Cooperative Double-IRS for Enhanced Wireless Capacity

• The topic defines cooperative double-IRS as two reflecting surfaces strategically deployed to manipulate the wireless channel and achieve multiplicative beamforming gains.
• It employs joint passive beamforming techniques such as alternating optimization, semidefinite relaxation, and manifold optimization to maximize the end-to-end SNR.
• Deployment guidelines emphasize optimal IRS placement, element allocation, and efficient channel estimation to overcome challenges and realize significant capacity improvements.

Cooperative Double Intelligent Reflecting Surface (Double-IRS)

A cooperative double intelligent reflecting surface (double-IRS) system consists of two IRSs deployed in series between a transmitter (often a multi-antenna base station) and a receiver (single- or multi-antenna user), collaboratively manipulating the wireless propagation environment to enhance end-to-end channel capacity, coverage, spatial multiplexing, and functional flexibility. In the canonical architecture, IRS 1 is located near the transmitter, IRS 2 near the user, and the direct path is typically blocked or severely attenuated. Joint optimization of the passive phase shifts at both IRSs enables the exploitation of multi-hop cascaded channels and the achievement of “multiplicative” array gain exceeding that of conventional single-IRS deployments.

1. Canonical System Model and Channel Representation

A double-IRS system (Wu et al., 15 Jan 2025) includes:

• A base station (BS) with $M$ transmit antennas;
• IRS 1 with $N_1$ reflecting elements and diagonal phase matrix $$\Theta_1 = \mathrm{diag}(e^{j\theta_{1,1}},\dotsc,e^{j\theta_{1,N_1}})$$;
• IRS 2 with $N_2$ elements and $$\Theta_2 = \mathrm{diag}(e^{j\theta_{2,1}},\dotsc,e^{j\theta_{2,N_2}})$$;
• A single-antenna user.

All channels are narrowband, quasi-static, and typically LoS-dominated:

• $$\mathbf{H}_{\mathrm{BS},1} \in \mathbb{C}^{N_1\times M}$$ : BS $\to$ IRS 1;
• $\mathbf{G}_{2,1} \in \mathbb{C}^{N_2\times N_1}$ : IRS 1 $\to$ IRS 2;
• $$\mathbf{h}_{2,\mathrm{U}}\in \mathbb{C}^{N_2\times1}$$ : IRS 2 $\to$ user.

The end-to-end (E2E) channel observed at the user is

$$h_{\rm eff} = \mathbf{h}_{2,\mathrm{U}}^T \Theta_2 \mathbf{G}_{2,1} \Theta_1 \mathbf{H}_{\mathrm{BS},1} \mathbf{w}$$

where $\mathbf{w}$ is the BS transmit beamformer ($\|\mathbf{w}\|^2 \leq P$).

Cascaded path-loss is captured via individual large-scale fading coefficients and path-loss exponents per hop, with the normalized small-scale components $\overline{\mathbf{H}}$ representing actual array responses (Wu et al., 15 Jan 2025).

2. Joint Passive Beamforming and Optimization Algorithms

The design challenge is to jointly optimize the BS beamformer $\mathbf{w}$ and the phase shifts $\Theta_1, \Theta_2$ to maximize the E2E SNR: $$\max_{\Theta_1, \Theta_2, \mathbf{w}} \left| \mathbf{h}_{2,\mathrm{U}}^T \Theta_2 \mathbf{G}_{2,1} \Theta_1 \mathbf{H}_{\mathrm{BS},1} \mathbf{w} \right|^2, \quad \text{s.t. } |[\Theta_i]_{n,n}|=1,\, \|\mathbf{w}\|^2\leq P.$$

Three principal algorithmic frameworks are employed (Wu et al., 15 Jan 2025, Niu et al., 2021):

• Alternating Optimization (AO): Iteratively fix $\Theta_1, \Theta_2$, optimize $\mathbf{w}$ (by maximum ratio transmission), then update $\Theta_1, \Theta_2$ element-wise to cancel the phase of the aggregate coefficient, repeating until convergence.
• Semidefinite Relaxation (SDR): Relax the non-convex quadratic phasing into a convex semidefinite program (SDP), extract feasible points by Gaussian randomization.
• Manifold Optimization: Treat the IRS phase vectors as points on the complex unit circle manifold, perform Riemannian gradient steps (including product manifold designs for MIMO-OFDM systems (Xiong et al., 27 Jan 2026)).

