Ferkans — Interactive Telecom Tutor

Collapsing One Side of the Double Fading

The double-fading theorem of Section 21.1 tells us that a passive RIS only helps when at least one of the two path lengths $d_1, d_2$ is small. In a conventional deployment, $d_1$ (transmitter to RIS) is typically a meaningful fraction of the cell radius — hundreds of metres at mmWave, still tens of metres at sub-THz. But nothing in the physics forbids us from placing the RIS directly in front of the transmitter. What if $d_1 \to 0$ ?

The architecture that makes this idea concrete is the array-fed RIS: a small active array of $N_a \ll N_{\text{RIS}}$ elements sits just a few wavelengths in front of a large passive RIS, illuminating it from the reactive-near-field side. The active array is a compact, amplified "injector" that couples power into the RIS; the RIS acts as a large- aperture lens or reflector that focuses the illumination toward the user. The end-to-end link pays only one path loss — the long one, RIS to user — and the nominal aperture gain $G_{\text{RIS}} \approx N_{\text{RIS}}$ survives intact.

This is the CommIT architecture of Caire and collaborators. It replaces the question "how do we make an RIS useful at long Tx $\to$ RIS distance?" with the cleaner question "how do we efficiently feed a large passive aperture with a small active array?". The answer turns a theoretical curiosity into a power- and hardware-efficient mmWave/sub-THz architecture.

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Definition:
Array-Fed RIS Architecture

An array-fed RIS consists of two co-located aperture stages:

An active array feed with $N_a$ RF-chain-backed elements that transmits the physical signal. The active array is compact (typical aperture $\sim 5$ – $20\lambda$ ) and sits at a short fixed distance $d_f$ from the RIS (typical $d_f \in [2, 10]\, \lambda$ , i.e. inside the reactive near field of the RIS).
A passive reflecting RIS with $N_{\text{RIS}} \gg N_a$ elements. Each RIS element has unit-modulus reflection $\Gamma_n = e^{j\phi_n}$ and is configured by the BS controller.

The compound signal model, in the narrowband single-user setting, is

$y = \underbrace{\mathbf{H}_{\text{RIS-Rx}}^H\, \boldsymbol{\Phi}\, \mathbf{G}_f\, \mathbf{v}}_{h_{\text{eff}}(\boldsymbol{\phi}, \mathbf{v})} \, s + \mathbf{w},$

where $\mathbf{G}_f \in \mathbb{C}^{N_{\text{RIS}} \times N_a}$ is the short-range (reactive/near-field) coupling matrix from the active feed to the RIS, $\boldsymbol{\Phi}$ is the diagonal RIS reflection, and $\mathbf{H}_{\text{RIS-Rx}}^H \in \mathbb{C}^{1 \times N_{\text{RIS}}}$ is the long-range reflected channel to the user. The Tx-side precoding $\mathbf{v} \in \mathbb{C}^{N_a}$ lives on the $N_a$ -dimensional active feed; the RIS phase profile $\boldsymbol{\phi}$ adds a further $N_{\text{RIS}}$ scalar degrees of freedom.

The key physical insight is that $\mathbf{G}_f$ is well-conditioned — essentially a lossless reactive-field transformer between the two apertures — so the effective single-hop path loss is set entirely by $\|\mathbf{H}_{\text{RIS-Rx}}\|^2$ , not by a product.

The architecture is often called a lens array in the antenna community: the passive surface acts as an RF lens focusing the active feed's pattern into a much larger effective aperture. The terminology "array-fed RIS" reflects its system-level framing — a standard hybrid architecture in which the analog stage happens to be a metasurface rather than a phase- shifter network.

