Text from the interactive explanation

RF Chain Count to Mitigate Beam Split in Wideband Hybrid Arrays: From Far-Field to Near-Field

Hussain, Ahmed, Abdallah, Asmaa, Celik, Abdulkadir, Eltawil, Ahmed M.

Interactive guide · wireless physics

Tomorrow's base stations may be walls of antennas that focus radio onto your phone the way a lens focuses sunlight. But a fast connection is a rainbow of frequencies — and the cheap electronics that steer these walls focus every colour in a different place. How much extra hardware does it take to pull the rainbow back into one point?

A visual guide to “RF Chain Count to Mitigate Beam Split in Wideband Hybrid Arrays: From Far-Field to Near-Field” by Ahmed Hussain, Asmaa Abdallah, Abdulkadir Celik and Ahmed M. Eltawil (arXiv, 2026).

Drag the phone

Drag the phone. Then add radio chains until the colours meet again.

Fully connected needs ⌈α⌉ = – · Sub-connected needs –

Live simulation: 96-antenna, 28 GHz panel with a 3 GHz band, seen from above. The paper's panel has 256 antennas. “In focus” means at least 90% of the full signal across the band; the fully connected count is the paper's formula ⌈α⌉, the sub-connected count is simulated.

A visual guide to “RF Chain Count to Mitigate Beam Split in Wideband Hybrid Arrays: From Far-Field to Near-Field” by Ahmed Hussain, Asmaa Abdallah, Abdulkadir Celik and Ahmed M. Eltawil (arXiv, 2026).

01Near field vs far field

Aim or focus?

A flashlight aims: its beam only has a direction. A magnifying glass focuses: it gathers light onto one spot. Which is an antenna panel? Make a guess, then slide the phone away from the panel.

One frequency (28 GHz), every antenna tuned individually, straight ahead of the same 96-antenna panel. The view zooms out as the phone moves away.

What you just saw

Close to the panel, the wave it sends out has to be visibly curved , like ripples converging on a point (the gold arc). The panel then concentrates energy in angle and in distance: a compact bright spot. Move the phone away and the required curve flattens; the spot stretches into a long beam that only has a direction. Beyond a certain distance — about 7 m straight ahead for this small panel — the spot has no finite depth at all: the panel can only aim.

Big panels and high frequencies stretch this focusing zone. For the paper's 1.37 m panel at 28 GHz, the classic near-field zone (the Rayleigh distance 2L²/λ) reaches about 350 m , and the panel can still focus in depth out to about 50 m straight ahead.

The 3-dB depth of the focus is the paper's eq. (14): BD = 8 η² r₀² rRD cos²φ₀ / ((rRD cos²φ₀)² − (4 η² r₀)²), with η = 1.318 and the Rayleigh distance rRD = 2L²/λ.

The depth becomes infinite when the denominator reaches zero, at rsp = rRD cos²φ₀ / (4η²) ≈ 0.14 rRD straight ahead (eq. 16). That boundary will matter again in section 3.

02Why cheap steering fails

The dial that only knows one frequency

A phase shifter is a dial that turns each antenna's wave by a fixed fraction of a cycle. Set correctly, all the waves arrive at the phone in step. But the right setting depends on frequency, and the dial can't know which frequency is passing through. What happens across the rest of the band?

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What you just saw

Each little arrow is one antenna's contribution at the phone (the paper's 256-antenna, 1.37 m panel). At the carrier, 28 GHz, every arrow points the same way: one long straight chain, the full signal. Move off the carrier and each arrow turns by an amount proportional to the frequency offset times that antenna's extra path length — so the chain curls up , and the straight gold “reach” line, the signal that's left, shrinks. Its length squared is the gain.

Off to the side, the extra path grows steadily across the panel, so the chain bends into a circle . Straight ahead and close, it grows with the square of the position, so the chain winds into a spiral (a Cornu spiral). Those two shapes are exactly why the paper's gain formulas contain sinc and Fresnel integrals. The band of frequencies that keeps at least half the power is the instantaneous bandwidth (IBW) — the frequency version of a beam's width.

The gain at the focus is eq. (18): g(f) = |(1/N) Σ exp(j2πf δₙ/c)|², with δₙ the extra path of antenna n. Keeping only the tilt term gives the sinc² law (eq. 20) and the angle-limited IBW 0.886 c/(L sin φ₀) (eq. 22); keeping only the curvature term gives the Fresnel law (eq. 24) and the depth-limited IBW ≈ 7 c r₀/(L² cos²φ₀) (eq. 25). The paper combines them as the minimum of the two (eq. 26).

