Ferkans — Interactive Telecom Tutor

Why Scaling Laws Matter for RIS

The fundamental question for RIS deployment is: how many reflecting elements $N$ are needed to achieve a given performance target? Scaling laws — asymptotic expressions for performance metrics as $N \to \infty$ — provide the answer. The key insight is that RIS achieves an $N^2$ power scaling with optimal phase alignment, compared to the $N$ -scaling of relays and massive MIMO arrays. This quadratic scaling is the hallmark result of RIS theory and the primary reason for the intense research interest.

However, the $N^2$ scaling comes with a crucial caveat: it applies to the coherent beamforming gain on top of the "double path loss" inherent to the cascaded channel. Whether the $N^2$ gain can overcome the double path loss depends on the deployment geometry, operating frequency, and the availability of a direct link.

Theorem: $N^2$ SNR Scaling Law for RIS

Consider an RIS-assisted single-antenna system (no direct link, $M = 1$ ) where the received signal is:

$y = \mathbf{h}_r^H \boldsymbol{\Theta} \mathbf{g} \, x + n$

with $\mathbf{g} = [g_1, \ldots, g_N]^T$ (BS-to-RIS) and $\mathbf{h}_r = [h_{r,1}, \ldots, h_{r,N}]^T$ (RIS-to-user), and $\boldsymbol{\Theta} = \mathrm{diag}(e^{j\theta_1}, \ldots, e^{j\theta_N})$ .

The received SNR with optimal phase alignment $\theta_n^* = -\angle(h_{r,n}^* g_n)$ is:

$\text{SNR}^* = \frac{P}{\sigma^2} \left(\sum_{n=1}^{N} |h_{r,n}| |g_n|\right)^2$

If the channel coefficients $\{h_{r,n}\}$ and $\{g_n\}$ are i.i.d. with $\mathbb{E}[|h_{r,n}|] = \mu_h$ and $\mathbb{E}[|g_n|] = \mu_g$ (e.g., for Rayleigh fading, $\mu_h = \mu_g = \sqrt{\pi/4}$ ), then as $N \to \infty$ :

$\text{SNR}^* \xrightarrow{\text{a.s.}} \frac{P}{\sigma^2} N^2 \mu_h^2 \mu_g^2$

That is, the SNR scales as $N^2$ — quadratically in the number of RIS elements.

In contrast:

$N$ -antenna decode-and-forward relay: $\text{SNR} \propto N$ (power/array gain, linear scaling)
$N$ -antenna massive MIMO BS: $\text{SNR} \propto N$ (array gain, linear scaling)

Each of the $N$ RIS elements contributes a reflected path with random complex gain $h_{r,n}^* g_n$ . With optimal phase alignment, all $N$ contributions add coherently in magnitude: $|\sum_n |h_{r,n}||g_n|| = \sum_n |h_{r,n}||g_n|$ . The power of this coherent sum is $(\sum_n |h_{r,n}||g_n|)^2 \approx N^2 \mu_h^2 \mu_g^2$ by the law of large numbers.

For a relay, each antenna provides an independent power gain (no coherent combining across the two hops), so the total gain is $N$ . For massive MIMO, the array gain at a single hop is $N$ (coherent combining at one end), but there is no second-hop gain. The RIS achieves $N^2$ because it performs coherent combining across both hops simultaneously via the product structure of the cascaded channel.

Proof

Received signal with optimal phases

The received signal (before noise) is:

$s = \sum_{n=1}^{N} h_{r,n}^* e^{j\theta_n} g_n \cdot \sqrt{P}$

Each term has magnitude $|h_{r,n}||g_n|\sqrt{P}$ and phase $\theta_n + \angle g_n - \angle h_{r,n}$ . To maximise $|s|$ , all terms must have the same phase. Setting:

$\theta_n^* = \angle h_{r,n} - \angle g_n = -\angle(h_{r,n}^* g_n)$

gives $h_{r,n}^* e^{j\theta_n^*} g_n = |h_{r,n}||g_n|$ (real, positive).

SNR expression

With optimal phases:

$|s|^2 = P \left(\sum_{n=1}^N |h_{r,n}| |g_n|\right)^2$

$\text{SNR}^* = \frac{P}{\sigma^2}\left(\sum_{n=1}^N |h_{r,n}| |g_n|\right)^2$

Asymptotic scaling via the law of large numbers

Define $X_n = |h_{r,n}| |g_n|$ with $\mathbb{E}[X_n] = \mu_h \mu_g$ (by independence). By the strong law of large numbers:

$\frac{1}{N}\sum_{n=1}^N X_n \xrightarrow{\text{a.s.}} \mu_h \mu_g$

Therefore:

$\left(\sum_{n=1}^N X_n\right)^2 = N^2 \left(\frac{1}{N}\sum_{n=1}^N X_n\right)^2 \xrightarrow{\text{a.s.}} N^2 \mu_h^2 \mu_g^2$

and $\text{SNR}^* \to (P/\sigma^2) N^2 \mu_h^2 \mu_g^2$ .

