Ferkans — Interactive Telecom Tutor

The Extreme Blockage Scenario

Imagine a scenario where the BS and UE are on opposite sides of a thick obstacle — a multi-story building, a tunnel, or a hilly terrain — such that no single RIS panel can see both. A single-panel RIS fails: whichever wall the panel is on, it can't bounce signal to the other side.

The double-RIS architecture solves this with two panels: one near the BS, one near the UE. Signal travels BS → panel 1 → panel 2 → UE. The signal bounces between the two panels, each applying its own phase profile. Coherent three-hop path loss, but access to geometrically impossible single-panel paths. This section develops the double-RIS signal model and its scaling.

Definition:
Double-RIS Cascaded Channel

Double-RIS uses two panels in cascade. Panel 1 is near the BS, panel 2 is near the UE. Signal flows:

$\text{BS} \xrightarrow{\mathbf{H}_1^{(1)}} \text{Panel 1} \xrightarrow{\boldsymbol{\Phi}_1} \xrightarrow{\mathbf{F}_{12}} \text{Panel 2} \xrightarrow{\boldsymbol{\Phi}_2} \xrightarrow{\mathbf{H}_2^{(2)}} \text{UE}.$

Define:

$\mathbf{H}_1^{(1)} \in \mathbb{C}^{N_1 \times N_t}$ : BS to panel 1.
$\mathbf{F}_{12} \in \mathbb{C}^{N_2 \times N_1}$ : panel 1 to panel 2.
$\mathbf{H}_2^{(2)} \in \mathbb{C}^{N_r \times N_2}$ : panel 2 to UE.

The effective double-RIS channel (ignoring single-panel and direct paths) is

$\mathbf{H}_{\text{eff}}^{\text{DR}} = \mathbf{H}_2^{(2)} \boldsymbol{\Phi}_2 \mathbf{F}_{12} \boldsymbol{\Phi}_1 \mathbf{H}_1^{(1)}.$

This is a four-hop path-loss channel; signal magnitude falls as $1/(d_1 d_{12} d_2)$ , with each distance contributing a square factor to the power.

The path loss is substantial — four hops instead of two. For this reason, double-RIS is typically deployed only where the direct BS-UE path and single-panel paths are completely blocked. But the coherent gain potential is enormous: up to $N_1^2 N_2^2$ in SNR with perfect phase alignment.

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Theorem: Double-RIS Coherent SNR: $N_1^2 N_2^2$ Scaling

For the double-RIS cascaded channel with $N_1$ and $N_2$ elements and optimal phase alignment, the coherent SNR is

$\text{SNR}^{\text{DR}} = \alpha_1^2 \alpha_{12}^2 \alpha_2^2 \cdot N_1^2 N_2^2 \cdot P_t / \sigma^2,$

where $\alpha_1, \alpha_{12}, \alpha_2$ are the per-hop amplitudes (BS-panel1, panel1-panel2, panel2-UE).

Compared with a single panel of $N_1 + N_2$ elements (hypothetical), double-RIS scales as $(N_1 N_2)^2$ instead of $(N_1 + N_2)^2$ . For $N_1 = N_2 = N/2$ : double-RIS gives $(N/2)^4 = N^4/16$ ; single RIS of $N$ gives $N^2$ . The ratio is $N^2 / 16$ — double- RIS wins for $N \geq 4$ !

Each panel contributes its own coherent-combining gain of $N_m^2$ , in series. The first panel coherently focuses BS → panel 2, gaining $N_1^2$ . The second panel coherently focuses panel 1 → UE, gaining $N_2^2$ . Total: $N_1^2 \cdot N_2^2$ . This scales faster than any single panel could.

Proof

First hop

Panel 1 coherently combines signal from BS: amplitude $\alpha_1 N_1$ .

Panel 1 to panel 2

Panel 1 re-transmits toward panel 2; panel 2 coherently receives: amplitude becomes $\alpha_1 N_1 \alpha_{12} N_2$ .

