Penalty Dual Decomposition (PDD) Algorithm

• The PDD algorithm is an iterative optimization method that decomposes complex constrained problems into tractable subproblems with penalty terms.
• It alternates updating primal variables and refining dual multipliers to achieve convergence in non-convex settings.
• Applied in vehicular networks, the PDD approach enhances spectral efficiency and system capacity with low overhead.

Intelligent Reflecting Vehicle Surface (IRVS) Paradigm

An Intelligent Reflecting Vehicle Surface (IRVS) is a vehicular embodiment of intelligent reflecting surface (IRS) technology, in which a massive number of passive, electronically reconfigurable elements are integrated onto the exterior surfaces of vehicles. These surfaces modulate the impinging electromagnetic wavefronts by dynamically controlling phase shifts, thus enabling passive beamforming and effective shaping of wireless propagation in highly mobile vehicular networks. IRVS extends the IRS concept from static infrastructure (e.g., building facades) to mobile platforms, fundamentally expanding the spatial degrees of freedom for link adaptation, capacity enhancement, and coverage restoration in dynamic vehicular environments (Jiang et al., 2023).

1. System Architecture and Channel Modeling

The canonical IRVS-aided vehicular network comprises three principal node classes: a multi-antenna base station (BS) (often vehicle-mounted), multiple single-antenna user equipments (UEs) (either human-carried or vehicular), and a (distributed) set of S moving vehicles, each equipped with a conformal IRVS of N_s passive reflecting elements. Communications proceed in a time-division multiple access (TDMA) frame, sequentially serving K users in orthogonal slots.

The channel model includes:

• Direct path, $f_k \in \mathbb{C}^{N_b \times 1}$: BS to UE_k, each entry $f_{k,n_b}\sim\mathcal{CN}(0, \sigma_k^2)$.
• BS to IRVS_s: $H_s\in\mathbb{C}^{N_s\times N_b}$, Rician with factor $\Gamma$, path loss $\sigma_s^2$.
• IRVS_s to UE_k: $g_{s,k}\in\mathbb{C}^{N_s\times 1}$, each $g_{s,k,n}\sim\mathcal{CN}(0,\sigma_{s,k}^2)$.

All S IRVSs are stacked into a "mega-IRS": $H\in\mathbb{C}^{N_r\times N_b}$, $g_k\in\mathbb{C}^{N_r}$, $N_r=\sum_s N_s$. The phase-shift matrix at slot $k$ is $$\Theta[k]=\operatorname{diag}(e^{j\varphi_1[k]},\ldots,e^{j\varphi_{N_r}[k]})$$.

2. Problem Formulation and Objective

The principal optimization target is sum spectral efficiency (SE) maximization, subject to both the linear active beamforming constraint at the BS and the passive phase programing of the IRVSs. The received signal at UE k in TDMA slot k is: $$r_k = \sqrt{P_d}\big[ g_k^T\,\Theta[k]\,H + f_k^T \big] w_k \, x_k + n_k, \quad n_k \sim \mathcal{CN}(0, \sigma_n^2)$$ The SINR is then: $$\mathrm{SINR}_k = \frac{P_d\,\big| [ g_k^T\,\Theta[k]\,H + f_k^T ] w_k \big|^2}{\sigma_n^2}$$ The maximization is: $$\max_{ \{w_k\}, \{\Theta[k]\} } \;\; \sum_{k=1}^K \log_2( 1 + \mathrm{SINR}_k )$$ subject to $\sum_{k=1}^K \|w_k\|^2 \leq 1$; $$\Theta[k]=\operatorname{diag}(e^{j\varphi_{1}[k]},\ldots,e^{j\varphi_{N_r}[k]})$$; $\varphi_i[k]\in[0,2\pi)$.

3. Solution Methodology: Alternating Optimization

Due to non-convex coupling, the IRVS paradigm uses iterative alternating optimization between active BS beamforming and passive IRVS phase adjustment:

1. Active Beamforming with Fixed IRVS Phases:
   • For each UE, define the equivalent channel $h_{eq,k}^T = g_k^T\,\Theta\,H + f_k^T$.
   • Optimal linear precoding adopts maximal ratio transmission (MRT): $w_k = h_{eq,k}^*/\|h_{eq,k}\|$.
2. Passive Phase Shift Tuning with Fixed Beamformers:
   • For element $i$, optimize $| \sum_i \chi_{k,i} q_{k,i} |$ subject to $|q_{k,i}|=1$, where $\chi_k = \operatorname{diag}(g_k^T) H w_k$.
   • The optimal continuous phase for each element: $\varphi_i = \arg( f_k^T w_k ) - \arg( \chi_{k,i} )$.
   • With discrete phase constraints (L levels), phase quantization is applied: $$\varphi_i^{\mathrm{disc}} = \Delta\theta \cdot \operatorname{round}( \varphi_i/\Delta\theta )$$, $\Delta\theta = 2\pi/L$.
   • For a random-phase baseline, $\varphi_i$ is chosen uniformly at random in $[0,2\pi)$ (or from the discrete set).

These steps are alternated until convergence, which is reported to typically occur within 2–3 iterations.

4. Comparative Performance and Design Guidelines

Comprehensive evaluation demonstrates:

• Even random-phase IRVS provides a $\approx7$ bps/Hz sum SE gain over TDMA/NOMA without IRVS.
• 1-bit per element adds a further $\approx7$ bps/Hz; 2-bit quantization closely matches continuous phase control, attaining $\approx98\%$ of the maximal gain.
• Sum rate gains are strongly increasing in the number of IRVS surfaces ($S$) and UEs ($K$); for $K=20, S=5$, the $95$th percentile sum rate is approximately quadrupled over no-IRVS TDMA.
• Two-bit phase quantization suffices for nearly optimal operation, supporting hardware simplification without major performance loss.
• Random-phase is a robust fallback with non-trivial benefit when real-time control is infeasible.
• Very fast PIN diode switching ($\sim0.2\;\mu$s) enables per-user adaptation by IRVSs, feasible for moderate slot sizes in TDD/TDMA.

5. Implementation and Practical Considerations

• Physical feasibility: IRVSs can be integrated as lightweight conformal panels on vehicle exteriors, which is compatible with vehicular constraints.
• Hardware complexity is mitigated by restricting phase resolution; practical 1–2 bit phase control yields substantial performance gains with low hardware requirements.
• The IRVS paradigm is amenable to medium access schemes beyond TDMA, and supports rapid adaptation due to the electronic speed of phase updates.

6. Potential Applications and Impact

IRVS significantly advances vehicle-to-everything (V2X) communications by transforming mobile vehicles into programmable wireless propagation agents capable of virtual LOS construction, coverage restoration, and spatial diversity augmentation. This capability is especially relevant for highly dynamic, dense, or obstructed vehicular environments, such as military tactical networks and emergency first-responder scenarios, where rapid and robust link adaptation is critical. The sum-capacity gains at marginal additional cost and power consumption point toward IRVS as a foundational enabler in future 6G vehicular networks (Jiang et al., 2023).

7. Summary and Research Outlook

The IRVS paradigm generalizes IRS-enabled capacity scaling to the vehicular mobility regime, combining joint active/passive beamforming (BS plus IRVS) with low-complexity, quantized phase control on massive mobile arrays. This technological advance promises transformative enhancements in spectral efficiency, adaptability, and robustness for vehicular networks, while imposing only marginal overhead in cost and power. Open research questions remain regarding optimal scheduling and distributed phase control under high vehicular mobility, advanced MAC protocols that leverage mobile IRS distributions, and integration with broader 6G vehicular architectures.