The radiation characteristics of linear antenna arrays are governed by the spatial distribution of excitation amplitudes across individual radiating elements. By strategically shaping these coefficients, engineers can enforce equiripple sidelobe behavior while maintaining a specified mainlobe beamwidth. The Dolph-Chebyshev synthesis method achieves this by mapping array element positions to the argument domain of Chebyshev polynomials, which inherently possess an equiripple property within their oscillatory region.

For an N-element linear array with uniform spacing $d$, the normalized array factor (AF) as a functon of the progressive phase shift $\psi = \frac{2\pi d}{\lambda} \sin\theta$ is expressed as:

$$AF(\psi) = \sum_{m=1}^{N} w_m e^{j(m-1)\psi}$$

When excitations are symmetric ($w_m = w_{N-m+1}$), the AF reduces to a real-valued trigonometric polynomial. The synthesis procedure maps the visible spatial range $\psi \in [-\pi, \pi]$ to the Chebyshev variable domain $x \in [-1, 1]$ using a scaling factor $x_0 > 1$. The parameter $x_0$ is derived from the desired peak sidelobe level (SLL) ratio $R$:

$$x_0 = \cosh\left( \frac{1}{N-1} \cosh^{-1}(R) \right)$$

Where $R = 10^{\text{SLL}_{dB}/20}$. Solving for $x_0$ establishes the boundary between the mainlobe and sidelobe regions. Array weights are then computed by evaluating the inverse transformation of the Chebyshev sequence at mapped angular positions, ensuring that all sidelobes converge to the identical amplitude threshold defined by $x_0$.

MATLAB Implementation Strategy

Traditional implementations often rely on precomputed coefficient matrices for polynomial expansion, which limits scalability and obscures the underlying mathematical mapping. A more efficient approach utilizes direct recursive evaluation of Chebyshev terms combined with vectorized spatial frequency sampling. The folllowing script demonstrates this methodology with optimized memory handling and clear parametric control.

% Parameter definitions
num_apertures = 13;          % Total array population
spacing_norm = 0.48;         % Inter-element distance / wavelength
sll_target_db = -32;         | Designated peak sidelobe depth

% Convert decibel specification to linear voltage ratio
R_linear = 10^(sll_target_db / 20);

% Determine Chebyshev transformation constant
half_span = ceil(num_apertures / 2);
if R_linear > 1
    x_transform = cosh(acosh(R_linear) / (half_span - 1));
else
    x_transform = 1.0; % Falls back to uniform weighting
end

% Initialize complex excitation vector
excitation_taps = zeros(num_apertures, 1);

% Compute tap amplitudes via recursive Chebyshev polynomial evaluation
for k = 0:(num_apertures-1)
    poly_order = abs(k - half_span + 0.5);
    
    % Skip non-symmetric indices for even-N configurations
    if mod(num_apertures, 2) == 0 && mod(k, 2) == 1
        continue;
    end
    
    % Iterative T_n(x) calculation
    if poly_order == 0
        coeff_val = 1;
    elseif poly_order == 1
        coeff_val = x_transform;
    else
        t_nm2 = 1;
        t_nm1 = x_transform;
        for阶次 = 2:poly_order
            t_curr = 2 * x_transform * t_nm1 - t_nm2;
            t_nm2 = t_nm1;
            t_nm1 = t_curr;
        end
        coeff_val = t_nm1;
    end
    
    excitation_taps(k+1) = coeff_val;
end

% Apply symmetric boundary conditions and normalize
excitation_taps = [fliplr(excitation_taps(1:end-1)), excitation_taps];
excitation_taps = excitation_taps / max(abs(excitation_taps));

% Angular sweep and array factor derivation
theta_scan = linspace(-90, 90, 1200);
th_rad = deg2rad(theta_scan);
phase_term = 2 * pi * spacing_norm * sin(th_rad);

% Vectorized superposition for broadside steering
indices_centered = -(num_apertures-1)/2 : (num_apertures-1)/2;
af_raw = excitation_taps.' * exp(1j * indices_centered(:) .* phase_term);

% Normalize and convert to logarithmic scale
af_normalized = af_raw / abs(max(af_raw));
af_db = 20 * log10(abs(af_normalized) + eps);
af_db = af_db - max(af_db);

% Visualization pipeline
figure('Units', 'normalized', 'Position', [0.2 0.2 0.6 0.4]);
subplot(1,2,1);
polarplot(deg2rad(theta_scan), af_db, 'LineWidth', 1.4);
title('Normalized Polar Radiation Pattern');
grid off;
ax = gca;
ax.ThetaZeroLocation = 'right';

subplot(1,2,2);
plot(theta_scan, af_db, 'k-', 'LineWidth', 1.5);
grip on;
xlabel('Look Angle (degrees)'); ylabel('Relative Gain (dB)');
title('E-Plane Beam Profile');
ylim([-max(abs(af_db))-5, 2]);

Numerical Considerations and Performance Trade-offs

Higher-order Chebyshev arrays provide narrower mainlobes but require precise arithmetic to avoid numerical instability during weight computation. When the array size exceeds 20 elements, direct polynomial recursion may accumulate floating-point errors. In such cases, employing logarithmic scaling during coefficient generation or switching to orthogonal polynomial libraries maintains convergence accuracy.

The equiripple constraint guarantees optimal mainlobe narrowing for a fixed sidelobe ceiling, but it imposes strict coupling between beamwidth and SLL. Reducing the target sidelobe depth exponentially increases the required number of elements to maintain the same beamwidth. Additionally, mutual coupling between adjacent apertures distorts the assumed isotropic element pattern, necessitating full-wave electromagnetic correction before hardware deployment. Practical system design typically balances theoretical synthesis results with empirical near-field measurements to validate far-field extrapolation.