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🎓CommIT Contribution(2022)

Power- and Hardware-Efficient Multiuser Multibeam Array-Fed RIS

G. Caire — CommIT group, technical report / IEEE Trans. Wireless Communications (preprint)

The array-fed RIS architecture developed by Caire and collaborators at the CommIT group reframes the RIS as an analog aperture stage of a hybrid architecture rather than as a stand-alone repeater. The central technical contributions, each of which is used somewhere in this chapter, are:

Co-located feed + RIS geometry. Place an $N_a$ -element active array at $d_f \sim 5\lambda$ from an $N_{\text{RIS}}$ -element passive RIS so that the forward link $\mathbf{G}_f$ is in the reactive near field. This eliminates the first path-loss factor and preserves the $N_{\text{RIS}}$ aperture gain of the reflected link. The result is an effective transmit aperture of $N_{\text{RIS}}$ elements with only $N_a$ RF chains — a cost reduction of roughly $N_{\text{RIS}}/N_a$ compared with a fully digital array of equal size.
Multiuser joint precoder design. The joint optimization of the active-array precoder $\mathbf{W} \in \mathbb{C}^{N_a \times K}$ and the RIS phase profile $\boldsymbol{\phi}$ is non-convex, but admits a computationally tractable alternating solution that we derive in Section 21.4. Unlike fully hybrid architectures, the RIS stage can synthesize $K$ simultaneous beams even when $N_a < K$ , provided $N_{\text{RIS}}$ is large.
Eigenmode analysis of the cascaded channel. The effective channel $\mathbf{H}_{\text{eff}}$ has rank at most $\min(N_r, N_a)$ regardless of $N_{\text{RIS}}$ , but its dominant singular values scale with $N_{\text{RIS}}$ , so the array gain is concentrated on a small number of high-quality eigenmodes. We derive this in Section 21.3.
Power and rate comparison with fully digital arrays. Caire and collaborators quantified the rate–power tradeoff: the array-fed RIS architecture typically achieves 75–85% of the sum rate of a fully digital array of equal physical aperture at 1/5 to 1/10 of the DC power — a factor set essentially by the $N_a/N_{\text{RIS}}$ ratio. Section 21.5 reproduces this comparison.

The architecture is particularly attractive for mmWave and sub-THz deployments, where even a modest RIS aperture ( $0.3$ – $1$ m) holds thousands of elements and where active-chain DC power dominates the equipment budget. In these bands, the array-fed RIS is plausibly the most power-efficient way to scale the transmit aperture.

rismmwavehybrid-beamformingmultiuserarray-fed

Array-fed RIS

A reflective transmitter architecture in which a small active array of $N_a$ RF-chain-backed elements illuminates a large passive RIS of $N_{\text{RIS}} \gg N_a$ elements at near-zero distance (within a few wavelengths). The active feed delivers transmit power locally; the RIS provides aperture gain. The architecture eliminates the double-fading loss of stand-alone passive RIS while keeping the RF-chain count proportional to the user count rather than the aperture size. Synonym: metasurface-fed antenna, lens-array antenna.

Double fading

The compounded path loss of a reflective link, in which a signal travels through two free-space hops $d_1$ (Tx $\to$ RIS) and $d_2$ (RIS $\to$ Rx). The total received power decays as $1/(d_1 d_2)^2$ rather than the single-hop $1/d_{\text{direct}}^2$ , which dominates when both hops are non-trivial. The phenomenon is the central obstacle to passive-RIS deployments and the motivation for the array-fed architecture.

Array-Fed RIS Block Diagram — Block diagram of an array-fed RIS transmitter. Digital precoding $\mathbf{W}$ maps $K$ user streams onto $N_a$ RF chains; the active array illuminates the passive RIS through a short-range near-field coupling $\mathbf{G}_f$ ; the RIS phase profile $\boldsymbol{\phi}$ reshapes the illumination into $K$ simultaneous beams steered toward the users. The double-fading path loss is avoided because the Tx-to-RIS distance $d_f$ is only a few wavelengths.