03Angular vs radial split

Two ways the rainbow splits

Now send the whole 3 GHz band at once, steered by phase shifters alone (one radio chain). Each colour lands in its own place. Answer the three questions, then explore with the sliders.

Spreads between the band edges (26.5 and 29.5 GHz), from the paper's split formulas, for the same 96-antenna panel.

A Where should the phone be so the colours don't spread sideways?

Straight ahead. Each frequency steers to its own angle, sin φ = sin φ₀ / (1 + f/fc): higher frequencies swing toward straight ahead, lower ones away. At 0° every frequency agrees, so the angular split vanishes; it grows toward the side.

B Straight ahead and close: do the colours still separate?

Yes — in depth. Higher frequencies focus farther, lower ones nearer. This radial split exists only in the near field, where the panel focuses in distance at all.

C What happens to the depth split if the phone moves away?

It fades. The farther the focus, the deeper each colour's spot, until they overlap completely. Beyond the boundary rsp (≈ 7 m here, ≈ 50 m for the paper's panel) every beam has unlimited depth and only the sideways split is left — the same “beam squint” known from the far field.

What you just saw

With a single radio chain, the rainbow splits two ways. Sideways (angular split): worst toward the side, absent straight ahead, and identical near and far. In depth (radial split): strongest close to the panel and straight ahead, and gone beyond rsp. Every frequency that lands off target is signal the phone doesn't get.

Angular split, eq. (11): sin φ = sin φ₀ / (1 + f̄), with f̄ = f/fc. Radial split, eq. (13): r = ξ r₀ with ξ = ((1 + f̄)² − sin²φ₀) / ((1 + f̄) cos²φ₀). The radial split stops mattering at rsp = rRD cos²φ₀ / (4η²) (eq. 16), where the beam depth becomes infinite.

04Counting the problem

A race of pulses

A 3 GHz signal can't tell apart arrival times closer than about 1/W = 0.33 ns — in that time light travels about 10 cm. So the question becomes: how much longer is the longest path from the panel to the phone than the shortest?

→ far away

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What you just saw

Phase shifters can't delay a pulse, so every antenna fires at the same moment and the pulses arrive spread out in time. Count how many time slots the arrivals span:

KFC = ⌈α⌉, α = W · (dmax − dmin) / c

That is the paper's formula for a fully connected array: bandwidth times delay spread across the panel. Each radio chain can take care of about one time slot. (Formally, α counts the dominant eigenvalues of the panel's space–frequency correlation matrix — you'll see them in the next section.)

The longest path always comes from the far edge of the panel. The shortest depends on where the line from the phone, dropped perpendicular to the panel, lands. If it lands beyond the panel's edge (Case A, r₀ ≥ rb = L/(2 sin φ₀)), the near edge is closest and α = W(d₊ − d₋)/c. If it lands on the panel (Case B), that foot point is closest, dmin = r₀ cos φ₀, and α = W(d₊ − r₀ cos φ₀)/c (eq. 49).

Far away, the spread tends to L sin φ₀, so α → W L sin φ₀ / c — the far-field count, which depends on angle alone (eq. 50).

05Sub-connected vs fully connected

Two ways to wire the fix

Radio chains are the expensive part. There are two ways to connect K of them to the paper's 256 antennas (3 GHz band). Pick a phone position, then find the fewest chains each wiring needs.

The panel is cut into K blocks, one chain each. Phase shifters: 256 .

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Every chain reaches every antenna through its own phase shifter. Phase shifters: – .

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What you just saw

Sub-connected: the digital processor re-aligns the blocks at every frequency, so what's left is the spread inside each block. Smaller blocks tolerate wider bands, so the paper's Algorithm 1 keeps adding chains until the worst (“binding”) block's instantaneous bandwidth covers the whole band.

Fully connected: the bars are the eigenvalues of the panel's space–frequency correlation matrix for this phone position. They form a staircase: about α tall steps, then a cliff to almost nothing. Each chain can capture one bar, so ⌈α⌉ chains capture nearly all of the signal.

Side-on and far (90°, 10 L): sub-connected needs 16 chains by the paper's 3-dB rule (worst band edge −2.8 dB, band average ≈ −0.8 dB). Fully connected needs 14 — fewer chains, but 14 × 256 = 3,584 phase shifters instead of 256, plus more insertion loss.

The bars are the eigenvalues σℓ/N of B = (1/W)∫ a(f) a(f)ᴴ df (eq. 37), computed here from 96 frequency samples across the band. The paper proves that the number of dominant eigenvalues ≈ α (eqs. 43–44), and that the best band-average gain with K chains is the sum of the K largest (eq. 42).

