Practical Phase Shift Model in Beamforming

• The Practical Phase Shift Model is a mathematical framework that incorporates hardware nonidealities such as amplitude–phase coupling, quantization, and power constraints.
• It integrates circuit-level dependencies and systemic limitations to enable accurate analysis and optimization of beamforming in IRS/RIS and photonic networks.
• This model guides algorithmic design and resource allocation, enhancing performance bounds and robustness in complex communications and optical applications.

A practical phase shift model refers to any mathematical representation of phase shift behavior that explicitly incorporates nonidealities, constraints, or physical effects overlooked by the ideal (constant-amplitude, freely-controllable) phase shifter assumption. These models are foundational in accurately characterizing, analyzing, and optimizing physical-layer beamforming in intelligent reflecting surfaces (IRS/RIS), PIN-diode arrays, photonic or other metasurface-based phase shifting networks, as well as in robust anomaly detection and phase control in other domains. By directly modeling device- and system-level limitations—such as amplitude–phase coupling, quantization, power constraints, or hardware-induced errors—practical phase shift models enable rigorous analysis of performance bounds, trade-offs, and the development of algorithms that are implementable on physical hardware.

1. Core Physical and Circuit Models of Practical Phase Shifters

The canonical practical phase shift model for IRS and RIS technologies deviates from the unit-modulus assumption $r(\theta) = e^{j\theta}$ by introducing a phase-dependent amplitude attenuation: $$r_n(\theta_n) = \beta(\theta_n) e^{j\theta_n}, \quad \beta(\theta_n) \in [0,1], \quad \theta_n \in [-\pi, \pi)$$ where $\beta(\cdot)$ is fitted to capture the physical and circuit-level dependencies arising in PIN-diode or other tunable meta-atom implementations (Abeywickrama et al., 2020, Abeywickrama et al., 2019, Papazafeiropoulos, 2021).

A widely-adopted parametric form for $\beta(\theta)$ is

$$\beta(\theta) = (1-\beta_{\min}) \left(\frac{\sin(\theta - \phi)+1}{2}\right)^{\alpha} + \beta_{\min}$$

with

• $\beta_{\min}\in[0,1]$: minimum achievable amplitude (e.g., due to resonance and ohmic loss maximization at certain phases),
• $\phi\in [0, 2\pi)$: phase offset aligning the amplitude minimum,
• $\alpha>0$: steepness control, tightly modeling the sharpness of amplitude drop at destructive interference points.

Such models capture critical hardware aspects including energy conservation, resonance detuning, and the lossy nature of practical device operation. Under specific architectures (e.g., 1-bit PIN-diode IRS), the model specializes to a binary state: “ON” (0° phase, nonzero power dissipation) and “OFF” (180°, near-zero power consumption), see (Wu et al., 2024).

2. Power Consumption, Quantization, and System Constraints

Practical phase shift models must account for systemic power and control constraints:

• Phase Shift Dependent Power Consumption (PS-DPC): For PIN-diode IRS, each ON-state diode (corresponding to a specific phase) consumes DC power $P_{\mathrm{PIN}}$, leading to total IRS power $P_{\mathrm{IRS,PS}} = P_{\mathrm{PIN}} \sum_n b_n$ (binary state $b_n$) (Wu et al., 2024).
• Quantized Phase Control: Phase shifters typically support a discrete set of $2^B$ levels, $$\theta_n \in \{0, 2\pi/2^B, ..., (2^B-1)\cdot 2\pi/2^B\}$$, which induces beamforming loss and requires discrete optimization (Shekhar et al., 2022, Fernandes et al., 6 Oct 2025).
• Feedback and Signaling Overheads: High-dimensional phase vectors are often compressed via tensor-factorization or low-dimensional codebooks, trading off between achievable rate and feedback payload (Sokal et al., 2022, Fernandes et al., 6 Oct 2025).
• Control Law Coupling (STAR-RIS): For simultaneous transmission/reflection RIS, reciprocal constraints enforce correlated phase laws, e.g., $$\phi_n^{\mathrm{T}} - \phi_n^{\mathrm{R}} = \pm \frac{\pi}{2}$$ when both transmission and reflection coefficients are nonzero, significantly reducing degrees of freedom (Zhong et al., 2022, Xu et al., 2021).

3. Algorithmic Integration in Beamforming and System Design

Modeling nonideal phase shifters fundamentally alters design and optimization methodologies:

• Mixed-Integer Nonconvex Optimization: The presence of binary variables (e.g., ON/OFF states), phase quantization, or nonlinear amplitude–phase constraints requires mixed-integer programming and advanced relaxations. Techniques such as Generalized Benders Decomposition (GBD), penalty-based Alternating Optimization, and majorization-minimization are employed for tractable solutions (Wu et al., 2024, Abeywickrama et al., 2020, Li et al., 2023).
• Weighted-MMSE Reformulations: For multi-user sum-rate maximization, amplitude–phase coupling is handled via auxiliary variables (e.g., per-user MSE weights, receive filters), recasting the sum-log-SINR problem jointly over discrete and continuous variables (Wu et al., 2024).
• Reinforcement and Deep Learning Approaches: DRL techniques (e.g., DDPG, DQN, TD3) operate over the physically-constrained action space (continuous or hybrid continuous/discrete), directly encoding amplitude–phase coupling in their state-action mappings, outperforming conventional decoupled optimization in simulation benchmarks (Hashemi et al., 2021, Chou et al., 30 Sep 2025, Zhong et al., 2022).
• Tensor and Codebook Compression: Feedback efficiency is maximized with rank-one tensor factorization of the phase vector, which can reduce feedback overhead by $\mathcal{O}(10-100\times)$ with negligible rate loss in LoS-dominated scenarios (Sokal et al., 2022).

