Ferkans — Interactive Telecom Tutor

Beyond the Unit-Modulus Ceiling

The passive RIS coherent gain scales as $N^2$ , but the product path loss $d_1^2 d_2^2$ is often so severe — especially at mmWave — that even $N = 1024$ is not enough. The fundamental limitation: passive RIS only redirects incident power, it cannot generate new power. The active RIS lifts this ceiling by incorporating a low-power amplifier into each element, boosting the reflected signal beyond unit modulus. The cost: added noise (from the amplifier), additional power consumption, and a more complex optimization.

The golden thread: the RIS still programs the channel, but now gain and phase are both programmable. The element's reflection coefficient has magnitude $|g_n|$ up to a bound $g_{\max}$ , not exactly 1. This expands the feasible set dramatically and often more than compensates for the added noise, especially when the passive RIS is operating deep below the link budget.

Active RIS

A reconfigurable intelligent surface in which each element contains a low-power amplifier, allowing the reflection coefficient magnitude to exceed unity: $|\psi_n| \leq g_{\max}$ with $g_{\max} > 1$ . Breaks the product-path-loss ceiling of passive RIS at the cost of added amplifier noise and DC power consumption.

Related: Passive Ris, Amplifier, Af Relay

Amplifier Noise Figure

The ratio $\text{NF} = \sigma^2_{\text{RIS}}/\sigma^2_{\text{ref}}$ of the amplifier-added noise power relative to a reference (usually thermal). Measured in dB. Determines the active-passive crossover: $d^\star \propto 1/\sqrt{\text{NF}}$ . Lower NF $\Rightarrow$ active RIS wins over a longer range.

Related: Active RIS, Crossover Distance

Definition:
Active RIS System Model

An active RIS replaces each passive element with an amplify- and-reflect element:

Incident signal at element $n$ : $x_n^{\text{in}}$ (complex scalar).
Amplifier gain: $g_n \in \mathbb{C}$ with $|g_n| \leq g_{\max}$ .
Added noise: $w_n^{\text{RIS}} \sim \mathcal{CN}(0, \sigma^2_{\text{RIS}})$ at the output of element $n$ .
Output: $x_n^{\text{out}} = g_n x_n^{\text{in}} + w_n^{\text{RIS}}$ .

Stacking $\boldsymbol{\Psi} = \text{diag}(g_1, \ldots, g_N)$ and $\mathbf{w}^{\text{RIS}} = [w_1^{\text{RIS}}, \ldots, w_N^{\text{RIS}}]^T$ , the received signal at the UE through the active RIS is

$y = \big(\mathbf{h}_d^H + \mathbf{h}_2^H \boldsymbol{\Psi} \mathbf{H}_1\big) \mathbf{v}\,s + \mathbf{h}_2^H \boldsymbol{\Psi} \mathbf{w}^{\text{RIS}} + w,$

where $w \sim \mathcal{CN}(0, \sigma^2)$ is the UE noise. Compared to passive RIS, two changes:

$|\psi_n| = |g_n|$ is free on $[0, g_{\max}]$ rather than fixed at 1.
An additional noise term $\mathbf{h}_2^H \boldsymbol{\Psi} \mathbf{w}^{\text{RIS}}$ appears — the amplified RIS noise.

,

Theorem: SINR for Active RIS

Under active RIS with coefficients $\boldsymbol{\Psi} = \text{diag}(\psi_n)$ satisfying $|\psi_n| \leq g_{\max}$ and total amplifier power constraint $P_t\sum_n |\psi_n|^2 |(\mathbf{H}_1\mathbf{v})_n|^2 + \sigma^2_{\text{RIS}} \sum_n |\psi_n|^2 \leq P_{\text{RIS}}$ , the received SINR at the UE is

$\text{SNR} = \frac{P_t\,|\mathbf{h}_{\text{eff}}^H \mathbf{v}|^2}{\sigma^2_{\text{RIS}} \|\mathbf{h}_2^H \boldsymbol{\Psi}\|^2 + \sigma^2},$

where $\mathbf{h}_{\text{eff}}^H = \mathbf{h}_d^H + \mathbf{h}_2^H \boldsymbol{\Psi} \mathbf{H}_1$ is the active-RIS effective channel. Compared with passive RIS, the denominator has the extra amplified noise term $\sigma^2_{\text{RIS}} \|\mathbf{h}_2^H \boldsymbol{\Psi}\|^2$ .