For multi-user MIMO or hybrid precoding systems (including mmWave), majorization-minimization surrogate construction and block-coordinate descent with convex subproblems for digital precoders and manifold-constrained optimization for IRS phases are applied (Niu et al., 2021).

3. Cooperative Double-IRS Gain Mechanisms and Theoretical Scaling

With two IRSs in cooperative cascaded configuration (i.e., both reflecting the signal in tandem via a strong LoS inter-IRS path), the passive beamforming gain under ideal alignment scales as

$\mathrm{Gain}_{\text{D-IRS}} \propto (N_1 N_2)^2,$

while a single-IRS system with $N$ elements achieves only $\mathcal{O}(N^2)$ scaling (Han et al., 2020, Wu et al., 15 Jan 2025).

• For $N_1 = N_2 = N/2$, double-IRS: $\mathcal{O}(N^4)$; single-IRS: $\mathcal{O}(N^2)$.
• This scaling translates into a rate increment per element-doubling of $4\,\mathrm{bps/Hz}$ for double-IRS, compared to $2\,\mathrm{bps/Hz}$ for single-IRS (Mei et al., 2021).
• The quartic gain $\mathcal{O}(N^4)$ is realizable only under strong, rank-1 LoS-dominated IRS$_1$–IRS$_2$ channels; otherwise, the gain interpolates toward $\mathcal{O}(N^2)$ as the channel becomes richer in scattering (Ding et al., 2022, Han et al., 2020).

Double-IRS architectures fundamentally enhance multiplexing capability in multiuser scenarios: for $K > \mathrm{rank}(\hat H_{\text{single-IRS}})$, the double-IRS system maintains high max-min rate, while the single-IRS system saturates due to rank-deficiency (Zheng et al., 2020).

4. Channel Estimation and Practical Design Considerations

Double-IRS channel state information (CSI) acquisition is more challenging due to the bilinear (or trilinear) cascaded structure and elevation of overall dimensionality. Effective CSI protocols exploit the algebraic relationships among single- and double-reflection cascades (You et al., 2020, Zheng et al., 2020):

• Decoupled training: Partition estimation into phases for the individual BS–IRS, IRS–IRS, and IRS–User links with ON/OFF switching of IRS elements.
• Rank-1 LoS exploitation: For LoS-dominant inter-IRS channel, only the left and right “signature” vectors need estimation, slashing pilot overhead from $M_1M_2$ to $M_1 + M_2$ pilots (You et al., 2020).
• Multiuser training overhead: For $N \geq M/2$, the total pilot cost is $\frac{3}{2}M + 2(K-1)$, sublinear in $M$ and $K$ (Zheng et al., 2020), much less than the “brute-force” $KM + \mathcal{O}(KM^2)$ cost for independent single-IRS estimation.

High-precision phase-shifters (2–3 bits) are preferred to minimize quantization loss. Realistic deployments require control-channel update intervals of $\leq 100$ ms to support slow user mobility (Wu et al., 15 Jan 2025). Optimal element allocation is near-equal between the two IRSs under strong LoS, but skewed toward the user-side IRS when amplifier noise at IRS 1 dominates (Kang et al., 2023).

5. Performance, Validation, and Empirical Insights

Table: Double-IRS Scaling, SNR, and Field Results (all for LoS inter-IRS unless specified)

Metric	Double-IRS	Single-IRS	Empirical Findings
Beamforming Gain	$\mathcal{O}(N^4)$	$\mathcal{O}(N^2)$	RSRP gain: +10–15 dB; DL throughput: +246–360% (Wu et al., 15 Jan 2025)
SNR scaling	$(N_1 N_2)^2$	$N^2$	Double-IRS outperforms single-IRS if $N$ above threshold ($\sim$K large) (Wu et al., 15 Jan 2025, Han et al., 2020)
Capacity	$C \sim 4\log_2 N$, multiplexing gain 2 if IRS paths are orthogonal (Han et al., 2021)	$C \sim 2 \log_2 N$, multiplexing gain 1	Double-IRS achieves superior rate at moderate/high $N$, empirical rate vs. $N$ approaches $4 \log_2 N$ slope (Han et al., 2021)

Numerical simulation and field deployments (e.g., at 26 GHz in Shanghai) confirm theory: RSRP increases up to $+15$ dB and downlink throughput up to $+360\%$ for double-IRS, with optimal placements at BS-side and user-side, and inter-IRS spacing within $20$ m ensuring LoS (Wu et al., 15 Jan 2025). Capacity and sum-rate improvements are also sustained in multi-user MIMO and mmWave massive MIMO settings (Zheng et al., 2020, Niu et al., 2021). In wideband MIMO-OFDM secrecy scenarios, manifold-based joint phase optimization yields $\sim 22$–$32\%$ secrecy rate improvement over single- or distributed-IRS architectures, and is robust to CSI imperfections (Xiong et al., 27 Jan 2026).