Theorem: Array-Fed RIS Link Budget (Single User, LOS)

Consider an array-fed RIS transmitter with active-array power budget $P_t$ , co-located with an $N_{\text{RIS}}$ -element passive RIS via a lossless reactive-field coupler $\mathbf{G}_f$ . A single user with one antenna is at distance $d_2$ in the far field of the RIS, on boresight $(\phi_u, \theta_u)$ . With optimal active precoding and RIS phase profile, the received power is

$P_r^{\text{AF}} = P_t\, \frac{\lambda^{2}}{(4\pi)^2}\, \frac{G_a\, G_{\text{RIS}}}{d_2^2},$

where $G_a$ is the gain of the active-array feed (with $G_a \lesssim N_a$ ) and $G_{\text{RIS}} \approx N_{\text{RIS}}$ is the directivity of the passive RIS on boresight.

The link budget now contains one inverse-square factor instead of two. The active feed's directivity $G_a \sim N_a$ harvests the available active power; the RIS's aperture $G_{\text{RIS}} \sim N_{\text{RIS}}$ focuses the outgoing wavefront. Compared with the passive-RIS double- fading formula of Theorem TDouble-Fading Path Loss, we save a factor of $d_1^2$ . At typical distances ( $d_1 \sim 20$ m), that is $\sim 26$ dB.

Proof

Active radiated power at the RIS

The active feed radiates a total of $P_t$ W. In the reactive near field, the RIS captures nearly all of it because $d_f \ll \sqrt{A_a A_{\text{RIS}}/\lambda}$ (the boundary between near and far field for two apertures). So the effective power illuminating the RIS is $P_t$ itself, up to a small coupling loss which we absorb into $G_a$ .

RIS re-radiation toward the user

The RIS with aligned phase profile $\phi_n = -\arg[\mathbf{a}_{\text{RIS}}(\phi_u,\theta_u)]_n$ reshapes the captured wavefront into a narrow beam of effective directivity $G_{\text{RIS}} \approx 4\pi A_{\text{RIS}}/\lambda^{2} \sim N_{\text{RIS}}$ for a half-wavelength-spaced surface.

Far-field Friis at the user

The user at range $d_2$ on boresight collects $P_t G_a G_{\text{RIS}} (\lambda/(4\pi d_2))^2$ W of power, which is the stated expression. $\blacksquare$

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Passive RIS vs Array-Fed RIS vs Fully Digital Array

Property	Passive RIS	Array-Fed RIS	Fully Digital Array
RF chains needed	0	$N_a$ (small)	$N_{\text{RIS}}$ (large)
Transmit power source	External Tx at $d_1$	Local feed $N_a$	Local feed $N_{\text{RIS}}$
Path-loss factors	$1/(d_1 d_2)^2$	$1/d_2^2$	$1/d_2^2$
Effective aperture	$\approx N_{\text{RIS}}$	$\approx N_{\text{RIS}}$	$\approx N_{\text{RIS}}$
DC power budget	$\sim$ mW (control only)	$N_a\cdot P_c^{\text{RF}}$	$N_{\text{RIS}}\cdot P_c^{\text{RF}}$
Multiuser flexibility	Single-beam (joint with Tx)	$\min(N_a, N_{\text{RIS}})$ beams	Up to $N_{\text{RIS}}$ beams
Per-user CSI needed	Cascaded $\mathbf{H}_{\text{eff}}$	Cascaded $\mathbf{H}_{\text{eff}}$	Direct $\mathbf{H}$
Phase resolution matters	Yes (critical)	Less (dominated by $\mathbf{W}$ )	N/A
Best for	Blocked-direct coverage holes	Large-aperture mmWave access	Sub-6 GHz massive MIMO

Example: A 1024-Element Array-Fed RIS vs 256-Element Digital Array

We want to deploy a mmWave base station at $f_0 = 28$ GHz with a $0.3 \times 0.3$ m aperture ( $N_{\text{RIS}} = 1024$ at half-wavelength spacing). Option A is a fully digital 1024-element array. Option B is an array-fed RIS with $N_a = 32$ active elements feeding the 1024-element passive surface. Each active RF chain costs $P_c^{\text{RF}} = 250$ mW. Compare the DC power of the two options (ignore baseband, cooling).

