35
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www.ATIcourses.com
Boost Your Skills with On-Site Courses Tailored to Your Needs The Applied Technology Institute specializes in training programs for technical professionals. Our courses keep you current in the state-of-the-art technology that is essential to keep your company on the cutting edge in today’s highly competitive marketplace. Since 1984, ATI has earned the trust of training departments nationwide, and has presented on-site training at the major Navy, Air Force and NASA centers, and for a large number of contractors. Our training increases effectiveness and productivity. Learn from the proven best. For a Free On-Site Quote Visit Us At: http://www.ATIcourses.com/free_onsite_quote.asp For Our Current Public Course Schedule Go To: http://www.ATIcourses.com/schedule.htm
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2
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Global Optimization
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.
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.
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3
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Optimization Methods
Global
Random Walk
Particle Swarm
Genetic Algorithms
Methods
Local
Conjugate Gradient Methods
Simplex
Quasi-Newton Methods
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4
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Optimization Methods
Conjugate Gradient
Random Walk
Genetic Algorithm
Global Optimization Poor Fair Good
Discontinuous Functions Poor Good Good
Non-differentiable Functions
Poor Good Good
Convergence Rate Good Poor Fair
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5
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Genetic Algorithms
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.
Economics, Ecology, Social Systems, Machine Learning, Chemistry, Physics, etc.
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6
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Terminology
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.
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7
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Chromosome Coding
GAs operate on a coding of the parameters, instead of the parameters themselves.
In binary coding, the parameters are each represented by a finite-length binary string.
Chromosomes are the combination of all the encoded parameters. (A string of ones and zeros)
Binary coding yields very simple binary operators.
0101 1001 1101 1010 0001 0011
R1 L1 C1 R2 L2 C2
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8
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Genetic Algorithm
Initialize Population
Selection of Parents
CrossOver and Mutation
Temp Population Full?
Replace Population
Termination Criteria Met?
End
Evaluate Fitness
Evaluate Fitness
NoNo
Yes
Yes
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9
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Initialize Population
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.
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10
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Parent Selection
Proportionate selection – Probability of selecting an individual is a function of the individual’s relative fitness.
123456789
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11
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Parent Selection
Tournament selection – N individuals are selected at random, the individual with the highest fitness in the sub population is selected.
PopulationN
randomly selectedchromosomes
Parent =Chromosome
with best Fitness
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12
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Crossover and Mutation
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.
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13
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Crossover and Mutation
Single Point Crossover
Mutation
a1 a2 a3 a4 a5Parent 1 Parent 2b1 b2 b3 b4 b5
a1 a2 b3 b4 b5 b1 b2 a3 a4 a5Child 1 Child 2
a1 a2 a3 a4 a5 a6 a7
a1 a2 A3 a4 a5 a6 a7
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14
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Population Replacement
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.
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15
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Fitness Function
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.
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16
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
GA Optimization Guidelines
Population Size: Typically 30 – 100 Large populations enable faster convergence by providing more genetic
diversity. Smaller populations yield faster execution, especially for complicated fitness functions.
Probability of Crossover: Typically 0.6 – 0.9 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 0.01 – 0.1 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.
Replacement Strategy: Generational vs. Steady-State Steady-state generally converges faster. Lower values of replacement
percentage usually converge faster.
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17
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Particle Swarm
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.
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18
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Harmonious Flight
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.
National Geographic 2007
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19
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Particle Swarm
Initialize Swarm
Update Velocities (Vn)
Update Positions (Xn)
Termination Criteria Met?
End
Evaluate FitnessNo
Yes
Evaluate Fitness
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20
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Initialize Swarm
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.
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21
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Update Velocities
Update the velocity of each particle toward the local and global best position.
Limit the velocity if necessary.
( )( )nnbestglobal
nnbestlocalnn
xxrandxxrandvv
−⋅⋅+
−⋅⋅+⋅=
,2
,1
κ
κω
maxmax , vvvvthenvvif
n
nnn
=>
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22
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Update Positions
Update position using unit acceleration.
Clip position if necessary.
nnn vxx +=
ddnddn
ddnddn
xxthenxxif
xxthenxxif
min,,min,,
max,,max,,
,
,
=<
=>
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23
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Particle Swarm Guidelines
(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.
(Memory & Cooperation) – Can be tuned for the particular problem. Common practice in literature to set both equal in the range 1 –2.
ω
21, κκATI Mat
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24
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
MATLAB Example
Find the minimum of the following function.
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25
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
MATLAB Example
Find the minimum of the follow function.
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26
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Antenna Pattern
Suppose we want to minimize the antenna gain in a particular direction due to an interfering source (Adaptive Nulling).
[ ]φθφθλπα sinsincossin2
mnmnmn yx +=
∑∑= =
=N
n
M
m
jjmn
mnmn eeIAF1 1
),( αβφθ
=mnI Amplitude coefficient for each element
=mnβ Phase shift for each element
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27
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Two Interfering Sources
16 x 16 element planar array
6 bit phase shifters, 3 bits used for nulling
2 interfering sources located at (θ = 18o, φ = 0o) and (θ = 26o, φ = 90o)
50 Chromosomes / Particles
200 Iterations
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28
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Two Interfering Sources
Location ofInterfering
Sources
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29
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Two Interfering Sources
Particle SwarmGenetic Algorithm
Nulls Placed in the Antenna Pattern in the Direction of the Interfering Sources
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30
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Two Interfering Sources
Interfering Source
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31
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Two Interfering Sources
Interfering Source
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32
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Two Interfering Sources
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33
Radar Signal Analysis and Processing with MATLAB ♦ Applied Technology Institute
Two Interfering Sources
Main Beam Loss 1.02 dB (Genetic Algorithm) 1.63 dB (Particle Swarm)
Beamwidth
Original GA PS
Φ = 0o
3 dB 6.29o 6.30o 6.35o
10 dB 10.48o 10.50o 10.60o
Φ = 90o
3 dB 6.29o 6.30o 6.35o
10 dB 10.48o 10.52o 10.59o
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