Synthesis of Linear and Non-Separable Planar Array csc2 Patterns

 Michael J. Buckley, Ph.D.

Radiating Element Magnitude

Synthesis of Linear and Non-Separable Planarcsc2 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

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

Shaped Beam Patterns shaped beam region

Orchard

Local search technique

Shaped Beam Pattern Parameters:Shaped Beam RegionSidelobe RegionPattern Ripple

Shaped Beam vs. Pencil BeamShaped beam element to element

phase shift is not progressive Pencil beam element to element phase shift is progressive

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 distribution2) large magnitude at array edge (not suitable for phased arrays)

• Local search algorithm 1) smooth distribution2) works well in linear or non-separable planar arrays (center peak, low at array edges)

Orchard et al.

local search techniqueTraveling wave

1996

Radiating Element Magnitude

Array Phase Distributions for Shaped Beam Patterns

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

localminima

global minimum with no local minima for -2 < x < 2

(a, f(a))

(b, f(b))

Non-Convex optimization - multiple solutions found, uniformly and non-uniformly spaced arrays - traveling wave and phased array solutions

Non-Convex Optimization Results 15 Element ArrayRadiating Element MagnitudeNon-Uniformly Spaced Array

Radiating Element MagnitudeUniformly Spaced Array

Non-Convex Optimization ResultsThree 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

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

22

( , ) 4sin( ) 6sin( )

( 1) ( 1)

f x y x y

x y

global minimum (1.47,1.48)

local minima

Simulated Annealing starting point

global minimumy = 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 Annealingiteration

Snapshots from Wolfram Mathematicaoptimization sequence

starting point

end point

global minimum (1.47,1.48)

DE user suppliedstarting point

end point globalminimum

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 Mathematicaoptimization sequence

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

Local Search Method

Local Search Method

Snapshots from Wolfram Mathematica optimization

Snapshots from Wolfram Mathematica optimization

Local Search AlgorithmNon-Convex Optimization

Powell’s Direction Set

• 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 nkn

n

f i e

Apply Local Search Algorithm to Linear Array csc2 Shaped Beam Synthesis θ

element in a 26 element linear array

• 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

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

Linear Array Sensitivity Analysis• optimum cost function distribution • kth radiating element

magnitude +10 % calculate cost functionmagnitude -10 % calculate cost functionphase +10 % calculate cost function phase -10 % calculate cost function

• do for all radiating elements

• Shaped beam pattern holdsup well

end pointcost function = 0

element 15 phase + 10 %cost function = 1380.

element 15 phase + 10 %cost function = 1380.

Local Search Algorithm Distribution in a Non-Separable Array

csc2 pattern alongthe x axis

Low sidelobe patternalong 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

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.

Summary

Starting point

End point

Local Search Algorithm

Differential Evolutionglobal minimum

localminima

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

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

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