Devising High-Performing Pseudo-Random Spreading Codes Using Genetic Algorithms

Tara Yasmin Mina and Grace Xingxin Gao

ION GNSS+ 2019

• Interest by Air Force to develop the future generation of GPS▪ 2016 RFI (NTS): Reinvigorate Navig. Tech. Satellite from 1970s initiative [1]

▪ 2019 SS (HoPS): Leverage multi-orbit commercial sats., reprogrammable payload [2]

• NTS-3 – objectives include exploring new techniques to: ▪ Enhance PNT resiliency / performance

▪ Increase number of signals broadcast on L1 frequency band

▪ Explore modifications to all signal layers (carrier, data, spreading codes)

• New era of satellite navigation – time to revisit design of GPS PRN codes

Air Force Initiatives to Modernize GPS

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[1] AFRL, FBO, 2016 (SN: RFI-RVKVE-NTS-3)

[2] Air Force SMC, FBO, 2019(SN: FA881414D0001)

[3] Lutz, AFRL SV Directorat [3]

NTS-1(1974)

NTS-2(1977)

NTS-3(Launch in 2022)

• Current GPS spreading codes algorithmically generable▪ i.e. Legacy GPS L1 C/A uses Gold codes [4] – utilizes two 10-bit LSFRs

▪ i.e. L1C uses Weil codes, with 7-bit pad [5]

• 1970s design considerations▪ Hardware memory limitations

▪ Limited to LFSR-based codes

• Advantages of memory codes:1. Greater range of possible families

2. Opportunity to find superior codes

3. Any desired seq. length possible

• Large space: complicates code design method

Use of Memory Codes for GNSS

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[6]

[4] Gold, ION GNSS, 1967 [5] Rushanan, IEEE ISIT, 2006[6] ICD-GPS-200C, DoD, 1993

• Genetic algorithms (GAs) – optimization / learning technique:▪ Population-based, candidate solutions represented as binary sequences

▪ Iteratively improve solution: choose / combine highly fit individuals

• Prior work utilizing GAs for Galileo application▪ Develop codes with ASZ property (0 auto-correlation at ±1 chip delays) [7]

▪ Parameters for selecting high-quality codes for GNSS applications [8] [9]

Genetic Algorithms for Random Codes

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[7]

[7] Wallner, Avila-Rodriguez & Hein, ION GNSS, 2007 [8] Soualle, et al, European GNSS, 2005[9] Winkel, US Patent No. 8.035.555, 2011

1. Utilizing a cumulative, combined cost function▪ 1-dimensional cost function to minimize [8]

▪ Ex: summation of costs to minimize for spreading codes [9]

▪ Must define relative priorities between objectives

▪ Cannot explore across complete multi-dimensional cost function space

2. Multi-dimensional cost▪ Progress non-dominated front

▪ Population-based methods useful

▪ Continuously improve local front

▪ Must maintain solution diversity:

o Avoid crowding of solutions

o Spread of Pareto-optimal points

Multi-Objective Techniques

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[8] Soualle, et al, European GNSS, 2005[9] Winkel, US Patent No. 8.035.555, 2011

• Develop a multi-objective optimization platform to construct high-quality families of spreading code sequences

• GA architecture – key features:

▪ Perform scattered crossover: increase recombination between parent chromosomes

▪ Incorporate Pareto-optimal elitism: ensure strong solutions maintained in population

▪ Ranking via non-dominated sorting for selecting high-performing individuals

▪ Utilize fitness sharing via niching: maintain well-spread range of Pareto-optimal points

• Demonstrate ability of our genetic algorithm to devise a set of memory code sequences which can:

▪ Achieve low auto- and cross-correlation side peaks

▪ Perform better than well-chosen families of equal-length Gold and Weil codes

Key Contributions

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• High-level Overview of GAs

• Design of GA Architecture

▪ Key Objectives for High-Quality Codes

▪ Fitness Ranking via Non-Dominated Sorting

▪ Increase Solution Diversity using Niching

▪ Multi-Objective Genetic Algorithm Architecture

• Experimental Validation

• Summary

Outline

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• Initially proposed by John Holland; currently many variations

• Inspired by Darwinian theory of evolution▪ Evolve population of solutions, improving each subsequent generation

▪ Utilizes notions of crossover to “breed” between highly fit individuals

Genetic Algorithms [10]

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[11]

[10] Holland, Univ. of Michigan Press, 1975[11] Khan, Genetic Linkage & Mapping, 2019

Biological crossover of chromosomes:

Key Steps:

1. Create initial population

2. Evaluate individual fitness

3. Select high-performing individuals

4. Form offspring solutions:

➢ Crossover / Recombination

➢ Mutation

5. Propagate next generation (return to step 2)

Basic / Standard Genetic Algorithm

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• High-level Approach using GAs

• Design of GA Architecture

▪ Key Objectives for High-Quality Codes

▪ Evaluate Unbiased Fitness via Non-Dominated Sorting

▪ Increase Solution Diversity using Fitness Sharing

▪ Multi-Objective Genetic Algorithm Architecture

• Experimental Validation

• Summary

Outline

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1. Mean, normalized absolute auto-correlation side peak

