Ferkans — Interactive Telecom Tutor

What Is the Ultimate Limit of RIS?

We have seen that RIS provides $N^2$ SNR gain in the coherent combining regime and $N$ in the random regime. But what is the capacity — the information-theoretically maximum rate — of a RIS-aided channel? This is the fundamental limit; all practical algorithms sit below it. The question was posed in 2019 and is still partially open.

Definition:
RIS-Aided Channel Capacity

Consider a point-to-point link with an active transmit precoder $\mathbf{w}$ and a passive RIS with phase-shift matrix $\boldsymbol{\Phi}$ . The input-output relation is $y = (\mathbf{h}_d^H + \mathbf{h}_2^H \boldsymbol{\Phi} \mathbf{H}_1) \mathbf{w} x + w,$ with $w \sim \mathcal{CN}(0, \sigma^2)$ . The RIS-aided capacity is $C_{\text{RIS}} = \sup_{|\phi_n|=1, \|\mathbf{w}\| \leq 1} \log_2\left(1 + \frac{|(\mathbf{h}_d^H + \mathbf{h}_2^H \boldsymbol{\Phi} \mathbf{H}_1)\mathbf{w}|^2 P_t} {\sigma^2}\right).$ The supremum is subject to the non-convex unit-modulus constraint.

Theorem: Capacity Upper Bound

The RIS-aided capacity satisfies $C_{\text{RIS}} \leq \log_2\left(1 + \frac{P_t (\|\mathbf{h}_d\| + \|\mathbf{h}_2\|\cdot\|\mathbf{H}_1\|_F)^2} {\sigma^2}\right).$ This upper bound assumes perfect coherent combining across all paths and is typically tight within 1-2 dB at large $N$ .

Proof

Cauchy-Schwarz

$|(\mathbf{h}_d^H + \mathbf{h}_2^H \boldsymbol{\Phi} \mathbf{H}_1) \mathbf{w}| \leq \|\mathbf{h}_d + \mathbf{H}_1^H \boldsymbol{\Phi}^H \mathbf{h}_2\| \cdot \|\mathbf{w}\|$ .

Triangle inequality

$\|\mathbf{h}_d + \mathbf{H}_1^H \boldsymbol{\Phi}^H \mathbf{h}_2\| \leq \|\mathbf{h}_d\| + \|\mathbf{H}_1^H \boldsymbol{\Phi}^H \mathbf{h}_2\| \leq \|\mathbf{h}_d\| + \|\mathbf{H}_1\|_F \|\mathbf{h}_2\|$ .

Capacity

Applying these to the capacity formula with $\|\mathbf{w}\| = 1$ : $C \leq \log_2(1 + P_t (\|\mathbf{h}_d\| + \|\mathbf{h}_2\|\|\mathbf{H}_1\|_F)^2 / \sigma^2)$ . $\blacksquare$

Theorem: Asymptotic Capacity Scaling

As $N \to \infty$ with fixed transmit power, the RIS-aided capacity satisfies $C_{\text{RIS}} = \log_2(N^2) + \log_2(\text{SNR}) + O(1) = 2 \log_2 N + O(1)$ with the $O(1)$ term accounting for the channel's angular-spread structure and the direct path. The key result is: capacity grows as $\log_2(N^2) = 2\log_2 N$ — the RIS doubles the degrees of freedom relative to a conventional point-to-point link.

Proof

Peak SNR

With optimal $\boldsymbol{\Phi}$ and $\mathbf{w}$ , the effective SNR is $P_t N^2 \alpha_{\text{RIS}}/\sigma^2$ .

