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What Does RIS Replace?

RIS advocates often claim it is "a new kind of smart surface." But in wireless engineering, there is nothing sacred about newness: what matters is whether the new technology wins along the axes the operator cares about — cost, power, spectral efficiency, complexity. A grown-up introduction to RIS must compare it honestly with its competitors: amplify-and-forward (AF) relays, decode-and-forward (DF) relays, passive repeaters, and small active antenna arrays. We do so here.

Definition:
Repeaters and Relays — A Brief Taxonomy

We compare four architectures that all aim to improve coverage without deploying a full BS:

Passive repeater / reflectarray: A fixed, non-reconfigurable surface (or antenna pair) that reflects or re-radiates the signal with a predetermined phase pattern. No tunability, no power consumption, but no ability to follow the UE.
Amplify-and-forward (AF) relay: An active device that receives the BS signal, amplifies it, and re-transmits. Adds noise amplification; requires RF front-end and power amplifier; usually half-duplex (loses a factor of 2 in spectral efficiency).
Decode-and-forward (DF) relay: An AF relay plus baseband processing: it demodulates, decodes, re-encodes, and re-transmits. Higher quality, higher latency, higher complexity and power.
Active antenna array (small cell / mini-BS): A full transceiver with its own baseband. The gold standard for coverage, but the most expensive option in both capital and operating cost.
RIS (passive, reconfigurable): Our object of study. Passive (no amplifier), reconfigurable (each element's phase is tunable), full-duplex (no half-duplex loss), no added noise.

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RIS vs. Alternatives: Axes of Comparison

Property	Passive Repeater	AF Relay	DF Relay	Small Active Array	RIS
Power consumption	zero	moderate (PA)	high (PA + DSP)	high	very low (bias only)
Added noise	none	yes ( $N_0^{\text{relay}}$ )	no (regenerated)	yes	none
Duplexing	full	half (usually)	half	full	full
Reconfigurable?	no	limited (gain)	yes	yes	yes (phase only)
SNR scaling with size	$\mathcal{O}(N)$	linear in $G_a$	linear in $G_a$	$\mathcal{O}(N_t N_r)$	$\mathcal{O}(N^2)$ coherent
Per-link path loss	$d_1^2 d_2^2$	$d_1^2 + d_2^2$ (equivalent)	$d_1^2 + d_2^2$	$d_0^2$	$d_1^2 d_2^2$
CSI requirement at the node	none	none	full	full (at transmitter)	cascaded (indirect)
Deployment cost	low	medium	high	very high	low–medium

Theorem: RIS vs. AF Relay: The Crossover Point

Under equal-magnitude two-hop channels with per-hop amplitudes $\alpha, \beta$ and a single AF relay of power amplifier gain $g$ with its own noise at amplitude $\sigma_{\text{relay}}$ , the received SNRs are

$\text{SNR}_{\text{RIS}} = \frac{P_t\, \alpha^2 \beta^2\, N^2}{\sigma^2}, \qquad \text{SNR}_{\text{AF}} = \frac{P_t\, \alpha^2\, g^2\, \beta^2}{\sigma^2 + g^2 \beta^2 \sigma_{\text{relay}}^2}.$

Under high-SNR relay operation (the second term in the AF denominator dominates), $\text{SNR}_{\text{AF}}$ saturates at $P_t\alpha^2 / \sigma_{\text{relay}}^2$ , independent of relay gain. The RIS overtakes the AF relay once

$N^2 > \frac{\sigma^2}{\sigma_{\text{relay}}^2 \beta^2}.$

An AF relay with power budget $P_{\text{relay}}$ can be viewed as providing a fixed power gain that is spread over its single antenna. An RIS with $N$ elements provides a coherent power gain of $N^2 \alpha^2 \beta^2 P_t$ using zero additional power. For small $N$ , the active relay wins; for large $N$ , the $N^2$ scaling of the RIS overtakes. The crossover point depends on the relay's amplifier gain and the per-element RIS path loss.

Proof

RIS SNR

From $\mathcal{O}(N^2)$ $O (N^{2})$ SNR Scaling" data-ref-type="theorem">TCoherent RIS: $\mathcal{O}(N^2)$ SNR Scaling, with $\mathbf{v}=1$ and equal magnitudes: $\text{SNR}_{\text{RIS}} = (P_t/\sigma^2) N^2 \alpha^2 \beta^2$ .

