Synthesis of Linear and Non-Separable Planar Array Patterns
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The document discusses techniques for synthesizing linear and non-separable planar array patterns with shaped beam characteristics. It describes using a local search algorithm to optimize the element magnitudes and phases to produce low sidelobe and ripple csc^2 patterns. For non-convex problems with multiple local minima, the local search algorithm converges rapidly to good solutions if initialized close to the optimal point. It is shown to outperform other methods like simulated annealing and differential evolution for large variable count problems. Application examples are provided for both linear and non-separable planar arrays.
![Convex versus Non-Convex Optimization
• Convex function ‘a function is convex on
an interval [a, b], if the values of the
function on this interval lie below the line
through the endpoints (a, f(a)) and (b,
f(b))’ (“Numerical Python …”, R.
Johansson)
• Non-convex function – multiple local
minima in an interval
• Recently a paper in AP Transactions
argued that shaped beam pattern
synthesis is best treated as a convex
optimization problem
global minimum
local
minima
global minimum with no
local minima for -2 < x < 2
(a, f(a))
(b, f(b))](https://image.slidesharecdn.com/slideshareexp1-161028194028/85/Synthesis-of-Linear-and-Non-Separable-Planar-Array-Patterns-8-320.jpg)
![References
[1] H.J. Orchard, R.S. Elliott, and G.J. Stern, “Optimizing the synthesis of shaped beam antenna patterns,” Proc. IEE, vol 132-H, pp. 63-68, 1985
[2] M.J. Buckley, “Synthesis of shaped beam antenna patterns using implicitly constrained current elements,” IEEE AP Trans, vol. 44, no. 2, Feb 1996, pp 192-
197
[3] D. Boeringer and D. Werner, “Particle swarm optimization versus genetic algorithms for phased array synthesis,” IEEE AP Trans, vol. 52, no. 3, Mar 2004,
pp 771-779
[4] D. Simon, Evolutionary Optimization Algorithms: Biologically Inspired and Population-Based Approaches to Computer Intelligence. Hoboken, New Jersey,
Wiley, 2013
[5] J. I. Echeveste, M. A. Gonzalez de Aza, J. Rubio, and J. Zapata, “Near-Optimal Shaped-Beam Synthesis of Real and Coupled Antenna Arrays via 3-D-FEM
and Phase Retrieval,” IEEE AP Trans, vol. 64, no. 6, June 2016, pp 2189-2196
[6] R. Johansson, Numerical Python: A Practical Techniques Approach for Industry. Urayau, Chiba, Japan, Apress 2015
[7] www.wolfram.com
[8] H. Ruskeepaa, Mathematica Navigator: Mathematics, Statistics, and Graphics. Amsterdam, Academic Press (Elsevier) 2009
[9] www1.icsi.berkeley.edu/~storn/code.html
[10] W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes: The Art of Scientific Computing, Cambridge, U.K.: Cambridge Univ. Press, 1986
© Copyright September 2016 Michael J Buckley, LLC All Rights Reserved](https://image.slidesharecdn.com/slideshareexp1-161028194028/85/Synthesis-of-Linear-and-Non-Separable-Planar-Array-Patterns-33-320.jpg)
Synthesis of Linear and Non-Separable Planar Array Patterns
- 1. Synthesis of Linear and Non-Separable Planar Array csc2 Patterns Michael J. Buckley, Ph.D. Radiating Element Magnitude
- 2. Synthesis of Linear and Non-Separable Planar csc2 Far Field Patterns • Define the csc2 shaped beam synthesis problem (ground mapping radars and target seeking systems) • Convex and Non-Convex Optimization • Local Search Algorithm Linear Array Synthesis • Local Search Algorithm – Non-Separable Planar Array Synthesis • Linear Array Sensitivity Analysis
- 4. Non-Separable Planar Array Definition • Non-separable planar array – array distribution cannot be represented as the product of two linear array distributions • Example: a linear array csc2 shaped beam distribution in a non-separable planar array csc2 element_pattern(x,y)≠ element_pattern(x)i element_pattern(y)
- 5. Shaped Beam Patterns shaped beam region Orchard Local search technique Shaped Beam Pattern Parameters: Shaped Beam Region Sidelobe Region Pattern Ripple Shaped Beam vs. Pencil Beam Shaped beam element to element phase shift is not progressive Pencil beam element to element phase shift is progressive
- 6. Array Magnitude Distributions for Shaped Beam Patterns • Orchard magnitude distribution 1) element to element magnitude variation 2) large magnitudes at array edges • Traveling wave magnitude distribution 1) very smooth distribution 2) large magnitude at array edge (not suitable for phased arrays) • Local search algorithm 1) smooth distribution 2) works well in linear or non-separable planar arrays (center peak, low at array edges) Orchard et al. local search technique Traveling wave 1996 Radiating Element Magnitude
- 7. Array Phase Distributions for Shaped Beam Patterns
