Learning Objectives

  • Understand the fundamental relationship between aperture distribution and far-field radiation pattern via Fourier Transform
  • Learn to synthesize desired antenna patterns using Fourier Transform Method
  • Apply sampling techniques to convert continuous source distributions to discrete array excitations
  • Analyze trade-offs between desired pattern and practical implementation constraints
  • Compare theoretical patterns with synthesized approximations
  • Develop skills in array factor calculation and pattern visualization
Key Concept: The far-field radiation pattern of an antenna is the Fourier Transform of its current distribution. This relationship forms the basis for pattern synthesis.

Prerequisites

  • Basic understanding of antenna fundamentals
  • Fourier Transform theory
  • Linear array theory and array factor
  • Complex numbers and phasor notation

Learning Outcomes

  • Calculate excitation coefficients for desired radiation patterns
  • Implement Fourier Transform synthesis for sector patterns
  • Evaluate pattern accuracy using mean squared error
  • Determine required array size for specified beamwidth

Fourier Transform Method - Theory

1. Continuous Line Source

For a continuous line source of length l positioned along the z-axis, the normalized space factor is given by:

SF(θ) = ∫-l/2l/2 I(z')ejkz'cosθdz'

where I(z') is the current distribution and k = 2π/λ is the wavenumber.

2. Inverse Relationship

The current distribution can be obtained from the desired pattern using the inverse Fourier Transform:

I(z') = (1/2π) ∫-∞∞ SF(θ)e-jkz'cosθdθ

3. Discrete Array Approximation

For an N-element linear array with uniform spacing d, the array factor is:

AF(θ) = Σn=1N Inej(n-1)kd cosθ

4. Sampling Theorem

The continuous current distribution is sampled at element positions to obtain discrete excitation coefficients:

In = I(z')|z'=(n-1)d - (N-1)d/2

5. Visible Region

Only patterns within the visible region (0° ≤ θ ≤ 180°) can be physically realized. The invisible region corresponds to evanescent waves.

Key Insight: The Fourier Transform Method provides the optimal least-mean-square approximation to the desired pattern for a given aperture size.

Example Patterns

  • Sector Pattern: SF(θ) = 1 for θ₁ ≤ θ ≤ θ₂, 0 otherwise
  • Cosine Pattern: SF(θ) = cosⁿ(θ) for beam shaping
  • Flat-top Pattern: Provides uniform illumination over sector

Experimental Procedure

Part 1: Sector Pattern Synthesis

Step 1: Define Desired Pattern

  • Set sector boundaries: θstart and θend (e.g., 60° to 120°)
  • Choose array length l in wavelengths (try 5λ and 10λ)
  • Calculate required number of elements for given spacing

Step 2: Compute Current Distribution

  • Use inverse Fourier Transform to calculate I(z')
  • Apply windowing if needed to reduce Gibbs phenomenon
  • Normalize current distribution to maximum value

Step 3: Discretize for Array

  • Sample continuous distribution at element positions
  • For N elements, positions: zn = (n - (N+1)/2)d
  • Record excitation coefficients In

Step 4: Calculate Array Factor

  • Compute AF(θ) for 0° ≤ θ ≤ 180°
  • Use fine angular resolution (Δθ ≤ 1°)
  • Convert to dB scale: AFdB = 20log₁₀|AF(θ)|

Part 2: Analysis Tasks

  • Vary array length l and observe pattern accuracy
  • Change number of elements N for fixed length
  • Investigate effect of element spacing d > λ/2
  • Compare different window functions
Important: When sampling the continuous distribution, ensure spacing d ≤ λ/2 to avoid grating lobes in visible region.

Performance Metrics

  • Mean Squared Error: MSE = ∫|SFdesired - SFsynthesized|²dθ
  • Half-Power Beamwidth: Angular width at -3dB points
  • Sidelobe Level: Maximum sidelobe amplitude relative to main beam
  • Efficiency: Ratio of power in main beam to total radiated power

Fourier Transform Synthesis Laboratory

Pattern Parameters

Array Configuration

Calculating...

Student Report Guidelines

1. Title & Abstract

  • Descriptive title including method and objectives
  • 150-200 word abstract summarizing goals, approach, and key findings
  • Keywords: Fourier Transform, antenna synthesis, array factor

2. Theoretical Background

  • Explain Fourier transform relationship between aperture and pattern
  • Derive equations for continuous line source
  • Discuss sampling theorem for discrete arrays
  • Explain windowing techniques and Gibbs phenomenon

3. Simulation Setup

  • Document all parameter values (N, d, l, pattern type)
  • Describe window functions used and rationale
  • Include block diagram of synthesis procedure
  • State performance metrics to be evaluated

4. Results & Analysis

  • Plot desired vs synthesized patterns for each case
  • Plot current distribution (excitation coefficients)
  • Calculate and tabulate performance metrics
  • Include polar plots and Cartesian plots
  • Compare different configurations

5. Discussion

  • Explain discrepancies between desired and synthesized patterns
  • Discuss effect of array length on pattern accuracy
  • Analyse impact of windowing on sidelobe levels
  • Comment on practical implementation challenges
  • Compare with other synthesis methods (Woodward-Lawson, etc.)

6. Conclusion

  • Summarize key findings and insights
  • State limitations of Fourier Transform Method
  • Suggest future improvements or extensions

7. Appendices

  • MATLAB/Python code (if used separately)
  • Detailed calculations
  • Additional plots
  • References with proper citations
Important: Screenshots from this virtual lab are acceptable, but ensure they are clearly labeled with all parameter values. Raw data should be included in appendices.

Grading Rubric

  • Theory Understanding (25%): Correct equations and concepts
  • Simulation Execution (25%): Proper parameter sweeps and configurations
  • Analysis Quality (25%): Depth of discussion and insight
  • Presentation (15%): Clarity, labeling, and organization
  • Conclusions (10%): Validity and completeness

Desired vs Synthesized Pattern

● Desired Pattern | ● Synthesized Pattern

Current Distribution

Polar Pattern

Interactive Features:
  • Adjust parameters using sliders in the Virtual Lab tab
  • Real-time pattern synthesis and visualization
  • Compare different windowing techniques
  • Observe Gibbs phenomenon with sharp transitions
  • Export data for report generation
Implementation Notes:
  • Array spacing automatically adjusts to maintain d ≤ λ/2
  • Current distribution is normalized to maximum excitation
  • Pattern calculated at 0.5° intervals for smooth curves
  • All calculations performed using JavaScript FFT