Ferkans — Interactive Telecom Tutor

A Surface That Costs Almost Nothing to Power

Every hardware-aware architecture we have seen so far — hybrid beamforming in Chapter 20, mixed-ADC receivers in Chapter 19, even the subarray-based XL-MIMO of Chapter 18 — starts from the same premise: a certain number of RF chains must illuminate the antennas. A reconfigurable intelligent surface (RIS) takes the opposite point of view. What if almost every element on the large array were passive — a tunable phase shifter reflecting whatever incoming wave it happens to see? The RF-chain count drops to zero on the surface itself, and the energy cost of adding another element is essentially the cost of its control line. Operationally, a 1024-element RIS runs on the power budget of a small LED.

The catch is the signal model. An RIS does not transmit; it reflects. A wave launched from a transmitter at range $d_1$ from the surface is scattered back toward a receiver at range $d_2$ , and the end-to-end power budget is $\propto 1/(d_1^2 d_2^2)$ — the double-fading loss. For large $d_1$ and $d_2$ , the nominal aperture gain $G_{\text{RIS}} \approx N_{\text{RIS}}$ is completely erased by this product of path losses. This section builds the RIS model carefully, so we can measure the double-fading penalty precisely and set the stage for Section 21.2, where the CommIT array-fed architecture of Caire and collaborators eliminates most of it.

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Definition:
Reconfigurable Intelligent Surface

A reconfigurable intelligent surface (RIS) is a planar array of $N_{\text{RIS}}$ sub-wavelength passive elements ("tiles"), each acting as a tunable phase shifter. Element $n$ reflects an incident narrowband wave $s_{\text{in},n}$ as

$s_{\text{out},n} = \Gamma_n\, s_{\text{in},n}, \qquad \Gamma_n = \alpha_n\, e^{j\phi_n},$

where $\phi_n \in [0, 2\pi)$ is programmable and $\alpha_n \in [0, 1]$ is the amplitude response (typically $\alpha_n \approx 1$ for a well-designed PIN-diode or varactor element). We assume unit-modulus reflection, $|\Gamma_n| = 1$ , throughout this chapter; amplitude-control RIS variants are discussed briefly in the references.

The RIS has no RF chain, no amplifier, no active transmission. Its only power draw is the DC bias that sets each element's phase state, on the order of a few mW for the entire surface. A continuous-phase RIS is an idealization; real hardware uses $b$ -bit phase quantization, $b \in \{1, 2, 3\}$ , so that $\phi_n \in \{0, 2\pi/2^b, \ldots, 2\pi(2^b-1)/2^b\}$ .

The name "intelligent" is marketing. The surface itself has no intelligence; it executes the phase profile $\boldsymbol{\phi}$ that a controller computes based on channel state information. "Reconfigurable passive reflector" would be a more honest name, and most of the early literature used exactly that term (reflectarray) before the rebranding around 2019.

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Reconfigurable Intelligent Surface (RIS)

A planar array of $N_{\text{RIS}}$ passive, sub-wavelength tiles with programmable phase shifters. The RIS reflects an incident wave with a chosen complex-valued pattern $\Gamma_n = \alpha_n e^{j\phi_n}$ per element. It carries no RF chain and its DC power budget is essentially the cost of the phase-shifter control line — a few mW for the entire surface. Synonyms: intelligent reflecting surface (IRS), software metasurface, programmable reflectarray.

Historical Note: From Reflectarrays to Intelligent Surfaces

1963–2019

Passive reflectarrays were proposed in 1963 by Berry, Malech, and Kennedy as a way to shape a reflected beam by choosing the phase delay at each element of a planar array without any active circuitry. For fifty years the idea sat inside the antenna-engineering community as a niche technique for satellite dishes and radar. The reconfigurable variant appeared in the 2000s with varactor- and MEMS-controlled elements, and by 2010 it was clear that an electronically steerable reflector was practical. What changed in 2019 was the system-level framing: Basar, Di Renzo, and others argued that if communication systems could co-design their propagation environment with the same optimization tools used for transmit beamforming, the resulting gains would be large enough to treat the metasurface as a first- class degree of freedom. The CommIT group, with its long background in massive MIMO and hybrid architectures, joined this wave by asking the right engineering question: can a metasurface actually transmit power efficiently, or is the double-fading loss a deal-breaker?