The fully connected curve uses the ideal, unconstrained eigenvector weights. With practical phase-shifter-only weights, the paper's simulations lose about 5% more (Fig. 12). The sub-connected curve and its band-average are exact simulations of the block precoder; the marked count comes from Algorithm 1 with the 3-dB threshold (ρ = 0.5).

06Near field vs far field, mapped

The map

Where does a user need the most chains? Predict first: a user 60° to the side walks from 100 L toward the panel. Will they need more or fewer chains along the way? And a user at 15° ?

60° Walking in from 100 L, the count will go…

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15° Walking in from 100 L, the count will go…

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What you just saw

Beyond about 10 L (the dashed line) nothing changes with distance: the near-field count equals the far-field count W L sin φ₀ / c, which depends on angle alone. Closer in, every line array has a floor: as the focus approaches the panel the count tends to W L / 2c at every angle — 7 chains for the paper's panel at 3 GHz, half the far-field maximum of 14.

That makes 30° special, because sin 30° = ½: there the floor and the far-field value coincide. Below 30°, near-field users need more chains than far ones; above 30°, fewer. The practical upshot from the paper: a hybrid array sized for far-field users beyond 30° already covers every near-field user — no extra radio chains are needed for near-field operation.

Computed live from the paper's closed forms: eq. (49) for the line (the white curve is the Case A/B boundary rb = L/(2 sin φ₀)), eq. (67) for a ring of the same 1.37 m diameter at elevation θ₀ (no near-field floor; it never needs more than W L/c), and Algorithm 1 with the stricter ρ = 0.95 threshold for sub-connected wiring, as in the paper's Fig. 11 (line array, 3 GHz). Compare Figs. 6, 11 and 13. The 28 GHz option is the paper's deliberately huge illustration: floor 64, maximum 128. Ranges near L are only met with very large panels.

07Design rules

Build a 6G panel

You're designing a panel for the hardest user — side-on and beyond 10 L. How many radio chains must you build? Then: do they all need to be switched on all the time?

Double the band or the panel → double the chains.

Only power what you need

The paper points out that when the user's position is known, the panel can switch on just ⌈α(φ₀, r₀)⌉ chains (a dynamic subarray architecture) and keep the 95% gain guarantee while saving energy. Follow one walker.

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Paper's panel (1.37 m, 256 antennas), 3 GHz band, 14 chains installed. Distances drawn on a log scale.

✦What the paper establishes

Back to the rainbow

The rainbow you split in the lab is real physics for every wideband antenna wall steered by phase shifters. The paper turns it into hardware numbers.

Two kinds of split

Near-field beam split separates into an angular part (the familiar far-field squint) and a radial part that vanishes beyond a derived boundary rsp.

Eqs. 11–17 · Fig. 2

Sub-connected count

Near-field instantaneous-bandwidth formulas give an algorithm: add blocks until the binding block's bandwidth covers the band.

Algorithm 1 · Figs. 9–11

α = W × delay spread

Fully connected arrays need ⌈α⌉ chains — the number of dominant eigenvalues of the space–frequency correlation matrix — for lines and rings.

Eqs. 44, 49, 67 · Fig. 12

Design rules

Beyond about 10 L the count equals the far-field one; sizing for far-field users beyond 30° covers the near field. Fully connected uses fewer chains but K × N phase shifters.

Figs. 13–16 · Table I

As 6G pushes bandwidths and panel sizes up, this gives designers a simple rule for how many radio chains to build — roughly bandwidth × panel size — and tells them when some can be switched off. The authors list what comes next: multipath channels instead of a clear line of sight, planar (two-dimensional) panels, curved “arc” arrays, and wideband near-field codebooks built from the eigenvectors you met in section 5.

Sources & credits

Ahmed Hussain, Asmaa Abdallah, Abdulkadir Celik, Ahmed M. Eltawil. “RF Chain Count to Mitigate Beam Split in Wideband Hybrid Arrays: From Far-Field to Near-Field.” arXiv:2610.05876 [eess.SP], v2, October 2026. arxiv.org/abs/2610.05876 · License: CC BY 4.0.

All visualisations are new renderings computed from the paper's equations; the hero and field explorers simulate a smaller 96-antenna panel so the waves are visible. Table values recomputed from ⌈WL/c⌉ and match the paper's Table I. Results in the paper are analytical and simulated for a free-space line-of-sight channel; the wave crests here are slowed down enormously so the eye can follow them.

Typefaces: Instrument Serif and Inter (SIL Open Font License).