For Rayleigh fading where $|h_{r,n}| \sim \mathrm{Rayleigh}$ and $|g_n| \sim \mathrm{Rayleigh}$ , we have $\mu_h = \mu_g = \sqrt{\pi/4}$ , so:

$\text{SNR}^* \to \frac{\pi^2 P}{16 \sigma^2} N^2$

Compared to the random phase baseline where the phases are uncontrolled:

$\mathbb{E}[\text{SNR}_{\text{random}}] = \frac{P}{\sigma^2} \sum_{n=1}^N \mathbb{E}[|h_{r,n}|^2] \mathbb{E}[|g_n|^2] = \frac{P}{\sigma^2} N$

The optimised RIS achieves a gain of $\pi N/4 \approx 0.785 N$ over random phases. $\blacksquare$

,

Example: Crossover Analysis: When Does RIS Beat a Relay?

A decode-and-forward relay with $N_r$ antennas provides SNR scaling $\text{SNR}_{\text{relay}} = (P/\sigma^2) \cdot \alpha_r N_r$ , where $\alpha_r$ accounts for the half-duplex loss and single-hop path loss. An $N$ -element RIS provides $\text{SNR}_{\text{RIS}} = (P/\sigma^2) \cdot \alpha_s N^2$ , where $\alpha_s$ accounts for the double path loss.

Suppose $\alpha_r = 10^{-3}$ (relay at 50 m, single-hop path loss) and $\alpha_s = 10^{-7}$ (RIS double path loss with 50 m BS-RIS and 50 m RIS-user at 3.5 GHz).

(a) Find the crossover point: the number of RIS elements $N^*$ at which the RIS matches the relay with $N_r = 4$ antennas.

(b) Is $N^*$ practically feasible?

Solution

Crossover equation

Setting $\text{SNR}_{\text{RIS}} = \text{SNR}_{\text{relay}}$ :

$\alpha_s N^{*2} = \alpha_r N_r$

$N^* = \sqrt{\frac{\alpha_r N_r}{\alpha_s}} = \sqrt{\frac{10^{-3} \times 4}{10^{-7}}} = \sqrt{4 \times 10^4} = 200$

Feasibility assessment

(b) $N^* = 200$ elements is practically feasible. At 3.5 GHz, the wavelength is $\lambda = 8.6$ cm and element spacing is $\lambda/2 = 4.3$ cm. A $14 \times 14$ element RIS occupies $(14 \times 4.3)^2 \approx 60 \times 60$ cm $^2$ — roughly the size of a small poster.

At 28 GHz ( $\lambda = 1.07$ cm), the same 200 elements fit in a $7.5 \times 7.5$ cm $^2$ surface.

For $N > 200$ , the RIS outperforms the relay due to $N^2$ scaling. For $N < 200$ , the relay wins because its single-hop path loss advantage dominates.

Practical caveat: this analysis assumes perfect CSI and continuous phase shifts. Phase quantisation, imperfect CSI, and mutual coupling reduce the effective $N^2$ gain. $\blacksquare$

$N^2$ vs $N$ Scaling on Log-Log Plot

Visualise the fundamental scaling difference between RIS (

N^2

), relays (

N

), and massive MIMO (

N

) on a log-log SNR vs

N

plot. The RIS line has slope 2, crossing the relay line at a crossover point

N^*

that depends on the path-loss geometry. For

N > N^*

, the quadratic RIS gain dominates.

RIS achieves slope-2 (20 dB/decade) scaling while relay and massive MIMO achieve slope-1 (10 dB/decade). Crossover at

N^* \approx 100

.

$N^2$ vs $N$ Scaling Comparison

Compare the SNR scaling of RIS (quadratic in $N$ ), relay (linear in $N$ ), and massive MIMO (linear in $N$ ) as a function of the number of elements/antennas. The plot shows the received SNR in dB versus $N$ on a log scale. The RIS curve has slope 2 (in the log-log plot) while the relay and massive MIMO curves have slope 1, illustrating the fundamental $N^2$ vs $N$ scaling difference. Adjust the base SNR (the single-element/antenna SNR) to shift all curves vertically. Observe that despite its steeper slope, RIS may require a large $N$ to overcome the initial path loss disadvantage.

