Phased Array Beamforming

How a phased array forms and steers a beam

Picture a row of small antennas, each a loudspeaker for radio. Drive them all with the exact same signal, perfectly in step, and the waves they radiate stack up strongest straight ahead — perpendicular to the row. That direction is called boresight. Off to the sides, the waves from different elements arrive at slightly different times and partially cancel, so the radiated power falls away. You've built a beam pointing dead ahead, and nothing moved.

Now delay each element a little more than the one before it — element 2 lags element 1 by a fixed phase, element 3 lags element 2 by the same amount, and so on. That progressive lag tilts the combined wavefront. The waves now add up in step along a slanted direction, and the beam swings off boresight toward the elements that fire later. Dial the per-element phase, and the beam points wherever you want. The antenna face never tilts; only the electrons do. That's the whole trick of a phased array.

The geometry is pure path-length bookkeeping. For two adjacent elements spaced a distance d apart, a wave heading off at angle θ from boresight reaches one element earlier than the other by an extra path of d·sin(θ). To make the two element-waves arrive in phase in that direction, you pre-load the earlier element with exactly that much phase delay:

Steer a beam to angle θ₀ with N elements at spacing d:

  k   = 2π / λ                     (wavenumber)
  β   = − k · d · sin(θ₀)          (phase step between adjacent elements)
  Δφ  = β  per element             (progressive phase)

Array factor (how the elements combine):

  AF(θ) = sin( N·ψ/2 ) / sin( ψ/2 ),   ψ = k·d·sin(θ) + β

The main beam sits where ψ = 0, i.e. sin(θ) = sin(θ₀).

The full radiation pattern is the single-element pattern multiplied by this array factor — the "pattern multiplication" theorem. The element sets the broad envelope; the array factor carves the sharp, steerable beam inside it.

The governing equations and real numbers

Three numbers dominate a phased-array design: how narrow the beam is, how far the phase must wrap to steer it, and how big each phase step must be. Take a practical X-band radar at 10 GHz, so λ = c/f = 30 mm. Use half-wavelength spacing, d = 15 mm, and a square array of 32 × 32 = 1024 elements (aperture ≈ 0.48 m on a side).

Wavelength:     λ = c/f = 3e8 / 10e9 = 0.030 m = 30 mm
Spacing:        d = λ/2 = 15 mm
Aperture (1D):  L = N·d = 32 × 15 mm = 480 mm

Beamwidth (broadside, half-power):
  θ_HPBW ≈ 0.886 · λ / L  (rad)  for a uniform array
         = 0.886 × 0.030 / 0.480 = 0.0554 rad ≈ 3.2°

Phase step to steer to θ₀ = 45°:
  β = − k·d·sin(45°)
    = −(2π/0.030)(0.015)(0.707)
    = −π·0.707 = −2.22 rad ≈ −127° per element

So to put the beam at 45°, each element runs about 127° behind its neighbor; across 32 elements the total phase ramp is about 11 full turns. A practical phase shifter only needs to cover 0–360° because phase is periodic — you store the ramp modulo 2π.

Two penalties appear off boresight. First, scan loss: the projected aperture shrinks by cos(θ₀), so gain drops by roughly cos(θ₀) and the beam fattens by 1/cos(θ₀). At 60° that's a factor of 2 — a 3 dB gain loss and a doubled beamwidth. Second, grating lobes: if spacing is too wide, a second in-phase direction appears. The no-grating-lobe rule for scanning out to θ_max is:

d  <  λ / (1 + |sin θ_max|)

Scan to 60°:  d < λ / (1 + 0.866) = 0.536 λ
Scan to 90° (theoretical):  d < λ / 2

This is why half-wavelength spacing is the industry default.

Array gain scales with element count: an ideal N-element array has gain ≈ N times a single element (about 10·log₁₀N dB). The 1024-element array above adds ~30 dB over one element. That gain, plus the microsecond steering, is what makes electronic scanning worth its cost.

Worked example: a 5G massive-MIMO panel

Take a 28 GHz millimeter-wave 5G base-station panel — a real deployment band (n261). Work the numbers for a 64-element array (8 × 8):

Frequency:    f = 28 GHz  →  λ = 10.7 mm
Spacing:      d = λ/2 = 5.36 mm
Array:        8 × 8 = 64 elements
Panel size:   ≈ 8 × 5.36 mm = 43 mm per side (matchbox-sized)

Boresight beamwidth (per axis):
  θ_HPBW ≈ 0.886 · λ / (N·d)
         = 0.886 × 10.7 / 42.9 mm = 0.221 rad ≈ 12.7°

Array gain over one element:
  G ≈ 10·log₁₀(64) = 18 dB

Steer the beam to a user at +30° azimuth:
  β = −k·d·sin(30°) = −(2π/10.7)(5.36)(0.5) = −π/2 = −90° per element

A 90°-per-element ramp is trivial for a 6-bit phase shifter (resolution 360°/64 ≈ 5.6°). The base station can park one beam on a phone at +30° and a second, independent beam on another phone at −15° in the same time slot, because digital beamforming lets each user get its own weighted sum of the 64 elements. That spatial reuse — many beams, one panel — is the headline feature of 5G massive MIMO, and it's pure phased-array math.

