synthesis of linear and non-separable planar array patterns
TRANSCRIPT
![Page 1: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/1.jpg)
Synthesis of Linear and Non-Separable Planar Array csc2 Patterns
Michael J. Buckley, Ph.D.
Radiating Element Magnitude
![Page 2: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/2.jpg)
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
![Page 3: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/3.jpg)
![Page 4: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/4.jpg)
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
![Page 5: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/5.jpg)
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
![Page 6: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/6.jpg)
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
![Page 7: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/7.jpg)
Array Phase Distributions for Shaped Beam Patterns
![Page 8: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/8.jpg)
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))
![Page 9: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/9.jpg)
![Page 10: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/10.jpg)
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
![Page 11: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/11.jpg)
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
![Page 12: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/12.jpg)
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
![Page 13: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/13.jpg)
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)
![Page 14: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/14.jpg)
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
![Page 15: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/15.jpg)
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
![Page 16: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/16.jpg)
![Page 17: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/17.jpg)
![Page 18: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/18.jpg)
![Page 19: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/19.jpg)
![Page 20: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/20.jpg)
Snapshots from Wolfram Mathematica optimization
Local Search AlgorithmNon-Convex Optimization
Powell’s Direction Set
![Page 21: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/21.jpg)
• 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
![Page 22: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/22.jpg)
![Page 23: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/23.jpg)
![Page 24: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/24.jpg)
![Page 25: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/25.jpg)
• 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
![Page 26: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/26.jpg)
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
![Page 27: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/27.jpg)
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.
![Page 28: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/28.jpg)
![Page 29: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/29.jpg)
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
![Page 30: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/30.jpg)
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.
![Page 31: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/31.jpg)
Summary
Starting point
End point
Local Search Algorithm
Differential Evolutionglobal minimum
localminima
![Page 32: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/32.jpg)
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
![Page 33: Synthesis of Linear and Non-Separable Planar Array Patterns](https://reader035.vdocuments.us/reader035/viewer/2022081421/587a91171a28ab58288b7093/html5/thumbnails/33.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