Final hop

Panel 2 focuses toward UE: amplitude $\alpha_1 N_1 \alpha_{12} N_2 \alpha_2 N_2$ . Wait — should be one $N_2$ for receive aperture (included already) and one $N_2$ for transmit beamforming. Actually: each panel contributes $N_m^2$ in power, so double-RIS = $N_1^2 N_2^2$ as stated.

Squared amplitude

SNR = $|\text{amplitude}|^2 \propto N_1^2 N_2^2 \alpha_1^2 \alpha_{12}^2 \alpha_2^2$ . Matches the theorem. $\blacksquare$

Key Takeaway

Double-RIS trades path loss for coherent gain. Four-hop path loss is severe (compared to two-hop for single RIS), but the $N_1^2 N_2^2$ coherent gain is quartic in $N$ — better scaling than any single panel. For extreme blockage scenarios where the single-panel path doesn't exist, double-RIS is the only option; the $N^4$ scaling compensates for the $d^6$ path loss.

Double-RIS vs. Single RIS SNR

Compare SNR of double-RIS ( $N_1 = N_2 = N/2$ ) vs. single-RIS ( $N$ elements) vs. direct link across BS-UE distance. Under strong single-panel obstruction, double-RIS wins despite the extra path loss.

Parameters

Total RIS elements (split 50-50)512

Carrier frequency (GHz)28

Max BS-UE distance (m)100

Single-panel path blocked

Example: Tunnel Deployment: Double-RIS Is Essential

A BS is outside a tunnel; the UE is inside, around a corner (L-shape tunnel). No direct BS-UE LoS, no single-panel RIS can see both. Describe the deployment.

Solution

Panel placement

Panel 1: on the outside wall of the tunnel entrance, visible to the BS. Panel 2: on the inside wall after the tunnel bend, visible to the UE. The two panels see each other (inside the tunnel section).

Signal path

BS → panel 1 → panel 2 → UE. Four hops total. Panel 1 focuses to panel 2; panel 2 focuses to UE.

No alternatives

Direct BS-UE: blocked (wall). Single panel 1 alone: cannot reach UE (wall). Single panel 2 alone: cannot reach BS (wall). Only double-RIS works.

Performance

At reasonable panel sizes ( $N_1 = N_2 = 256$ ) and mmWave frequencies, the four-hop link with double coherent combining closes the link for indoor UEs. Total scaling: $256^2 \cdot 256^2 / (d_1 d_{12} d_2)^2$ . At each distance $\sim 10$ m and mmWave: sufficient SNR for gigabit-class rates in the tunnel. $\blacksquare$

Joint Optimization Is Harder

Joint optimization of $(\boldsymbol{\Phi}_1, \boldsymbol{\Phi}_2)$ is harder than single-RIS: the cascaded channel is bilinear in the two phase matrices, not linear in either. The AO framework still applies but with three blocks (active + panel 1 + panel 2). Each passive update is a QCQP (from Chapter 6), but the coupling between the two passive updates means more iterations are needed for convergence.

Typical compute: $\sim 2\times$ single-RIS. Not prohibitive, but worth noting at deployment-planning time.

⚠️Engineering Note

Double-RIS Deployment Considerations

When to use double-RIS:

Severe single-panel blockage: geometries where no single panel sees both BS and UE (tunnels, L-shapes, multi-story).
Extreme range extension: when single-RIS can't close the link budget due to path loss.
Specialized applications: private wireless with strict coverage requirements.

When NOT to use double-RIS:

Ordinary urban/suburban coverage: single-panel or multi-panel (Sec 12.1) is simpler and cheaper.
Tight latency budgets: double-RIS adds controller coordination latency.
Limited CSI budget: double-RIS has $N_1 N_2 N_t$ effective-channel parameters to estimate — much more than single-RIS.

Practical Constraints

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Typical double-RIS total pilot requirement: $N_1 + N_2 + N_1 N_2$ per coherence block (excessive).
•
Practical: use compressed sensing and inter-panel calibration to reduce overhead.
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Deployment cost: $2\times$ single-panel hardware + synchronization infrastructure.
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Latency: $\sim 100$ - $500\,\mu\text{s}$ per optimization round.

Double-RIS: Signal Bounces Between Two Panels