Chebyshev Synthesis Antenna Array Beam Pattern Design

Core Theory

Antenna arrays are critical components in modern wireless communication systems, using multiple radiating elements to generate directional radiation profiles that directly impact overall system performance. Chebyshev synthesis is a widely adopted design method for producing beam patterns with controlled main lobe width and side lobe levels.

Chebyshev polynomials form the mathematical foundation of this approach, defined as:
$$T_n(x) = \cos(n \cdot \arccos(x))$$
where $n$ is the polynomial order and $x \in [-1, 1]$.

Design Workflow

The standard workflow for Chebyshev-based beam pattern design is:

1. Specify Beam Requirements: Define target parameters including main lobe width, desired side lobe suppression ratio, and side lobe decay characteristics.
2. Select Polynomial Order: Choose an appropriate Chebyshev polynomial order aligned with the specified beam requirements.
3. Compute Element Weights: Calcualte the excitation amplitude weights for each array element using the Chebyshev polynomial and design parameters.
4. Synthesize Array: Assemble the antenna array using the computed weights to generate the targeted beam pattern.

Method Advantages

• Flexible beam shaping: Supports creation of narrow main lobes, controlled low side lobe levels, and rapid side lobe attenuation.
• Rigorous mathematical basis: Grounded in orthogonal polynomial theory, enabling accurate performance prediction.
• Simplified implementation: Straightforward weight calculation and array synthesis processes.

Method Limitations

• Fixed side lobe constraints: Achievable side lobe levels are tied to polynomial order and design parameters.
• Increased computational complexity for high-order polynomials.
• Limited flexibility for highly specialized beam profile requirements.

Typical Applications

Chebyshev synthesis is widely used in radar, satellite communications, and mobile wireless systems, particularly for applications requiring precise beam control and low side lobe suppression.

MATLAB Simulation Code

% Chebyshev antenna array weight synthesis script
clearvars; clc; close all;

% System configuration parameters
num_elements = 13;          % Total number of array elements
signal_wavelength = 10;     % Transmit signal wavelength
elem_spacing = 0.6 * signal_wavelength; % Inter-element spacing
scan_angle = deg2rad(60);   % Beam scanning angle relative to array boresight
target_sll_dB = 26;         % Target side lobe suppression level (dB)

% Calculate number of summation terms based on array element count parity
if mod(num_elements, 2) == 0
    sum_terms = num_elements / 2;
else
    sum_terms = (num_elements - 1)/2 + 1;
end

% Hardcoded Chebyshev polynomial coefficient matrix
% Rows correspond to polynomial orders 0 through 13
% Columns correspond to x^0 to x^13 polynomial terms
cheby_coeff_matrix = [
    1,0,0,0,0,0,0,0,0,0,0,0,0,0;
    0,1,0,0,0,0,0,0,0,0,0,0,0,0;
    -1,0,2,0,0,0,0,0,0,0,0,0,0,0;
    0,-3,0,4,0,0,0,0,0,0,0,0,0,0;
    1,0,-8,0,8,0,0,0,0,0,0,0,0,0;
    0,5,0,-20,0,16,0,0,0,0,0,0,0,0;
    -1,0,18,0,-48,0,32,0,0,0,0,0,0,0;
    0,-7,0,56,0,-112,0,64,0,0,0,0,0,0;
    1,0,-32,0,160,0,-256,0,128,0,0,0,0,0;
    0,9,0,-120,0,432,0,-576,0,256,0,0,0,0;
    -1,0,50,0,-400,0,1120,0,-1280,0,512,0,0;
    0,-11,0,220,0,-1232,0,2816,0,-2816,0,1024,0;
    1,0,-72,0,840,0,-3584,0,6912,0,-6144,0,2048,0;
    0,13,0,-364,0,2912,0,-9984,0,16640,0,-13312,0,4096
];

Simulation Results

The simulation output will generate a plotted beam pattern matching the specified side lobe suppression level, with clear separation between the main lobe and controlled side lobes.

Reference

[1] Gao, J. (2009). Chebyshev Linear Array Antenna Beamforming Considering Mutual Coupling. Electronic Test, (4), 5. https://doi.org/10.3969/j.issn.1000-8519.2009.04.002