- 8. Convex versus Non-Convex Optimization • Convex function ‘a function is convex on an interval [a, b], if the values of the function on this interval lie below the line through the endpoints (a, f(a)) and (b, f(b))’ (“Numerical Python …”, R. Johansson) • Non-convex function – multiple local minima in an interval • Recently a paper in AP Transactions argued that shaped beam pattern synthesis is best treated as a convex optimization problem global minimum local minima global minimum with no local minima for -2 < x < 2 (a, f(a)) (b, f(b))
- 10. Non-Convex optimization - multiple solutions found, uniformly and non-uniformly spaced arrays - traveling wave and phased array solutions Non-Convex Optimization Results 15 Element Array Radiating Element Magnitude Non-Uniformly Spaced Array Radiating Element Magnitude Uniformly Spaced Array
- 11. Non-Convex Optimization Results Three Different Distributions • Three different far field pattern and pattern ripple plots shown • Pattern ripple non-convex optimization +/- .06 dB convex optimization +/- .5 dB • Multiple non-convex solutions are better than the convex solution Shaped beam synthesis is best treated as a non-convex optimization problem Non-Convex Result
- 12. Non-Convex Optimization Approaches • a two variable non-convex function • the function has a global minimum at (1.47,1.48) and multiple local minima • Three optimization techniques : 1) Simulated Annealing 2) Differential Evolution 3) the local search algorithm • Plot from Wolfram Mathematica 2 2 ( , ) 4sin( ) 6sin( ) ( 1) ( 1) f x y x y x y π π= + + − + − global minimum (1.47,1.48) local minima
- 13. Simulated Annealing starting point global minimum y = 1.47, y = 1.48 Simulated Annealing end point Simulated Annealing Algorithm • Stochastic global optimization algorithm (canned routine from Wolfram Mathematica) • Starting point is close to the global minimum, it converges to a local minimum • Note that the number of Simulated Annealing iterations would be problematic in a 50 variable case Simulated Annealing iteration Snapshots from Wolfram Mathematica optimization sequence starting point end point global minimum (1.47,1.48)
- 14. DE user supplied starting point end point global minimum Differential Evolution Algorithm • Stochastic global optimization algorithm (canned routine from Wolfram Mathematica) • Genetic type algorithm developed at UC Berkley • It does converge to the global minimum • Note that the number of Differential Evolution iterations would be problematic in a 50 variable case Snapshots from Wolfram Mathematica optimization sequence
- 15. Local Search Algorithm • Direct Search Algorithm • Minimal number of iterations • In order for this algorithm to work: -domain knowledge is required -the search volume must be constrained 0.5 1.0 1.5 2.0 2.5 1.0 1.5 2.0 2.5 3.0 x y Local Search Method Davidon- Fletcher- Powell Local Search Method Local Search Method Snapshots from Wolfram Mathematica optimization
- 20. 0.5 1.0 1.5 2.0 2.5 1.0 1.5 2.0 2.5 3.0 x y Local Search Method Davidon- Fletcher- Powell Snapshots from Wolfram Mathematica optimization Local Search Algorithm Non-Convex Optimization Powell’s Direction Set
- 21. • 26 element uniformly spaced (.7λ) linear array • Compared to a result published by Orchard et al. • Orchard et al. write the array pattern with no element pattern in order to optimize using the Schelkunoff unit circle ( ) ( ) 26 ( (.7 )sin ) 1 nj nk n n f i e λ θ ϕ θ − = = ∑ Apply Local Search Algorithm to Linear Array csc2 Shaped Beam Synthesis θ element in a 26 element linear array
- 25. • Local search algorithm has less pattern ripple than Orchard et al. results • Local search algorithm array distributions – less element to element magnitude and phase variation with radiating element magnitude maximum in the interior of the array Local Search Algorithm vs. Orchard et al. csc2 Linear Array Synthesis Orchard Local Search Algorithm
- 26. Convergence of Local Search Algorithm • Convergence is rapid for the local search algorithm • For the 26 element csc2 example the algorithm converges in 15 iterations • An iteration is defined as one optimization step, either quasi- Newton or Powell’s Direction Set Method quasi-Newton Reset Solution Space Powell’s Direction Set 1st iteration 2nd iteration csc2 initial
- 27. Linear Array Sensitivity Analysis • optimum cost function distribution • kth radiating element magnitude +10 % calculate cost function magnitude -10 % calculate cost function phase +10 % calculate cost function phase -10 % calculate cost function • do for all radiating elements • Shaped beam pattern holds up well end point cost function = 0 element 15 phase + 10 % cost function = 1380. element 15 phase + 10 % cost function = 1380.