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Definition:
End-to-End RIS-Aided Narrowband Signal Model

A transmitter with $N_t$ antennas sends a symbol $s \in \mathbb{C}$ using precoding vector $\mathbf{v} \in \mathbb{C}^{N_t}$ . The signal reaches the RIS through the forward channel $\mathbf{H}_{\text{Tx-RIS}} \in \mathbb{C}^{N_{\text{RIS}} \times N_t}$ , is reshaped elementwise by the RIS diagonal reflection matrix $\boldsymbol{\Phi} = \text{diag}(e^{j\phi_1}, \ldots, e^{j\phi_{N_{\text{RIS}}}})$ , then travels to a single-antenna receiver through the reflected channel $\mathbf{H}_{\text{RIS-Rx}}^H \in \mathbb{C}^{1 \times N_{\text{RIS}}}$ . The baseband received signal is

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

where $\mathbf{w} \sim \mathcal{CN}(0, \sigma^2)$ . The end-to-end scalar channel factorizes through the RIS phase profile $\boldsymbol{\phi}$ , so every rate metric in this chapter is ultimately an optimization problem over both $\mathbf{v}$ (the Tx precoder) and $\boldsymbol{\phi}$ (the RIS configuration).

When the RIS and the Rx have multiple antennas, the same expression generalizes to $\mathbf{H}_{\text{eff}} = \mathbf{H}_{\text{RIS-Rx}}\, \boldsymbol{\Phi}\, \mathbf{H}_{\text{Tx-RIS}} \in \mathbb{C}^{N_r \times N_t}$ .

This is a bilinear model in $\boldsymbol{\phi}$ and $\mathbf{v}$ : fixing one makes the other a linear optimization, but joint optimization is non-convex. Alternating optimization and SDR-based relaxations are the standard tools; we explore them in Section 21.4.

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Theorem: Double-Fading Path Loss

Consider a far-field RIS-aided link with a transmitter of isotropic power $P_t$ located at distance $d_1$ from the center of an RIS of $N_{\text{RIS}}$ elements, and a receiver at distance $d_2$ from the RIS. Assume the RIS phase profile is aligned with the user (constructive combining). The received signal power is

$P_r = P_t\, \beta_0^2\, N_{\text{RIS}}^2\, \frac{1}{(d_1\, d_2)^2},$

where $\beta_0$ is a constant that lumps the carrier wavelength, element gain, and normalization. Equivalently, in dB,

$10 \log_{10}(P_r/P_t) = 10 \log_{10}(\beta_0^2 N_{\text{RIS}}^2) - 20\log_{10}(d_1\, d_2).$

Three things are competing here. The RIS contributes an array gain of $N_{\text{RIS}}^2$ (not $N_{\text{RIS}}$ ) because constructive phase alignment in the reflected link squares the amplitude gain. But the price for being passive is that the path loss multiplies: the signal decays like $1/d_1^2$ from the Tx to the RIS and like $1/d_2^2$ from the RIS to the Rx. The product $(d_1 d_2)^2$ grows fast enough that for typical cell sizes, even a 1024-element RIS barely matches a short direct link. The double-fading is the central obstacle RIS architectures must overcome.

Proof

Radiated and received amplitude per element

Each RIS element receives an amplitude $\propto \sqrt{P_t}\, \beta_0/d_1$ (free-space $1/r$ falloff in the far field). After unit-modulus reflection, the element re-radiates the same amplitude, which reaches the receiver scaled by a further $\beta_0/d_2$ . Element $n$ contributes amplitude $\propto \sqrt{P_t}\, \beta_0^2/(d_1 d_2)$ times a phase factor $e^{j\phi_n + j\psi_n}$ , where $\psi_n$ is the round-trip propagation phase.

Coherent combining over $N_{ ext{RIS}}$ elements

The optimal phase profile sets $\phi_n = -\psi_n$ so every element contributes in phase at the receiver. The total received amplitude is $N_{\text{RIS}}$ times the per-element amplitude, so received power scales as $N_{\text{RIS}}^2$ . Plugging in gives $P_r \propto P_t (\beta_0^2 N_{\text{RIS}} / (d_1 d_2))^2 = P_t \beta_0^4 N_{\text{RIS}}^2 / (d_1 d_2)^2$ .