2. Mean, normalized absolute cross-correlation peak

Key Objectives Utilized

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Auto-Correlation Side Peaks Cross-Correlation Peaks

Peak correlation

• Ranking utilizing Non-Dominated Sorting, or Pareto ranking▪ Pareto ranking – ranking via layers of dominating points [12]

▪ Define layers of non-dominated fronts to provide scalar fitness score

▪ Probability of selection: proportional to final fitness score

• Fitness score evaluation:1. Compute multi-dimensional cost

2. Find layers of non-dominated fronts:

𝑃1 , 𝑃2 , 𝑃3 , …

3. Assign Indiv. rank 𝑟𝑝 via layer order:

𝑝 ∈ 𝑃𝑖 ⇒ 𝑟𝑝 = 𝑖

4. Compute unbiased fitness score:

𝑓𝑝 =1

𝑟𝑝

Ranking to Evaluate Fitness of Points

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[12] Goldberg, Addison-Wesley, 1989

• Fitness sharing – maintains diversity of Pareto-optimal points▪ Via computing niche count 𝜂 – degree of crowding in neighboring area [13]

▪ Parameter 𝜎 chosen for defining neighboring area of consideration

• Key steps for fitness sharing:1. Compute multi-dimensional cost

2. Normalize cost (0 to 1)

3. Compute pairwise distances 𝑑𝑖𝑗4. For each candidate 𝑖,

sum scaled distances within 𝜎:

𝜂𝑖 =

∀ 𝑗

max 1 −𝑑𝑖𝑗

𝜎, 0

5. Bias fitness score by niche count:

𝑓𝑖′ =

𝑓𝑖

𝜂𝑖

Use of Fitness Sharing via Niching

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[13] Fonseca & Fleming, ICGA, 1993

• Recombination step: scattered crossover▪ Uniformly, randomly select multiple crossover points b/w 2 individuals

▪ Initial observations: improved speed and performance of algorithm

Incorporate Scattered Crossover / Pareto-Optimal Elitism

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• Pareto-optimal elitism: ▪ Directly pass high-performing individuals to next generation of solutions

▪ Elite points: set of dominating, Pareto-optimal points

▪ Ensures continuous improvement, preserves best solutions

Multi-Objective GA Architecture

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Initialize Population

Evaluate Raw Cost Function Vector

Determine Pareto-Optimal Points

Create Next Generation

Randomly Select High-Performing Candidates

Perform Scattered Crossover

Mutation

Evaluate Niche Counts

Max Generation Reached?

No

Yes

Terminate Process

Perform Non-Dominated Sorting

Bias / Re-Scale Fitness

• High-level Approach using GAs

• Design of GA Architecture

▪ Key Objectives for High-Quality Codes

▪ Evaluate Unbiased Fitness via Non-Dominated Sorting

▪ Increase Solution Diversity using Fitness Sharing

▪ Multi-Objective Genetic Algorithm Architecture

• Experimental Validation

• Summary

Outline

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Objective: validate results with equal-length Gold / Weil codes

▪ Validation of GA: comparing with 31-length Gold codes [4] and Weil codes [5]

▪ Trade-off between computation requirements and sequence length

▪ Total (3 codes, length-31): 2(3 ⋅ 31) ∼ 1027 possible solutions

▪ Gold char. polynomials: 𝒎𝟏 = 𝟏 + 𝒙𝟑 + 𝒙𝟓 ; 𝒎𝟐 = 𝟏 + 𝒙𝟏 + 𝒙𝟐 + 𝒙𝟑 + 𝒙𝟓 [14]

▪ Various family sizes tested, compare with best set of Gold / Weil codes

Experimental Validation

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[4] Gold, IEEE TIT, 1967[5] Rushanan, IEEE ISIT, 2006[14] Misra & Enge, Ganga-Jamuna Press, 2011

Optimization Params. Values

Mutation rate 10−5

Population size 300

Max Elite Members 150

Max generation 10,000

Niche radius (𝜎) 0.1

Families of 3 and 5 Sequences

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Better Performing Sequences than Gold / Weil Codes Achieved with GA

Families of 7 and 10 Sequences

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Better Performing Sequences than Gold / Weil Codes Achieved with GA

Families of 20 Sequences

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Better Performing Sequences than Gold / Weil Codes Achieved with GA

• Developed multi-objective GA architecture to devise family of high-performing code sequences

▪ Achieves low mean, circular auto- and cross-correlations

▪ Produces range of Pareto-optimal points

▪ Incorporates the following key features and multi-objective techniques:

o Scattered crossover for recombination / Pareto-optimal elitism

o Ranking via non-dominated sorting

o Fitness sharing via niching

• Demonstrated algorithm constructs higher quality code families, compared to best combinations of current, equal-length codes

▪ Range of code solutions dominate Pareto-optimal Gold / Weil codes

▪ Next Steps: Expand architecture for GPS-length code sequences

Summary

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Thank You!

This material is based upon work supported by Kirtland Air Force Research Lab (AFRL).