AF relay SNR

The AF relay receives signal power $P_r^{\text{relay}} = P_t \alpha^2$ and noise power $\sigma_{\text{relay}}^2$ . After amplification by gain $g$ and second-hop loss $\beta^2$ , the signal at the UE is $g^2 \beta^2 P_t \alpha^2$ and the noise is $g^2 \beta^2 \sigma_{\text{relay}}^2 + \sigma^2$ . Taking the ratio gives the stated expression.

Crossover condition

In the regime where relay noise dominates ( $g^2 \beta^2 \sigma_{\text{relay}}^2 \gg \sigma^2$ ), $\text{SNR}_{\text{AF}} \to P_t\alpha^2 / \sigma_{\text{relay}}^2$ . Setting $\text{SNR}_{\text{RIS}} > \text{SNR}_{\text{AF}}$ and solving for $N^2$ yields the stated inequality. The crossover $N$ depends on the second-hop path loss $\beta$ : weaker $\beta$ means larger $N$ needed. $\blacksquare$

The Björnson Threshold

Björnson, Özdogan, and Larsson (2020) framed the RIS-vs-relay question in clean engineering terms: how large must $N$ be for a passive RIS to outperform a decent AF relay? Their answer, under realistic mmWave parameters, is hundreds to thousands of elements. This is the "Björnson threshold," and it is the main reason why early RIS deployments tend to use $N \geq 256$ or $N = 1024$ . Below the threshold, a well-designed AF relay wins.

Common Mistake: RIS Is Not Always Better

Mistake:

Some papers conclude that RIS "dominates" relays because they plot the coherent $N^2$ gain without modelling the two-hop path loss or the relay's alternative active gain. In those plots, RIS always wins.

Correction:

A fair comparison includes the full propagation model and the alternative technology's power budget. At moderate $N$ ( $\leq 100$ in most scenarios) and without severe blockage, an AF or DF relay of the same cost typically beats a passive RIS. RIS wins when (i) $N$ is genuinely large ( $\geq 256$ ), (ii) the direct path is blocked (so the relay would also need to exist — i.e., relay vs RIS comparison, not relay-vs-direct), and (iii) sustained power availability at a relay site is an issue.

⚠️Engineering Note

When to Deploy an RIS

A checklist for deciding between an RIS and alternatives:

Is the direct path reliably blocked? If yes, both RIS and relay are options. If no, neither is likely to be cost-effective.
Is there grid power at the candidate site? If no, RIS wins decisively (it needs only micro-power for bias).
Is the target area dense in UEs? If yes, a small cell or active array may be more spectrum-efficient.
Is the carrier frequency high? At mmWave/sub-THz, the small element size lets RIS achieve large $N$ in reasonable physical area — the $N^2$ gain kicks in more easily.
Is the UE mobile? Fast mobility strains the RIS control loop (Chapter 18). Low-mobility (fixed wireless access, indoor IoT) is the current sweet spot.

Practical Constraints

•
Typical break-even $N$ vs. an AF relay at 28 GHz: $N \sim 400$ – $1000$ depending on geometry.
•
RIS deployment cost per element (2024 estimate): $\sim \$0.10$ – $\$1$ at volume; full panel $\sim \$100$ – $\$1\,000$ .
•
Power consumption per element: $\sim 10\ \mu\text{W}$ idle, $\sim 100\ \mu\text{W}$ during reconfiguration.

RIS vs. AF Relay SNR as $N$ Grows

Animation of the crossover: the RIS coherent SNR grows quadratically with

N

while the AF relay's SNR is flat. The intersection — the Björnson threshold — is where passive RIS finally beats an active relay.

Why This Matters: Relationship to Massive MIMO

Massive MIMO and RIS are not competitors — they are complementary. A massive MIMO BS forms narrow beams but cannot cover UEs in a shadow. An RIS mounted on a wall near the shadow can catch the BS beam and re-direct it. The array-fed RIS of Chapter 11 is the purest expression of this combination: a small active array (Chapter 11 of MIMO book handles this side) illuminates a large passive RIS, and the two together give aperture plus spatial multiplexing. In this sense, RIS is less a new thing and more an aperture extension for existing MIMO systems.

See full treatment in Eigenmode Analysis of the BS-RIS Channel

RIS vs. Relays, Repeaters, and Active Antennas