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What is the array factor?

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The array factor AF(θ) = Σ anejn(kd·cosθ + β) describes the pattern of an array of isotropic elements with spacing d, amplitudes an and progressive phase β. For N uniform elements, the normalized magnitude is |AF| = |sin(Nψ/2)/(N·sin(ψ/2))| with ψ = kd·cosθ + β, giving a first sidelobe at −13.26 dB and nulls where Nψ/2 = ±mπ.[1][2]

How do you compute beam direction and nulls?

The main beam lies where ψ = 0, i.e., cosθ0 = −β/(kd). Nulls occur at ψ = ±2πm/N (m = 1, 2, …, not a multiple of N). For a broadside array (β = 0) of length Nd, the half-power beamwidth is approximately 0.886λ/(Nd) radians for large N.[1]

What does an example look like?

Eight elements at d = λ/2, broadside: first nulls at sinθ = ±0.25 from broadside (±14.5°), HPBW ≈ 0.886/4 rad ≈ 12.7°, first sidelobe −12.8 dB (approaching −13.26 dB as N grows), directivity close to N = 8 (9 dBi) for isotropic elements at λ/2 spacing.[1][2]

How do amplitude weights change the array factor?

Non-uniform amplitudes trade beamwidth for sidelobes: binomial weights eliminate sidelobes at λ/2 spacing but broaden the beam a lot; Dolph–Chebyshev gives the narrowest beam for a specified equal sidelobe level; Taylor gives near-in sidelobes at a chosen level with decaying far-out sidelobes.[3][4] See array tapering.

Frequently asked questions

Why −13.26 dB?

It is the first sidelobe of the sin(x)/x function that describes a large uniform array.

Does the array factor include element patterns?

No; multiply by the element pattern for the total.

What is ψ?

The progressive phase difference between adjacent elements’ contributions in a given direction.

References

Online sources accessed October 21, 2026; printed works cited by edition. See all sources.

  1. “Antenna Theory: Analysis and Design, 4th ed.”. C. A. Balanis, Wiley. 2016.
  2. “Phased Array Antenna Patterns — Part 1: Linear Array Beam Characteristics and Array Factor”. P. Delos, B. Broughton and J. Kraft, Analog Dialogue (Analog Devices). 2020.
  3. “A Current Distribution for Broadside Arrays Which Optimizes the Relationship Between Beam Width and Side-Lobe Level”. C. L. Dolph, Proceedings of the IRE, vol. 34, no. 6, pp. 335-348. June 1946.
  4. “Design of Line-Source Antennas for Narrow Beamwidth and Low Side Lobes”. T. T. Taylor, IRE Transactions on Antennas and Propagation, vol. 3, no. 1, pp. 16-28. January 1955.
Categories: Arrays and beamforming