, reflects the genius of

19 Harmonious Flight 18 The ability of animal groups such as this flock of starlings to shift shape as one, even when they have no leader, reflects the genius of collective behavior something scientists are now tapping to solve human problems. ATI Material National Geographic 2007 ATI Material Material Do Not Duplicate

20 Particle Swarm 19 No Initialize Swarm Update Velocities (V n ) Update Positions (X n ) Termination Criteria Met? End Yes ATI Material Evaluate Fitness Evaluate Fitness Material Do Not Duplicate ATI Material

21 Initialize Swarm 20 Random Fill The initial swarm is created by giving each particle a random position and random velocity. A Priori Particles in the initial swarm are created with information about the solution. ATI Material ATI Material Material Do Not Duplicate

22 Update Velocities 21 Update the velocity of each particle toward the local and global best position. Limit the velocity if necessary. v n = ω v if v n n > + κ rand 1 + κ rand v 2 then ( x ) local best, n xn ( x x ) v global best, n = v n max, n v max vn ATI Material n Material Do Not Duplicate ATI Material

23 Update Positions 22 Update position using unit acceleration. Clip position if necessary. if if x x n, d n, d > < x x x n max, d min, d =,, x n + v then then ATI Material n x x n, d n, d = = x x max, d min, d Material Do Not Duplicate ATI Material

24 Particle Swarm Guidelines 23 ω (Inertia) Typical values between 0 1. This may be allowed to vary randomly for each iteration or decrease with each iteration to encourage local searching at the end of the process. κ 1, κ 2 (Memory & Cooperation) Can be tuned for the particular problem. Common practice in literature to set both equal in the range 1 2. ATI Material ATI Material Material Do Not Duplicate

25 MATLAB Example 24 Find the minimum of the following function. ATI Material ATI Material Material Do Not Duplicate

26 MATLAB Example 25 Find the minimum of the follow function. ATI Material ATI Material Material Do Not Duplicate

27 Antenna Pattern 26 Suppose we want to minimize the antenna gain in a particular direction due to an interfering source (Adaptive Nulling). N M AF ( θ, φ) = n= 1 m= 1 α mn ATI Material I β mn mn = = = 2π λ [ x sinθ cosφ + y sinθ sinφ] mn I mn e jβ mn e mn jα Amplitude coefficient for each element Phase shift for each element Material Do Not Duplicate ATI Material mn

28 Two Interfering Sources 16 x 16 element planar array 27 6 bit phase shifters, 3 bits used for nulling 2 interfering sources located at (θ = 18 o, φ = 0 o ) and (θ = 26 o, φ = 90 o ) 50 Chromosomes / Particles 200 Iterations ATI Material ATI Material Material Do Not Duplicate

29 Two Interfering Sources 28 ATI Material Location of Interfering Sources Material Do Not Duplicate ATI Material

30 Two Interfering Sources 29 Genetic Algorithm ATI Material Particle Swarm Nulls Placed in the Antenna Pattern in the Direction of the Interfering Sources ATI Material Material Do Not Duplicate

31 Two Interfering Sources 30 ATI Material Interfering Source Material Do Not Duplicate ATI Material

32 Two Interfering Sources 31 ATI Material Interfering Source Material Do Not Duplicate ATI Material

33 Two Interfering Sources 32 ATI Material ATI Material Material Do Not Duplicate

34 Two Interfering Sources 33 Main Beam Loss 1.02 db (Genetic Algorithm) 1.63 db (Particle Swarm) Beamwidth Φ = 0 o Original GA PS 3 db 6.29 o 6.30 o 6.35 o 10 db o o o ATI Material Φ = 90 o 3 db 6.29 o 6.30 o 6.35 o 10 db o o o ATI Material Material Do Not Duplicate

35 ATI Material Material Do Not Duplicate ATI Material

Generation of Ultra Side lobe levels in Circular Array Antennas using Evolutionary Algorithms

Generation of Ultra Side lobe levels in Circular Array Antennas using Evolutionary Algorithms D. Prabhakar Associate Professor, Dept of ECE DVR & Dr. HS MIC College of Technology Kanchikacherla, AP, India.