Learning Objectives

  • Understand the Woodward-Lawson sampling method for antenna pattern synthesis
  • Learn to decompose desired patterns into orthogonal uniform-array beams
  • Apply pattern sampling techniques to determine array excitation coefficients
  • Analyze the relationship between sample points and array parameters (N, d, λ)
  • Evaluate pattern accuracy and limitations of the sampling approach
  • Compare Woodward-Lawson with Fourier Transform and other synthesis methods
Key Concept: Woodward-Lawson method synthesizes patterns by sampling the desired pattern at specific angular locations and superimposing orthogonal uniform-array beams centered at these samples.

Prerequisites

  • Linear array theory and array factor calculation
  • Understanding of orthogonal functions and superposition
  • Complex numbers and phasor notation
  • Uniform array pattern characteristics (sinNx/x type)

Learning Outcomes

  • Calculate sample point locations for given array parameters
  • Determine excitation coefficients from pattern samples
  • Synthesize sector, flat-top, and shaped patterns
  • Assess pattern accuracy and Gibbs phenomenon effects
  • Optimize array size for specified pattern requirements

Woodward-Lawson Method - Theory

1. Fundamental Principle

The Woodward-Lawson method synthesizes a desired pattern by sampling it at discrete angular locations and superimposing orthogonal uniform-array beams centered at these sample points [^6^].

2. Composing Functions

Each sample is realized using a uniform array pattern (sinNx/x type) called a composing function:

AFm(θ) = sin[Nπd/λ(cosθ - cosθm)] / [N sin[πd/λ(cosθ - cosθm)]]

3. Sample Point Locations

For an N-element linear array with spacing d, sample locations are determined by:

cosθm = m·λ/(Nd)    (for odd N, m = 0, ±1, ±2,..., ±(N-1)/2)
cosθm = (m ± 0.5)·λ/(Nd)    (for even N)

The sample points are separated by Δcosθ = λ/(Nd) = λ/L in u-space, where L = Nd is the array length [^8^].

4. Total Array Factor

The synthesized pattern is the weighted sum of all composing functions:

AF(θ) = Σm=-MM bm · AFm(θ)
where bm = SFdesired(θm) (sample values at sample points)

5. Excitation Coefficients

The element excitation coefficients are given by:

an = (1/N) Σm=-MM bm · e-j·k·d·n·cosθm

where n = ±1, ±2,..., ±(N-1)/2 for odd N [^6^].

6. Orthogonal Beam Property

Woodward-Lawson beams are orthogonal, enabling lossless network implementation and independent control of pattern at each sample point [^8^].

Important Limitation: The method works best for patterns that can be approximated by superposition of uniform array patterns. Sharp transitions may cause Gibbs phenomenon-like oscillations.

Comparison with Other Methods

  • Advantages: Simple implementation, local pattern control, direct sample-to-coefficient mapping
  • Disadvantages: Ripple in synthesized pattern, less efficient than iterative methods for large arrays [^7^]
  • vs Fourier: Woodward-Lawson samples pattern in angle domain; Fourier samples aperture distribution

Experimental Procedure

Part 1: Sector Pattern Synthesis

Step 1: Specify Array Parameters

  • Set number of elements N (odd: 7-25, even: 8-26)
  • Choose element spacing d (typically 0.5λ to avoid grating lobes)
  • Calculate array length L = N·d
  • Verify sample spacing Δcosθ = λ/(Nd) covers visible region

Step 2: Calculate Sample Points

  • For odd N: Calculate cosθm = m·λ/(Nd) for m = 0, ±1,..., ±(N-1)/2
  • For even N: Use cosθm = (m+0.5)·λ/(Nd) scheme
  • Convert to angles: θm = arccos(cosθm) in degrees
  • Identify samples within visible region [0°, 180°]

Step 3: Determine Sample Values

  • Define desired pattern function SFdesired(θ)
  • Evaluate pattern at each sample point: bm = SFdesired(θm)
  • For sector patterns: bm = 1 within sector, 0 outside
  • For flat-top patterns: Use amplitude weighting bm = f(θm)

Step 4: Compute Excitation Coefficients

  • Use formula: an = (1/N) Σ bme-jkdn·cosθm
  • Calculate for all elements n = 1 to N
  • Normalize coefficients to maximum value
  • Record amplitude and phase of each coefficient

Step 5: Synthesize and Analyze Pattern

  • Compute AF(θ) = Σ bm·AFm(θ) for 0° ≤ θ ≤ 180°
  • Use fine angular resolution (Δθ ≤ 0.5°)
  • Plot desired vs synthesized patterns
  • Calculate performance metrics (ripple, beamwidth, sidelobes)

Part 2: Design Studies

  • Effect of Array Size: Vary N while keeping L constant
  • Spacing Impact: Investigate d > λ/2 on pattern quality
  • Pattern Complexity: Try sector, cosecant-squared, and custom shapes
  • Even vs Odd N: Compare symmetry properties
Analysis Tip: Pay attention to ripple amplitude and period in synthesized patterns. Compare with Fourier Transform method results for same specifications.