Wrap the constants and take dB

Collecting $\beta_0^4$ into $\beta_0^2$ by redefining the constant and taking $10 \log_{10}(\cdot)$ yields the stated expression. $\blacksquare$

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Key Takeaway

Double fading is the central obstacle. Although the RIS adds $20 \log_{10}(N_{\text{RIS}})$ dB of array gain, it also pays $20 \log_{10}(d_1 d_2)$ dB of cascaded path loss. For typical mmWave cell geometries this nearly cancels the gain, so a passive RIS only beats a direct link when either (a) the direct link is blocked or (b) the Tx and RIS sit very close, so that $d_1$ effectively disappears. Option (b) is the architectural insight behind the array-fed RIS of Section 21.2.

SNR Gain vs RIS Size: Aperture vs Double-Fading

Compare the expected SNR of an RIS-aided link to that of a direct Tx $\to$ Rx link as the number of RIS elements $N_{\text{RIS}}$ grows. The direct SNR is fixed; the RIS SNR grows like $N_{\text{RIS}}^2$ but pays $(d_1 d_2)^2$ of path loss. Vary the two distances to see where the crossover happens. At short $d_1$ (the array-fed regime of Section 21.2) the crossover moves dramatically to the left.

Parameters

d_1

(m, Tx

\to

RIS)20

d_2

(m, RIS

\to

Rx)20

d_{\text{direct}}

(m)40

f_0

(GHz)28

P_t

(dBm)30

Example: How Many RIS Elements to Match a Direct Link?

A Tx-Rx pair has a clear line-of-sight direct link of length $d_{\text{direct}} = 40$ m at $f_0 = 28$ GHz with path-loss exponent $2$ . We want to add an RIS at equal distances $d_1 = d_2 = 20$ m to boost the received power. Approximately how many RIS elements $N_{\text{RIS}}$ are required for the reflected-link SNR to match the direct-link SNR, ignoring directivity differences?

Solution

Equate SNRs

Direct SNR $\propto 1/d_{\text{direct}}^2 = 1/1600$ . Reflected SNR $\propto N_{\text{RIS}}^2 / (d_1 d_2)^2 = N_{\text{RIS}}^2/160000$ . Equating: $N_{\text{RIS}}^2/160000 = 1/1600$ , so $N_{\text{RIS}}^2 = 100$ — thus $N_{\text{RIS}} \approx 10$ .

Add a fair wavelength-dependent constant

A more careful calculation including a factor of $\lambda^{2}/(4\pi)^2$ per hop and $\lambda^2/(4\pi)$ for the direct link (Friis) reveals that the minimum element count is roughly $N_{\text{RIS}} \approx 4 \pi d_1 d_2 / (\lambda\, d_{\text{direct}}) \cdot \text{const}$ . For $\lambda = 1.07$ cm and $d_1 = d_2 = 20$ m with $d_{\text{direct}} = 40$ m, this gives $N_{\text{RIS}}$ in the hundreds, not ten.

Interpretation

A passive RIS only pays off when it is either very large (thousands of elements) or positioned very asymmetrically (one of $d_1, d_2$ small). The second condition is exactly what the array-fed RIS of Section 21.2 engineers deliberately. $\blacksquare$

Common Mistake: The $N^2$ Gain Is Not Free

Mistake:

A common slide claims that an RIS delivers a quadratic $N_{\text{RIS}}^2$ gain, vastly better than the linear $N_{\text{RIS}}$ gain of a conventional phased array. Readers conclude that even a modest RIS beats any active array of the same size.

Correction:

The $N_{\text{RIS}}^2$ scaling is relative to the per-element reflected power, not relative to an active array. After you account for the double path loss $(d_1 d_2)^{-2}$ , the RIS-aided SNR is $N_{\text{RIS}}^2/(d_1 d_2)^2$ , while a conventional phased array of the same size at distance $d$ from the user achieves $N_{\text{RIS}}/d^2$ . For comparable distances, the RIS loses by $d^2/(d_1 d_2)^2 \cdot N_{\text{RIS}}$ which is a large number for any reasonable geometry. The quadratic scaling is a mathematical artifact of measuring from the wrong baseline. The right baseline is power-into-Rx per unit Tx power, where the RIS pays double fading.