Capacity in log

$C = \log_2(1 + P_t N^2 \alpha_{\text{RIS}}/\sigma^2) = \log_2(P_t \alpha_{\text{RIS}}/\sigma^2) + \log_2(N^2) + O(1/N^2)$

Rewriting

$C = 2\log_2 N + \log_2(\text{SNR} \alpha_{\text{RIS}}) + O(1)$ . The $2\log_2 N$ is the asymptotic RIS rate scaling. $\blacksquare$

What 'Doubled DoF' Means

A conventional point-to-point link with $N$ transmit antennas has a per-user capacity scaling of $\log_2 N$ (single-user beamforming gain). A RIS-aided link with $N$ elements gives $2 \log_2 N$ . The factor 2 comes from: (i) coherent combining at the RIS, (ii) coherent beamforming at the BS. Both work together, effectively doubling the spatial degrees of freedom. This is the fundamental statement of why RIS is valuable in the capacity-scaling regime.

Example: Capacity Gain at 28 GHz

A 28 GHz link with $\text{SNR} = 10$ dB has capacity $C_{\text{direct}} = \log_2(1 + 10) \approx 3.46$ bits/s/Hz. Adding a RIS with $N = 256$ elements, path-loss factor $\alpha_{\text{RIS}} = 10^{-7}$ . Compute the RIS-aided capacity.

Solution

Effective SNR

$\text{SNR}_{\text{RIS}} = 10 \cdot 256^2 \cdot 10^{-7} = 6.55 \times 10^{-2}$ . Wait — the product path loss is too strong. The direct path still dominates. Let's recompute with $\alpha_{\text{RIS}} = 10^{-4}$ (closer deployment): $\text{SNR}_{\text{RIS}} = 10 \cdot 256^2 \cdot 10^{-4} = 65.5$ .

Capacity

$C_{\text{RIS}} = \log_2(1 + 10 + 65.5) = \log_2(76.5) \approx 6.26$ bits/s/Hz. Gain: $6.26 - 3.46 = 2.8$ bits/s/Hz — a modest but real improvement.

Large N

Doubling to $N = 512$ gives $\text{SNR}_{\text{RIS}} = 10 \cdot 512^2 \cdot 10^{-4} = 262$ . $C = \log_2(1 + 10 + 262) = \log_2(273) \approx 8.09$ bits/s/Hz. An additional $1.83$ bits/s/Hz. The $2 \log_2 N$ scaling means each doubling of $N$ adds 2 bits/s/Hz asymptotically; this example is near-asymptotic.

RIS-Aided Capacity vs. $N$

Plot $C_{\text{RIS}}(N)$ vs. $N$ on log-log axes. Compare to the asymptotic $2\log_2 N$ line and the direct-link capacity. The slope in the high- $N$ regime should approach 2 (doubled DoF).

Parameters

Max

N

1024

Direct-path SNR (dB)10

Product-path-loss ratio (dB)-40

Capacity Open Problems

Despite the asymptotic result, several capacity-related open problems remain:

Exact capacity under imperfect CSI: the upper bound assumes perfect $\mathbf{h}_{\text{eff}}$ knowledge. Real systems have noise-limited channel estimates. The capacity-achieving input distribution is unknown.
Multi-user RIS capacity region: the set of all achievable rate tuples for $K$ users is unknown beyond special cases.
RIS-aided MIMO capacity: full degrees-of-freedom analysis for multi-stream transmission through RIS is incomplete.
Wideband RIS capacity: the capacity under frequency-selective RIS response (Section 18.1) is an open problem.
Robust capacity: capacity when the worst-case $\boldsymbol{\Phi}$ is used by an adversarial environment (partial CSI attack) is unknown.

⚠️Engineering Note

The Practical Gap to Capacity

Practical RIS-aided systems operate at $C_{\text{practical}} = C_{\text{capacity}} - \delta$ , where $\delta$ accounts for:

imperfect CSI (typically 1-2 bits/s/Hz)
2-bit phase quantization (typically 0.5 bits/s/Hz)
control-loop latency aging (0-3 bits/s/Hz depending on mobility)
hardware efficiencies (Chapter 16)

Total $\delta \approx 2-6$ bits/s/Hz. For a 10 bits/s/Hz capacity link, the practical delivered rate is 4-8 bits/s/Hz. This gap is the commercial bottom line and drives the research agenda of the coming decade.

