To steer the beam to angle θ₀, we apply progressive phase shifts across the array. Element n receives phase shift: Δφₙ = n × k × d × sin(θ₀) where k = 2π/λ.

Watch how the wavefronts align in different directions as you adjust the steering angle. The equiphase front (shown as dashed line) indicates the beam direction.

The array factor describes the radiation pattern: AF = sin(Nψ/2) / (N×sin(ψ/2)) where ψ = kd(sinθ - sinθ₀).

Grating lobes appear when element spacing d > λ/2, creating additional main lobes that waste power and cause false targets. The condition for grating lobes: d/λ > 1/(1 + |sinθ₀|)

As beam steering angle increases, scan loss occurs due to the projected aperture reduction: Loss = cos(θ₀).

Passive arrays receive signals without transmitting—used for electronic intelligence (ELINT), direction finding, and electronic warfare. They detect emitters while maintaining radio silence.

Direction of Arrival (DOA) is determined by measuring phase differences between elements. For a signal at angle θ, the phase at element n relative to element 0 is: φₙ = 2π·n·d·sin(θ)/λ

Beamforming applies phase shifts to steer receive sensitivity: signals arriving from the steered direction add coherently; others are suppressed.