⚠️Engineering Note

Phase Quantization and Element Coupling in Real Hardware

Real metasurface elements are not ideal continuous phase shifters. Three hardware realities distort the clean signal model of this section:

Phase quantization. PIN-diode tiles typically offer $b = 1$ bit ( $\phi_n \in \{0, \pi\}$ ), varactor tiles go up to $b = 3$ – $4$ bits, and continuous-phase MEMS prototypes exist. The SNR loss from $b$ -bit quantization is approximately $10 \log_{10}(\text{sinc}^2(\pi/2^b))$ dB, so $b = 2$ costs about $0.9$ dB, $b = 3$ only $0.22$ dB.
Amplitude-phase coupling. A realistic element reflects with $\Gamma_n(\phi) = \alpha(\phi) e^{j\phi}$ where $\alpha(\phi)$ dips near the transitions between quantization levels. The surface is slightly lossier when tuned mid-cell.
Mutual coupling. At sub-wavelength spacing, neighbouring elements interact electromagnetically. The reflection pattern is not the element-wise product assumed in the ideal model. Measured mutual- coupling matrices can be incorporated as a precomputed transformation, but they violate the separability of the optimization over $\boldsymbol{\phi}$ .

For the chapter's analysis we stay in the ideal model; the array-fed RIS of Section 21.2 is actually more robust to these imperfections because its gain comes primarily from the active array, not from sub-wavelength phase precision on the RIS.

Practical Constraints

•
Real RIS phase quantization is 1–3 bits, not continuous
•
Amplitude varies with phase state: loss of ~0.3–1 dB depending on element technology
•
Sub-wavelength spacing induces mutual coupling: element-wise model is an approximation

📋 Ref: ETSI GR RIS 003 (draft), 3GPP TR 38.843 (study on smart radio environments)

Why This Matters: Is an RIS Just a Relay in Disguise?

A natural question is how a passive RIS differs from a half-duplex amplify- and-forward relay. Superficially they look similar: both take a signal that has travelled $d_1$ , boost/reshape it, and send it another $d_2$ . The difference is that a relay amplifies — it contains an RF chain and a power amplifier, so its output power is set by its own budget, not by the incoming wave. A passive RIS just reflects, so its output power is limited by whatever incoming power the Tx managed to deliver. Björnson, Özdogan, and Larsson (2020) showed that a 1 W relay with 100 receive antennas typically outperforms a 1000-element passive RIS at realistic distances. The array-fed RIS architecture of Section 21.2 is best understood as a concentrated relay + a passive aperture — it combines the active-transmit advantages of a relay with the low-cost aperture of an RIS, without using either as a standalone architecture.

Quick Check

A passive RIS with $N_{\text{RIS}} = 1024$ elements is placed at $d_1 = d_2 = 50$ m between Tx and Rx; the direct link has length $d_{\text{direct}} = 100$ m. Using the simplified model $P_r^{\text{direct}} \propto 1/d_{\text{direct}}^2$ and $P_r^{\text{RIS}} \propto N_{\text{RIS}}^2/(d_1 d_2)^2$ (equal constants, optimal phase profile), by how many dB is the RIS-aided link better or worse than the direct link?

$+60$ dB better

$+32$ dB better

$\approx 0$ dB, roughly equal

$-20$ dB worse

Correction:

+32

dB better

$P_r^{\text{direct}} \propto 1/10^4$ . $P_r^{\text{RIS}} \propto 1024^2/(50\cdot 50)^2 = 10^6/(6.25\cdot 10^6) \approx 0.16$ . The ratio is $0.16 \cdot 10^4 = 1600$ , i.e. $10\log_{10}(1600) \approx 32$ dB. The 1024-element RIS outperforms the direct link here, but only because the simplified model ignores wavelength factors — a full Friis accounting (Example EHow Many RIS Elements to Match a Direct Link?) brings the gain down dramatically and the crossover pushes into the thousands of elements.

RIS Fundamentals and the Double-Fading Problem

A Surface That Costs Almost Nothing to Power

Definition: Reconfigurable Intelligent Surface

Reconfigurable Intelligent Surface (RIS)

Historical Note: From Reflectarrays to Intelligent Surfaces

Definition: End-to-End RIS-Aided Narrowband Signal Model

Theorem: Double-Fading Path Loss

Radiated and received amplitude per element

Coherent combining over $N_{ ext{RIS}}$ elements

Wrap the constants and take dB

Key Takeaway

SNR Gain vs RIS Size: Aperture vs Double-Fading

Parameters

Example: How Many RIS Elements to Match a Direct Link?

Equate SNRs

Add a fair wavelength-dependent constant

Interpretation

Common Mistake: The N2N^2N2 Gain Is Not Free

Phase Quantization and Element Coupling in Real Hardware

Why This Matters: Is an RIS Just a Relay in Disguise?

Quick Check

Definition:
Reconfigurable Intelligent Surface

Definition:
End-to-End RIS-Aided Narrowband Signal Model

Common Mistake: The $N^2$ Gain Is Not Free