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.

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
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