The Research Agenda for the Next Decade

The field that began with Wu & Zhang (2019) and reached commercial rollout in 2024-2025 now confronts three grand challenges:

Performance: close the 4-6 dB theory-practice gap through better calibration, wideband elements, and low-latency control.
Cost: reduce per-panel cost from $\$200$ - $500$ to $\$50$ - $100$ for dense rollout.
Integration: native support in 6G architecture, full control-plane integration, and fluid ISAC.

The CommIT Group's research program — from array-fed RIS (Ch 11), through RIS-ISAC (Ch 13), to deployment optimization (Ch 17) — is positioned to address all three. This book's role is to train the engineers and researchers who will make it happen.

True-Time-Delay (TTD) Element

A RIS element providing frequency-flat delay $\tau$ rather than a frequency-flat phase. With $\Gamma(f) = e^{-j2\pi f \tau}$ , the element's phase response is linear in $f$ , enabling coherent combining across wide bandwidths. Essential for 6G sub-THz signals where fractional bandwidth exceeds $5\%$ . Costs $3$ - $5\times$ more per element than simple phase-only reflectors.

Channel Aging (RIS)

The decorrelation of the effective cascaded channel $\mathbf{h}_{\text{eff}}$ between when the RIS is configured and when the configured phase is applied. Aging efficiency $\eta_{\text{age}}(\tau) = J_0^2(2\pi \nu_D \tau)$ at Doppler $\nu_D$ and latency $\tau$ . Critical for mobile-user scenarios; sets the maximum RIS control-loop latency at $\tau_{\max} \leq 0.3/\nu_D$ for 90% efficiency.

Related: Coherence Time, Control Loop

Quick Check

The asymptotic RIS-aided channel capacity scales with the number of elements $N$ as:

$\log_2 N$

$2 \log_2 N$

$\log_2 N^N$

Linear in $N$

Correction:

2 \log_2 N

SNR scales as $N^2$ , so capacity $\log_2(1 + N^2 \text{SNR}_0) \to 2\log_2 N + O(1)$ . The factor 2 reflects the doubled degrees of freedom (coherent combining at BS side + coherent combining at RIS side).

Why This Matters: RIS in the 6G Architecture

The ITU-R IMT-2030 vision puts RIS at the center of six 6G pillars: peak rate (via aperture gain), connection density (via coverage fill), energy efficiency (passive reflection), AI-native (ML-driven RIS control), integrated sensing (RIS-ISAC, Chapter 13), and fluid network (distributed RIS panels as infrastructure). Unlike 5G (RIS is an overlay), 6G incorporates RIS natively from day one. Commercial rollout is projected 2028-2030 with dense deployments of 500 panels/km² in urban areas. The research agenda for 2025-2030 is to close the remaining $2$ - $6$ bits/s/Hz gap between theory and practice — the grand challenge for researchers reading this book.

Fundamental Capacity Limits of RIS-Aided Channels

What Is the Ultimate Limit of RIS?

Definition: RIS-Aided Channel Capacity

Theorem: Capacity Upper Bound

Cauchy-Schwarz

Triangle inequality

Capacity

Theorem: Asymptotic Capacity Scaling

Peak SNR

Capacity in log

Rewriting

What 'Doubled DoF' Means

Example: Capacity Gain at 28 GHz

Effective SNR

Capacity

Large N

RIS-Aided Capacity vs. NNN

Parameters

Capacity Open Problems

The Practical Gap to Capacity

The Research Agenda for the Next Decade

True-Time-Delay (TTD) Element

Channel Aging (RIS)

Quick Check

Why This Matters: RIS in the 6G Architecture

Definition:
RIS-Aided Channel Capacity

RIS-Aided Capacity vs. $N$