Parameters

N_\text{max}

256

Base SNR (dB)-10

Rate Comparison Across Technologies

Compare the achievable rate (bits/s/Hz) of four technologies as a function of the number of elements/antennas $N$ : (1) RIS with optimal phases, (2) half-duplex decode-and-forward relay, (3) massive MIMO (single-hop), and (4) direct link only (no assistance). The comparison accounts for path loss (using the specified distances), half-duplex penalty for relays, and the double path loss for RIS. Observe the crossover point where the $N^2$ scaling of RIS overtakes the $N$ scaling of relays. The direct link distance and RIS distance can be varied to explore different deployment geometries.

Parameters

P

(dBm)20

d_\text{direct}

(m)100

d_\text{RIS}

(m)60

Practical Caveats to the $N^2$ Scaling Law

The ideal $N^2$ scaling requires several conditions that may not hold in practice:

1. Perfect CSI. The phase alignment $\theta_n^* = -\angle(h_{r,n}^* g_n)$ requires exact knowledge of each element's cascaded channel. With noisy CSI, the effective scaling degrades. If the channel estimation error variance is $\sigma_e^2$ per element, the SNR scales as:

$\text{SNR} \approx \frac{P}{\sigma^2} \left(N^2 \mu_h^2 \mu_g^2 - N \sigma_e^2\right)$

The $N^2$ scaling dominates for large $N$ only if $\sigma_e^2 \to 0$ sufficiently fast; otherwise, the error term grows linearly and eventually dominates.

2. Continuous phase shifts. Quantising to $b$ bits per element reduces the gain by a factor of $\mathrm{sinc}^2(\pi/2^b)$ (see Section 28.5), which is independent of $N$ and thus preserves the $N^2$ scaling asymptotically.

3. Far-field assumption. The $N^2$ scaling assumes all elements are in the far field of both the BS and user. For very large $N$ (thousands of elements), the surface may extend into the near-field regime, where the plane-wave approximation breaks down and the per-element channel gains become position-dependent.

4. Channel correlation. The i.i.d. channel model used in the proof is pessimistic for LOS channels (where the gain can be even larger) but optimistic for spatially correlated NLOS channels (where the effective rank of the channel may be less than $N$ ).

Common Mistake: The $N^2$ Gain Does Not Automatically Beat Direct Links

Mistake:

Assuming that the $N^2$ SNR scaling of RIS always outperforms a direct link or relay, regardless of the deployment geometry.

Correction:

The $N^2$ scaling applies to the coherent beamforming gain on top of the cascaded channel's double path loss. With path-loss exponent $\alpha$ , the RIS reflected SNR scales as:

$\text{SNR}_{\text{RIS}} \propto \frac{N^2}{(d_1 d_2)^\alpha}$

while the direct link scales as $1/d^{\alpha}$ . If $d_1 d_2 \gg d$ , the double path loss can be so severe that thousands of elements are needed before the $N^2$ gain compensates. Always compute the crossover point $N^* = \sqrt{\alpha_r N_r / \alpha_s}$ before concluding that RIS is beneficial for a specific deployment.

Common Mistake: Ignoring Channel Estimation Overhead in Scaling Analysis

Mistake:

Citing the $N^2$ scaling law while neglecting that achieving it requires $O(N)$ pilot training slots, each of which consumes coherence-interval resources.

Correction:

With $T = N + 1$ pilot slots out of a coherence interval of $\tau_c$ symbols, only $\tau_c - N - 1$ symbols remain for data. The net throughput including training overhead is:

$R_{\text{net}} = \frac{\tau_c - N - 1}{\tau_c} \log_2(1 + \text{SNR}(N))$

For large $N$ , the training fraction $N/\tau_c$ can dominate, creating a throughput ceiling even though $\text{SNR}(N)$ grows as $N^2$ . The optimal $N$ balances beamforming gain against training overhead.

Common Mistake: RIS is Not a Free Lunch

Mistake:

Treating RIS as a zero-cost addition to the network, since it consumes "no power" and "no bandwidth."

Correction:

While RIS elements are passive (no RF power), an RIS deployment entails real costs:

Control signalling: the BS must communicate phase configurations to the RIS controller, consuming control channel resources
Channel estimation: $O(N)$ pilot overhead per coherence interval
Backhaul/control link: wired or wireless link to the RIS controller
Hardware cost: even simple PIN-diode elements cost money at scale
Site acquisition: the RIS must be placed at a useful location

A fair comparison with alternatives (relay, additional BS antennas) must account for all these costs, not just the RF power.

Key Takeaway

The $N^2$ SNR scaling law is the signature result of RIS theory: with optimal phase alignment, $N$ passive reflecting elements provide a coherent beamforming gain that scales quadratically, compared to the linear $N$ -scaling of relays and massive MIMO arrays. This quadratic gain arises because the RIS performs coherent combining across both hops of the cascaded channel simultaneously. However, the $N^2$ gain sits on top of a double path loss, so large $N$ is needed to achieve practical benefit.