The amplified RIS noise $\mathbf{h}_2^H \boldsymbol{\Psi} \mathbf{w}^{\text{RIS}}$ passes through $\mathbf{h}_2^H$ and the amplifier diagonal. Its variance at the UE is $\sigma^2_{\text{RIS}} \|\mathbf{h}_2^H \boldsymbol{\Psi}\|^2$ , which grows with the amplifier gains. This is the fundamental tradeoff: higher $|g_n|$ amplifies signal and noise.

Proof

Signal power

Same as passive: $P_t |\mathbf{h}_{\text{eff}}^H \mathbf{v}|^2$ , just with $\boldsymbol{\Psi}$ instead of $\boldsymbol{\Phi}$ .

Noise power at UE

UE receives two noise sources:

Amplified RIS noise propagated through $\mathbf{h}_2^H$ : variance $\sigma^2_{\text{RIS}} \|\mathbf{h}_2^H \boldsymbol{\Psi}\|^2 = \sigma^2_{\text{RIS}} \sum_n |(\mathbf{h}_2)_n|^2 |\psi_n|^2$ .
Local UE noise: $\sigma^2$ . Independent sum: total $= \sigma^2_{\text{RIS}} \|\mathbf{h}_2^H \boldsymbol{\Psi}\|^2 + \sigma^2$ .

SINR

Ratio of signal to total noise gives the stated formula. $\blacksquare$

Key Takeaway

Active RIS trades noise for gain. The passive RIS has $|\phi_n| = 1$ and no RIS-side noise; the active RIS has $|\psi_n| \leq g_{\max}$ and amplifier noise scaling with $|\psi_n|^2$ . The sweet spot lies between the two extremes: small $|\psi_n|$ (near passive) gives coherent gain similar to passive RIS; large $|\psi_n|$ gives more signal but also more noise, potentially degrading SINR. The optimization finds the sweet spot per element.

Theorem: Active RIS Removes the Passive SNR Ceiling

Under asymptotic active RIS gain $g_{\max} \to \infty$ with coherent alignment and fixed $N$ , the received SNR saturates to

$\text{SNR}^{\text{active}} \to \frac{P_t\, |\mathbf{h}_{\text{eff}}^H \mathbf{v}|^2 / (N^2 g_{\max}^2)}{\sigma^2_{\text{RIS}} \|\mathbf{h}_2\|^2 / N} = \frac{P_t\,N |\mathbf{h}_1 \mathbf{h}_2|^2}{\sigma^2_{\text{RIS}} \|\mathbf{h}_2\|^2}.$

Under equal-amplitude channels this is $\approx P_t\,\alpha^2 N / \sigma^2_{\text{RIS}}$ , independent of $\beta^2$ — the RIS-UE path loss no longer dominates when amplifier gain is large enough. The active RIS thus breaks the $d_1^2 d_2^2$ product path-loss ceiling.

Passive RIS coherent SNR ceiling is $N^2 \alpha^2 \beta^2 \cdot P_t/\sigma^2$ . With active RIS, each element boosts its contribution by $|g_n|^2$ to the signal and $|g_n|^2$ to the noise. Under noise limited regime, this shifts the effective SNR upward: the amplifier noise floor is a new lower bound, and at high amplifier gain, the SNR approaches $P_t/\sigma^2_{\text{RIS}}$ — independent of the BS-UE distance!

Proof

High-gain limit

At large $g_{\max}$ , both signal and RIS-noise terms are proportional to $g_{\max}^2$ . The $\beta^2$ in the signal cancels against the $\beta^2$ in the noise (through $\|\mathbf{h}_2\|^2 \propto \beta^2$ ), leaving only $\alpha^2$ -dependence.

UE noise becomes negligible

At high gain, the amplified RIS noise dominates UE noise: $\sigma^2_{\text{RIS}} g_{\max}^2 \beta^2 \gg \sigma^2$ for large $g_{\max}$ .