Solution

Fully digital power budget

$P_{\text{DC}}^{\text{dig}} = 1024 \cdot 0.25 = 256$ W. For comparison, a typical mmWave small-cell BS budget is on the order of 10–20 W — so 256 W is infeasible.

Array-fed RIS power budget

$P_{\text{DC}}^{\text{AF}} = 32 \cdot 0.25 + P_{\text{RIS}}^{\text{static}} \approx 8 + 0.2 \approx 8.2$ W. The RIS control power is at most a few hundred mW for 1024 elements.

Savings ratio

$256/8.2 \approx 31 \times$ lower DC power for the array-fed option. As we will see in Section 21.5, this comes at the cost of roughly 10–20% lower sum rate, an excellent tradeoff whenever DC power is the binding constraint. $\blacksquare$

Reflected Beam Pattern as the RIS Phase Profile Varies

Fix the active feed in a co-located array-fed RIS configuration and steer the outgoing beam by changing only the RIS phase profile. The angular pattern is computed analytically from the array manifold of a ULA RIS illuminated by a single-mode feed. Adjust the steering angle and the element count to see how beamwidth scales as $1/N_{\text{RIS}}$ .

Parameters

N_{\text{RIS}}

256

Steering angle (deg)20

N_a

(active feed)8

RIS phase resolution

Array-Fed RIS Forward Signal Synthesis

Complexity:

\mathcal{O}(N_{\text{RIS}}\, N_a + K\, N_{\text{RIS}})

Input: Active precoder

\mathbf{W} \in \mathbb{C}^{N_a \times K}

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RIS phase profile

\boldsymbol{\phi} \in [0,2\pi)^{N_{\text{RIS}}}

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user symbols

\mathbf{s} \in \mathbb{C}^{K}

, feed–RIS coupling

\mathbf{G}_f

Output: Transmitted field pattern at the RIS, and per-user received signal

1.

\mathbf{x}_a \leftarrow \mathbf{W}\, \mathbf{s}

// baseband signal on the active array

2.

\mathbf{u} \leftarrow \mathbf{G}_f\, \mathbf{x}_a

// captured by the RIS

3.

\boldsymbol{\Phi} \leftarrow \text{diag}(e^{j\phi_1}, \ldots, e^{j\phi_{N_{\text{RIS}}}})

4.

\mathbf{z} \leftarrow \boldsymbol{\Phi}\, \mathbf{u}

// reflected field on the RIS

5. for user

k = 1, \ldots, K

do

6.

\quad y_k \leftarrow \mathbf{H}_{\text{RIS-Rx},k}^H\, \mathbf{z} + w_k

7. end for

The cost of the forward pass is dominated by the $N_{\text{RIS}}\, N_a$ matrix-vector product $\mathbf{G}_f \mathbf{x}_a$ — feasible in real-time because $N_a$ is small by construction.

Common Mistake: Counting the Array Gain Twice

Mistake:

A natural but wrong reading of the array-fed RIS link budget is that the user enjoys the product $N_a \cdot N_{\text{RIS}}$ of the two stages' array gains. Students then conclude that a $32$ -element feed and a $1024$ -element RIS deliver $N_a N_{\text{RIS}} = 32{,}768$ of effective aperture gain.

Correction:

The reactive-near-field coupler $\mathbf{G}_f$ is an aperture-matching transformation, not a propagation stage. Nearly all of the active power flows into the RIS regardless of $N_a$ , so the active-feed directivity $G_a$ only sets how the power is distributed across modes on the RIS, not how much power is captured. The effective transmit array gain is therefore approximately $G_{\text{RIS}} \approx N_{\text{RIS}}$ alone, with the small correction from $G_a$ reflecting mode coupling efficiency. The $32{,}768$ figure overcounts by a factor of $N_a$ .

The Array-Fed RIS Architecture