- 29. Local Search Algorithm Distribution in a Non-Separable Array csc2 pattern along the x axis Low sidelobe pattern along the y axis •Small amplitudes at edges of array minimize the non-separable distribution - low side lobes - small pattern ripple •This was a 558 variable problem requiring minimal optimization, the optimization starting point was close to the goal end point •This was a non-convex problem, since the starting point was close to the endpoint, it converged to a good solution. element magnitude = .19 element magnitude = 1.0
- 30. Notes on Non-separable Array Synthesis • Low sidelobe plane cuts look good • Convergence was quite rapid – one iteration (47 line searches) • Array has larger amplitudes in the interior of the array than on the edge of the array (see previous slide). Interior array elements have better element patterns and higher gain than edge elements.
- 31. Summary 0.5 1.0 1.5 2.0 2.5 1.0 1.5 2.0 2.5 3.0 x y Local Search Method Davidon- Fletcher- Powell Starting point End point Local Search Algorithm Differential Evolution global minimum local minima
- 32. Michael J. Buckley, LLC focuses on the design and testing of antennas, manifolds, and radomes and on planar array synthesis, including shaped beam synthesis for non-separable planar arrays. Mike Buckley developed higher order Floquet mode scattering radiating elements to address the packaging, cost, and performance requirements of low cost AESA systems, antenna radome integration, and small array systems. He also developed local search algorithm techniques for large variable non-convex shaped beam synthesis problems. He previously worked at Rockwell Collins, Northrop-Grumman, Lockheed- Martin, and Texas Instruments. He has numerous patents and publications. He has a Ph.D. in electrical engineering and is a member of Phi Beta Kappa. HFSS Floquet Modes increasing number of modes
- 33. References [1] H.J. Orchard, R.S. Elliott, and G.J. Stern, “Optimizing the synthesis of shaped beam antenna patterns,” Proc. IEE, vol 132-H, pp. 63-68, 1985 [2] M.J. Buckley, “Synthesis of shaped beam antenna patterns using implicitly constrained current elements,” IEEE AP Trans, vol. 44, no. 2, Feb 1996, pp 192- 197 [3] D. Boeringer and D. Werner, “Particle swarm optimization versus genetic algorithms for phased array synthesis,” IEEE AP Trans, vol. 52, no. 3, Mar 2004, pp 771-779 [4] D. Simon, Evolutionary Optimization Algorithms: Biologically Inspired and Population-Based Approaches to Computer Intelligence. Hoboken, New Jersey, Wiley, 2013 [5] J. I. Echeveste, M. A. Gonzalez de Aza, J. Rubio, and J. Zapata, “Near-Optimal Shaped-Beam Synthesis of Real and Coupled Antenna Arrays via 3-D-FEM and Phase Retrieval,” IEEE AP Trans, vol. 64, no. 6, June 2016, pp 2189-2196 [6] R. Johansson, Numerical Python: A Practical Techniques Approach for Industry. Urayau, Chiba, Japan, Apress 2015 [7] www.wolfram.com [8] H. Ruskeepaa, Mathematica Navigator: Mathematics, Statistics, and Graphics. Amsterdam, Academic Press (Elsevier) 2009 [9] www1.icsi.berkeley.edu/~storn/code.html [10] W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes: The Art of Scientific Computing, Cambridge, U.K.: Cambridge Univ. Press, 1986 © Copyright September 2016 Michael J Buckley, LLC All Rights Reserved