Real systems and their specs

System	Band / frequency	Elements	What the array buys
AN/APG-81 (F-35 fighter AESA)	X-band (~8–12 GHz)	~1,600 T/R modules	Microsecond target switching, low-probability-of-intercept search
AN/SPY-1 (Aegis, passive PESA)	S-band (~3 GHz)	~4,350 phase shifters/face	Hemispheric track of hundreds of targets, no rotating dish
AN/SPY-6 (modern AMDR, AESA)	S-band	~5,000+ per face (RMA bricks)	Scalable sensitivity, simultaneous air + missile defense
5G mmWave base station (n261)	28 GHz	64–256	Per-user beams, spatial multiplexing (massive MIMO)
Starlink user terminal ("Dishy")	Ku-band (~12 GHz)	~1,280 (flat panel)	Electronic tracking of fast LEO satellites, no gimbal
Automotive radar	77 GHz	~12–48 (MIMO virtual array)	Angle resolution for adaptive cruise & collision avoidance
Radio-astronomy array (e.g. LOFAR/SKA)	HF–GHz	Thousands–millions	Multiple simultaneous sky beams from one aperture

Phased array vs other steering methods

	Phased array (AESA)	Phased array (PESA)	Mechanical dish	Fixed antenna	Lens / reflector array
How it steers	Per-element phase + amplitude (digital/analog)	One source, phase shifters per element	Motor tilts the whole dish	Does not steer	Feed moves / switched feeds
Re-point time	Microseconds	Microseconds	Seconds (inertia, motors)	—	Milliseconds–seconds
Simultaneous beams	Many (digital BF)	One at a time	One	One	Few
Moving parts	None	None	Gimbal, slip rings, bearings	None	Feed actuator
Graceful degradation	High — lose a few T/R modules, slight gain drop	Low — single source fails the whole array	Single point of failure (motor)	—	Moderate
Relative cost	Highest (a T/R module per element)	High	Low–moderate	Lowest	Moderate
Typical home	Fighter/ship radar, 5G, Starlink	Legacy radar (Patriot, Aegis)	Satellite ground stations, old air-traffic radar	Broadcast, fixed links	Some satcom, imaging

Design tradeoffs and failure modes

• Grating lobes from wide spacing. Push elements past ~λ/2 to save cost or T/R modules and a phantom main beam appears off to the side, wasting power and confusing a radar with ghost targets. The fix — keeping d < λ/(1+|sin θ_max|) — is non-negotiable, which is why element count explodes at high frequency where λ is tiny.
• Beam squint over bandwidth. Phase shifters are correct only at one frequency. Send a wideband chirp through a phase-steered array and the beam smears to slightly different angles at the band edges. Wideband systems pay for true time delay units (switched delay lines or photonic delays) that steer every frequency to the same angle.
• Scan loss off boresight. Gain falls by ~cos(θ) and beamwidth grows by 1/cos(θ). Most arrays are spec'd only to ±60°. Covering a full hemisphere means tilting the panel, using three or four faces (as on Aegis ships and the SPY-6 deckhouse), or a conformal array on a curved skin.
• Quantization lobes from coarse phase shifters. A cheap 3-bit shifter (45° steps) rounds the ideal phase ramp into a staircase, raising side lobes and nudging the beam off the commanded angle. Designers either use more bits (5–6 is common) or dither the quantization across elements to smear the error into noise.
• Mutual coupling and the element pattern. Packed at λ/2, elements electromagnetically couple to their neighbors, distorting impedance and the embedded element pattern. Calibration tables and array-edge taper manage it; ignore it and the real pattern won't match the textbook array factor.
• Thermal and reliability load (AESA). An active array packs a transmit/receive (T/R) module behind every element — a tiny power amplifier, low-noise amplifier, phase shifter, and switch — dissipating watts each. A 1,000-element face can draw kilowatts and needs liquid cooling. The upside is graceful degradation: lose 5% of modules and you lose a fraction of a dB, not the radar.

When a phased array earns its cost

• You must re-point faster than mechanics allow — tracking many fast targets, hopping between search and track, or following LEO satellites that cross the sky in minutes.
• You need multiple simultaneous beams — one panel serving many users (5G) or one aperture forming several sky beams at once (radio astronomy).
• Reliability through redundancy matters — an AESA's distributed T/R modules degrade gracefully where a single motor or tube is a hard failure.
• A flat, low-profile aperture is required — conformal arrays on aircraft skin or a flat Starlink panel that can't house a gimbal.

Reach for something simpler when the target barely moves and one beam suffices (a fixed antenna or a slow dish is far cheaper), when the budget can't absorb a phase shifter — let alone a T/R module — per element, or when ultra-wide bandwidth would demand true-time-delay hardware that outweighs the benefit.

Common misconceptions and pitfalls

• "The signals are different frequencies." No — every element radiates the same frequency. Only the relative phase (timing) differs. Mixing frequencies would be a different technique entirely.
• "More elements means a stronger beam, end of story." More elements narrow the beam and raise gain, but they don't change where scan loss, grating lobes, or quantization lobes bite. A poorly spaced 1,000-element array can perform worse than a clean 256-element one.
• "Phase shifting and time delay are the same." They coincide only at the center frequency. Confusing them is exactly what causes beam squint and ruins wideband performance.
• "AESA and PESA are the same thing." A passive array (PESA) has one central transmitter feeding phase shifters; an active array (AESA) has an amplifier at every element. AESA gives more power, lower noise, graceful degradation, and far higher cost.
• "The beam can point anywhere." Practical arrays scan to about ±60°. Past that, scan loss and grating-lobe risk make the beam useless without extra faces or a tilted panel.
• "Receive beamforming needs hardware phase shifters." In a digital array each element is sampled by its own ADC, and the phase ramp is applied in software — letting you form many receive beams at once and re-steer instantly without touching any hardware.