Performance Metrics

  • Pattern Ripple: Δ = (max - min) within desired sector
  • Mean Squared Error: MSE = ∫|SFdesired - AF|2dθ
  • Transition Width: Angle between 10% and 90% of desired level
  • Sidelobe Level: Peak sidelobe outside main beam region

Woodward-Lawson Synthesis Laboratory

Pattern Definition

Array Configuration

Pattern Sampling

Calculating...
Synthesizing pattern...
Current Algorithm: Woodward-Lawson sampling with orthogonal beam superposition

Student Report Guidelines

1. Title & Abstract

  • Title: "Antenna Pattern Synthesis Using Woodward-Lawson Method"
  • Abstract: 150-200 words summarizing objectives, method, and key findings
  • Include array parameters (N, d) and pattern type
  • Keywords: Woodward-Lawson, pattern synthesis, array sampling

2. Theoretical Background

  • Derive Woodward-Lawson sample point equations
  • Explain orthogonal beam property and its importance
  • Compare with Fourier Transform and Schelkunoff methods
  • Discuss limitations: ripple, transition width, sidelobes
  • Include equations for composing functions and coefficients

3. Design Procedure

  • Document array specifications (N, d, λ)
  • Show sample point calculation for your configuration
  • Present desired pattern function and sample values bm
  • Include MATLAB/Python code or describe virtual lab settings
  • Calculate excitation coefficients an (show sample calculations)

4. Results & Analysis

  • Plot desired pattern with sample points marked
  • Plot synthesized pattern superimposed on desired
  • Plot excitation coefficient distribution (amplitude/phase)
  • Include polar pattern visualization
  • Calculate and tabulate performance metrics
  • Show sample point table: m, θm, cosθm, bm

5. Parametric Studies

  • Vary N (keep L constant): Analyze ripple vs N trade-off
  • Vary N (keep d constant): Show beamwidth control
  • Compare even vs odd N: Discuss symmetry differences
  • Effect of d > λ/2: Grating lobe appearance
  • Pattern complexity limits: Sharp edges vs smooth transitions

6. Discussion

  • Explain ripples: Cause (Gibbs phenomenon) and mitigation
  • Compare with Fourier Transform method results
  • Discuss practical implementation: Feed network complexity
  • Limitations: When does method fail? Suggest alternatives
  • Optimization: Could sample weighting improve results?

7. Conclusion

  • Summarize key findings about Woodward-Lawson performance
  • State design rules: Minimum N for given pattern accuracy
  • Recommendations: Best applications for this method
  • Future work: Hybrid methods combining WL with optimization

8. Appendices

  • Complete code (MATLAB/Python/JavaScript)
  • Detailed calculation sheets
  • Additional plots for all parametric studies
  • References: Textbooks, papers (cite properly)
Important: Ensure all plots are clearly labeled with N, d, pattern type. Include sample point locations marked on desired pattern plots.

Grading Criteria

  • Theory (20%): Correct equations and understanding of orthogonal beams
  • Calculation (25%): Accurate sample points and coefficients
  • Visualization (20%): Clear plots with proper labeling
  • Analysis (25%): Insightful discussion of results and limitations
  • Presentation (10%): Organization, clarity, and completeness

Desired Pattern with Sample Points

● Desired Pattern | ● Synthesized Pattern | ▼ Sample Points

Excitation Coefficients

Polar Radiation Pattern

Implementation Notes:
  • Sample points automatically calculated based on N and d
  • Only samples in visible region (0°-180°) are used
  • Uniform array patterns used as composing functions
  • Excitation coefficients include amplitude and phase
  • Pattern calculated at 0.5° intervals for smooth curves
Design Constraints:
  • Minimum spacing: d ≥ 0.25λ for practical implementation
  • Visible region: |cosθ| ≤ 1 limits usable samples
  • Ripple period: Related to array length L = N·d
  • Transition width: Inversely proportional to array size