6. Deployment Guidelines and Open Challenges

Key design principles for cooperative double-IRS deployment (Wu et al., 15 Jan 2025, Mei et al., 2021, Kang et al., 2023):

• IRS placement: Locate IRS 1 near BS (to shape the transmitter-side field), IRS 2 near user (to enhance receive-side SNR); inter-IRS spacing $\lesssim 20$ m for urban LoS.
• Element allocation: Equal split ($N_1 \approx N_2$) is optimal under strong LoS; otherwise, assign more elements to user-side IRS to counter amplification noise.
• Environmental control: Mount IRSs on facades or rooftops to ensure clear LoS inter-IRS; avoid blockages and consider half-space illumination constraints.
• Phase control: High quantization levels (2–3 bits) to avoid SNR loss; control update latency must be within the adaptation window of channel variation (e.g., $<100$ ms).
• Scalability and Complexity: AO and manifold optimizations scale polynomially with $N_i$; closed-form solutions are feasible for LoS-dominated rank-1 inter-IRS links.

Open research challenges identified include joint placement and clustering for large networks, robust beamforming under statistical or partial CSI, FDD/broadband training design, hybrid active–passive IRS architectures, and dynamic adaptation under user mobility (Wu et al., 15 Jan 2025, Mei et al., 2021). In active double-IRS settings, the SNR scaling plateaus at $\mathcal{O}(M^2)$ as amplification noise dominates at high element count or per-element gain (Kang et al., 2023).

7. Multi-Reflection Extensions and Advanced Architectures

The double-IRS configuration serves as the foundation for generalized multi-IRS architectures and beam routing schemes. In such systems, the transmit beam can be split among multiple orthogonal paths, each routed via a selected chain of IRSs, and coherently combined at the receiver to exploit path diversity and LoS multiplicity (Mei et al., 2021). Clique-based graph algorithms enable efficient selection of non-overlapping IRS reflection paths, applicable in dense deployments with environmental blockages.

Extensions to satellite/terrestrial hybrid networks leverage double-IRS designs with distributed channel estimation and tracking protocols, exploiting rank-1 LoS decompositions to decouple high-dimensional MIMO channel estimation into tractable subspaces on each network side (Zheng et al., 2022).

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References:

• "Intelligent Reflecting Surfaces for Wireless Networks: Deployment Architectures, Key Solutions, and Field Trials" (Wu et al., 15 Jan 2025)
• "Cooperative Double IRS aided Secure Communication for MIMO-OFDM Systems" (Xiong et al., 27 Jan 2026)
• "Double Intelligent Reflecting Surface-assisted Multi-User MIMO mmWave Systems with Hybrid Precoding" (Niu et al., 2021)
• "Double-IRS Assisted Multi-User MIMO: Cooperative Passive Beamforming Design" (Zheng et al., 2020)
• "Intelligent Reflecting Surface Aided Wireless Networks: From Single-Reflection to Multi-Reflection Design and Optimization" (Mei et al., 2021)
• "Analysis and Optimization of A Double-IRS Cooperatively Assisted System with A Quasi-Static Phase Shift Design" (Ding et al., 2022)
• "Cooperative Double-IRS Aided Communication: Beamforming Design and Power Scaling" (Han et al., 2020)
• "Double-IRS Aided MIMO Communication under LoS Channels: Capacity Maximization and Scaling" (Han et al., 2021)
• "Wireless Communication via Double IRS: Channel Estimation and Passive Beamforming Designs" (You et al., 2020)
• "Uplink Channel Estimation for Double-IRS Assisted Multi-User MIMO" (Zheng et al., 2020)
• "Coverage Probability of Double-IRS Assisted Communication Systems" (Papazafeiropoulos et al., 2021)
• "Intelligent Reflecting Surface for Multi-Path Beam Routing with Active/Passive Beam Splitting and Combining" (Mei et al., 2021)
• "Intelligent Reflecting Surface-Aided LEO Satellite Communication: Cooperative Passive Beamforming and Distributed Channel Estimation" (Zheng et al., 2022)
• "Double-Active-IRS Aided Wireless Communication: Deployment Optimization and Capacity Scaling" (Kang et al., 2023)