Resulting SNR

Signal: $P_t \cdot N^2 g_{\max}^2 \alpha^2 \beta^2$ . Noise: $\sigma^2_{\text{RIS}} g_{\max}^2 N \beta^2$ . Ratio: $P_t N \alpha^2 / \sigma^2_{\text{RIS}}$ . Independent of $\beta$ . $\blacksquare$

Noise-Limited vs. Power-Limited Regimes

Active RIS operates in two regimes:

Low amplifier gain ( $g_{\max} \approx 1$ ): near-passive operation. RIS-noise is small, behavior similar to passive.
High amplifier gain ( $g_{\max} \gg 1$ ): noise-limited. $\sigma^2_{\text{RIS}} |g_n|^2 \gg \sigma^2$ , and the amplifier noise dominates at the UE. Increasing $g_n$ no longer helps — we are at the "noise ceiling."

The optimization finds the transition point automatically: each element chooses $|g_n|$ to balance signal gain against noise contribution. Typical optimum: per-element $|g_n|^2 \sim$ signal strength / RIS noise power, with total power constrained to $P_{\text{RIS}}$ .

Definition:
Active RIS Power Budget

The active RIS consumes RF power to amplify signals. The total transmitted RIS power is

$P_{\text{RIS}}^{\text{tx}} = P_t\sum_n |\psi_n|^2 |(\mathbf{H}_1\mathbf{v})_n|^2 + \sigma^2_{\text{RIS}} \sum_n |\psi_n|^2,$

where the two terms are amplified signal power and amplified noise. Budget: $P_{\text{RIS}}^{\text{tx}} \leq P_{\text{RIS}}$ .

Under efficiency $\eta_{\text{ampl}}$ (DC-to-RF), the DC power consumption is $P_{\text{DC}} = P_{\text{RIS}}/\eta_{\text{ampl}}$ . Each element consumes $P_{\text{DC}}/N$ on average — a small number (mW at mmWave) but non-negligible compared with the nearly-zero passive case.

Active vs. Passive SNR as a Function of Distance

Sweep the BS-UE distance $d_0$ (or equivalently the product path loss) and plot SNR for passive RIS, active RIS, and direct link. Active RIS breaks the $d^2 d^2$ product-path-loss ceiling at sufficient amplifier gain; increase $g_{\max}$ to see the crossover shift to longer distances.

Parameters

RIS elements

N

256

Max amplifier gain

g_{\max}

(dB)15

RIS power budget

P_{\text{RIS}}

(dBm)10

Amplifier noise figure (dB)5

Common Mistake: Don't Ignore the Amplified RIS Noise

Mistake:

"Active RIS just multiplies the signal by $g_{\max}$ . So we get $g_{\max}^2$ SNR boost over passive."

Correction:

Ignoring the amplified RIS noise is a common conceptual error. The boost is $g_{\max}^2$ for signal and for noise. The net SNR improvement depends on which source dominated before amplification: if UE noise dominated, active RIS helps; if passive-coherent combining was already noise-free (rare), active RIS only hurts. Always compute both signal and noise terms in the active-RIS SINR.

🔧Engineering Note

Active RIS Hardware

Active RIS elements combine a passive reflecting structure with a low-power amplifier:

Amplifier: class-A or class-AB monolithic microwave IC (MMIC), $\sim 10$ - $20$ dB gain per element, $\sim 3$ - $5$ dB noise figure.
Bias power: $\sim 5$ - $20$ mW per element at mmWave. For $N = 256$ : total $\sim 1$ - $5$ W. Much less than a full active array ( $\sim 100$ W) but more than passive ( $\sim 10$ mW).
Saturation: amplifiers have a max output power; input power at the RIS must stay below the $1$ -dB compression point to avoid nonlinear distortion.
Stability: feedback from reflection at the same element can cause oscillation. Careful isolation between amp input and output required.

Practical Constraints

•
Per-element amplifier gain at mmWave (2024): $\sim 10$ - $20$ dB.
•
Noise figure: $3$ - $5$ dB (sub-6 GHz), $5$ - $8$ dB (mmWave).
•
Amplifier efficiency: $\sim 20$ - $40\%$ (class A), $\sim 40$ - $60\%$ (class AB).
•
Total DC power for $N = 256$ panel: $\sim 1$ - $5$ W. Compared to $\sim 0.01$ W for passive.

The Active RIS Signal Model