RIS vs. Relay vs. Massive MIMO

Property	RIS (passive)	DF Relay	Massive MIMO BS
SNR scaling	$N^2$	$N$	$N$
Path loss	Double (cascaded)	Single hop	Single hop
RF chains	None	$N$ (full)	$N$ (full)
Power consumption	mW (control only)	Watts (TX + processing)	Watts (TX + processing)
Noise amplification	No	No (DF) / Yes (AF)	No
Duplexing	Full-duplex	Half-duplex (DF)	Full-duplex
Channel estimation	$O(N)$ overhead	Standard pilots	Standard pilots
Self-interference	None	Possible (FD relay)	None
Deployment flexibility	Thin surface, conformable	Standalone node	Tower/rooftop

Why This Matters: RIS in 6G Standards and Research

RIS is a leading candidate technology for 6G wireless systems, with active research in standardisation bodies:

3GPP: Study items on "smart repeaters" (a form of active/passive RIS) have been discussed in Release 18 and beyond.
ETSI ISG RIS: A dedicated Industry Specification Group was established in 2021 to define use cases, deployment scenarios, and evaluation methodologies for RIS-aided networks.
ITU-R: The IMT-2030 framework identifies "intelligent radio environment" as a key 6G enabling technology.

Prototype demonstrations have shown:

Varactor-based RIS with 256 elements at 5.8 GHz (NTT, 2020)
PIN-diode RIS with 1024 elements at 28 GHz (Samsung, 2022)
Liquid-crystal RIS for sub-THz frequencies (Fraunhofer, 2023)

The transition from laboratory prototypes to commercial products requires solving the channel estimation, control signalling, and deployment optimisation challenges discussed in this chapter.

See full treatment in Chapter 33

Quick Check

Why does an RIS achieve $N^2$ SNR scaling while a massive MIMO array with $N$ antennas at the BS achieves only $N$ scaling?

Because the RIS has more physical aperture than a massive MIMO array

Because the RIS reflects the signal twice (once on each side), doubling the gain in dB

Because the RIS coherently combines $N$ reflected paths, and the squared magnitude of the coherent sum gives $(\sum |a_n|)^2 \approx N^2$ , whereas massive MIMO coherently combines at one end giving $\sum |a_n|^2 \approx N$

Because the RIS operates in full-duplex mode, gaining a factor of 2 ( $N^2 = 2 \times N$ )

Correction:

Because the RIS coherently combines

N

reflected paths, and the squared magnitude of the coherent sum gives

(\sum |a_n|)^2 \approx N^2

, whereas massive MIMO coherently combines at one end giving

\sum |a_n|^2 \approx N

In massive MIMO, the array gain comes from coherently combining $N$ received signals, each with power $|a_n|^2$ . The total power after combining is $|\sum a_n e^{j\phi_n}|^2 \approx N$ (since $\sum |a_n|^2 \approx N$ , and the cross-terms average to zero in the NLOS case).

For an RIS, the gain comes from the squared magnitude of the coherent sum of products $|h_{r,n}||g_n|$ . After phase alignment, the signal power is $(\sum |h_{r,n}||g_n|)^2$ . By the law of large numbers, the sum grows as $N\mu$ , so the squared sum grows as $N^2\mu^2$ . The crucial difference is that the squaring operation produces the $N^2$ factor.

Performance Analysis and Scaling Laws

Why Scaling Laws Matter for RIS

Theorem: N2N^2N2 SNR Scaling Law for RIS

Received signal with optimal phases

SNR expression

Asymptotic scaling via the law of large numbers

Example: Crossover Analysis: When Does RIS Beat a Relay?

Crossover equation

Feasibility assessment

N2N^2N2 vs NNN Scaling on Log-Log Plot

N2N^2N2 vs NNN Scaling Comparison

Parameters

Rate Comparison Across Technologies

Parameters

Practical Caveats to the N2N^2N2 Scaling Law

Common Mistake: The N2N^2N2 Gain Does Not Automatically Beat Direct Links

Common Mistake: Ignoring Channel Estimation Overhead in Scaling Analysis

Common Mistake: RIS is Not a Free Lunch

Key Takeaway

RIS vs. Relay vs. Massive MIMO

Why This Matters: RIS in 6G Standards and Research

Quick Check

Theorem: $N^2$ SNR Scaling Law for RIS

$N^2$ vs $N$ Scaling on Log-Log Plot

$N^2$ vs $N$ Scaling Comparison

Practical Caveats to the $N^2$ Scaling Law

Common Mistake: The $N^2$ Gain Does Not Automatically Beat Direct Links