4. Impact of Hardware Impairments and Error Modeling

Practical phase shift models also serve as frameworks for robustness and reliability analyses:

• Phase-Shift Error Categories: RIS elements experience errors from quantization, PIN-diode failure, or mechanical deformation. These are modeled as (A) globally i.i.d. random errors—leading to an expected gain loss $$\delta = \mathbb{E}[\cos \Delta\phi]^2 + \mathbb{E}[\sin\Delta\phi]^2$$—(B) grouped i.i.d. errors, or (C) deterministically grouped errors (panel deformation) (Yang et al., 2023).
• Performance Degradation: Closed-form expressions quantify SNR/beamforming-gain loss due to finite-bit quantization ($\approx\,3.9$ dB for $2$-bit systems), hardware failures (tolerating up to $29\%$ diode failures for $<3$ dB loss), or wideband "squint" effects (Yang et al., 2023).
• Design Guidelines: Tight phase quantization ($q \geq 2$ bits), mechanical flatness ($\max$ deviation $< 0.1 \lambda$), and robust feedback/health monitoring are recommended for maintaining performance targets.

5. Application to Power-Constrained and Joint Resource Optimization

The explicit modeling of phase-dependent amplitude and power leads to new paradigms in power allocation and cross-layer optimization:

• BS-IRS Power Split: The total system budget must be judiciously allocated between BS transmit power and IRS diode states. Under stringent power limits, more diodes are turned OFF to enable higher BS transmission, sacrificing passive beamforming gain (Wu et al., 2024).
• Alternating Resource Allocation: In WP-MEC and energy-constrained networks, the presence of amplitude–phase coupling requires combined use of penalty methods, SCA, and fractional programming for joint optimization of passive beamforming, transmit precoding, task allocation, and energy harvesting (Li et al., 2023).
• Codebook and Feature Compression: Joint design of phase shift compression (quantization, feature selection) and WMMSE beamforming via model-based deep learning can achieve near-optimal sum-rate with a fraction of the bit budget needed for naive approaches (Fernandes et al., 6 Oct 2025).

6. Extensions to Photonics, Topological Control, and Non-RF Domains

Practical phase shift models also extend to integrated photonics and non-Hermitian optics:

• Topological Phase Control at Constant Amplitude: Resonator-based photonic systems can achieve quantized $2\pi$ phase shifts without amplitude variation by dynamic pole-zero engineering in the complex-frequency plane, either by modulating the excitation wavelength or by tuning resonator parameters so that the reflection zero/pole encircles the working frequency. The topology of the complex-plane trajectory guarantees quantized, lossless phase change, as formalized by the Cauchy argument principle (Krasnok, 22 May 2025).
• Statistical Modeling of Phase Errors: For nano-photonic circuits, spatially correlated phase errors from fabrication are accurately predicted by Gaussian process models. Linear functionals over these fields yield closed-form means and variances for phase differences, offering orders-of-magnitude speedup over Monte Carlo while retaining full error accuracy (Zhang, 8 Apr 2025).

7. Summary Table: Key Practical Phase Shift Models

Model Category	Mathematical Form/Constraint	References
Amplitude–phase coupling	$$\beta(\theta) = (1-\beta_{\min})((\sin(\theta-\phi)+1)/2)^{\alpha}+\beta_{\min}$$	(Abeywickrama et al., 2020, Papazafeiropoulos, 2021)
PIN-diode PS-DPC	$\mathrm{Power} = P_{\mathrm{PIN}} \sum b_n$ (ON/OFF)	(Wu et al., 2024)
Quantized phase shifters	$\theta_n \in \{0, 2\pi/2^B, \dots\}$	(Shekhar et al., 2022, Fernandes et al., 6 Oct 2025)
STAR-RIS phase correlation	$\phi_n^{\mathrm{T}}-\phi_n^{\mathrm{R}} = \pm\pi/2$	(Zhong et al., 2022, Xu et al., 2021)
Gaussian process phase error	$$\Delta\Phi \sim \mathcal{N}(\mu_{\Delta}, \sigma_{\Delta}^2)$$	(Zhang, 8 Apr 2025)
Topological phase control	$\Delta\varphi = 2\pi$ per encircled zero (constant $|r|$ path)	(Krasnok, 22 May 2025)

These models, by replacing idealizations with explicit hardware and system realities, drive the modern theory, design, and optimization of metasurface-based communications and photonic phase manipulation.