Discrete Phase Shift Optimization Framework

• The discrete phase shift optimization framework is a set of mathematical and algorithmic principles designed to select optimal phase values from finite codebooks in RIS and meta-surface applications.
• It integrates diverse methods—such as DTPQ, EIPQ, coordinate descent, sphere decoding, and deep reinforcement learning—to effectively balance performance with computational complexity.
• The framework quantifies quantization loss and provides practical guidelines, enhancing energy efficiency and beamforming accuracy in single- and multi-user MIMO systems.

Discrete Phase Shift Optimization Framework

A discrete phase shift optimization framework refers to algorithmic principles, mathematical formulations, and practical methods for configuring reconfigurable intelligent surfaces (RIS) or related passive electromagnetic arrays under the constraint that only a finite, discrete set of phase shift values can be selected at each element. This is a defining hardware limitation in modern RIS, meta-surfaces, and PIN-diode based arrays, and arises from cost, power, or speed restrictions in practical phase shifter implementation. The framework encompasses optimization formulations, solution algorithms, complexity analyses, and the impact of phase discretization on system-level objectives—including received power, sum rate, positioning error, or beam pattern fidelity.

1. Mathematical Formulation and Signal Model

In canonical SISO and MIMO RIS-assisted systems, the discrete phase shift optimization problem centers on maximizing a quadratic (or fractional quadratic) metric with respect to element-wise phase shift selections. Consider an RIS-aided SISO link where all energy transfer is via the RIS; the received signal is

$$y = h_{\rm RIS} x + w, \quad h_{\rm RIS} = \sum_{n=1}^{N}\sum_{m=1}^{M} a_{n,m} e^{-j(\frac{2\pi}{\lambda}(r^t_{n,m} + r^r_{n,m}) - \Delta_{n,m})},$$

where $\Delta_{n,m}$ is the discrete phase applied at element $(n,m)$, drawn from a $2^q$-level phase codebook $\mathcal Q$. The core optimization problem is

$$\max_{\{\Delta_{n,m}\subset\mathcal{Q}\}} |h_{\rm RIS}(\{\Delta_{n,m}\})|^2.$$

In MIMO/MISO/RIS applications with direct and cascaded links, the analogous problem is posed for maximizing

$|h_0 + \sum_{n=1}^N h_n e^{j\theta_n}|^2$

over discrete $\theta_n$ drawn from $\{0,2\pi/K,\dots,2\pi(K-1)/K\}$, $K = 2^b$. In MIMO, joint precoder and RIS optimization is often considered, with constraints of the form $\theta_n \in \mathcal D$, where $\mathcal D$ is a discrete phase codebook (Sang et al., 2023, Wu et al., 2018, Wu et al., 2019, Pekcan et al., 2023, Ramezani et al., 2023).

Given the combinatorial nature ($2^{bN}$ configurations for $N$ elements), the framework pursues efficient algorithms exploiting problem structure, and quantifies the performance loss incurred by discretization.

2. Phase Distribution Laws and Degrees of Freedom

Optimizing discrete RIS phases requires understanding how superposition and propagation conditions determine the mapping from continuous to discrete phase domains. When both transmitter and receiver are in the far field with respect to RIS, the composite phase terms are nearly identical across elements. This allows all phase shifts to be coarsely aligned with minimal loss and drastically reduces the effective optimization search space.

Conversely, in the near field, path length differences yield a spread of underlying phase distributions, populating the entire $[0,2\pi)$ interval. The foundational result (Sang et al., 2023) is:

Theorem 1: In optimal discrete-phase alignment, all quantized phases must lie within a single quantization interval—i.e.,

$$\max_{(j,i),(n,m)} |\phi^q_{j,i} - \phi^q_{n,m}| \leq \Omega$$

where $\Omega = 2\pi/2^q$. This reduces the effective DoF to the number of RIS elements, as only $MN$ candidate quantization thresholds $\gamma$ (one per phase value) need be considered for global optimization, rather than $2^{qMN}$ brute force.

3. Algorithmic Solutions for Discrete Phase Optimization

Multiple methods have been developed to handle the inherent nonconvexity and integer constraints:

3.1 Dynamic Threshold Phase Quantization (DTPQ)

DTPQ selects the quantization threshold from the set of continuous phase values, mapping each element's phase to the closest allowable discrete value within the optimal active interval. DTPQ achieves global optimality in $O(MN)$ operations via the aforementioned phase distribution law (Sang et al., 2023).

3.2 Equal Interval Phase Quantization (EIPQ)

EIPQ partitions the phase interval into $K$ subintervals and evaluates only $K$ candidate quantization thresholds, incurring $O(K)$ complexity and attaining sub-optimal but still superior performance to conventional fixed-threshold approaches.

3.3 Coordinate-Descent / Successive Refinement

Many frameworks employ cyclic per-element updates: for each element, select the discrete phase closest to the alignment direction implied by the summed coupling to all other elements and the target vector. This monotonic scheme achieves fast convergence to a stationary point (Wu et al., 2018, Wu et al., 2019, Pekcan et al., 2023).

3.4 Sphere Decoding and Integer Programming

For multiuser MIMO, the phase configuration is cast as a mixed-integer least squares problem and solved via Schnorr–Euchner sphere decoding, which efficiently explores the tree of possible phase assignments while pruning suboptimal branches (Ramezani et al., 2023).

3.5 Probabilistic and Physics-Inspired Methods

Probabilistic reformulation techniques reinterpret discrete phase variables as categorical random vectors, optimizing over expectation parameters and employing stochastic REINFORCE-type sampling or analytical moment-based gradient descent (Pradhan et al., 2023). Large-scale PIN-diode arrays further benefit from ratio-to-QUBO transformation, leveraging simulated annealing for combinatorial beamforming (Kim et al., 2023).

3.6 Heuristics and Metaheuristics

Particle swarm optimization (PSO) (Shekhar et al., 2022), grey wolf optimization (GWO) (Cheng et al., 9 Aug 2025), and cross-entropy–based sampling (Li et al., 2024) are directly applied for general nonlinear problems, especially where the quantized integer nature precludes analytic derivatives.

3.7 Deep Reinforcement Learning (DRL)

Recent advances exploit column-wise DDQN agents for large-scale RISs, integrating greedy elementwise refinements at each DRL step to achieve competitive performance with polynomial complexity (Wang et al., 7 May 2025).

These approaches dramatically reduce practical complexity, from exponential in $N$ ($O(2^{bN})$) for brute-force search to linear or near-linear in $N$, depending on quantization levels and channel conditions.

4. Quantization Loss, Path-Loss Scaling, and Performance Bounds

Discrete phase shift restriction introduces quantization error. Rigorous analysis demonstrates that, for RIS systems operating at moderate ($b=2$–$3$) bit-depth, the loss compared to continuous phase control is minimal.

In the large-$N$ regime, the received power scales as $O(N^2)$ for both continuous and discrete designs; quantization introduces a constant degradation given by

$$\eta(b) = \left( \frac{2^b}{\pi} \sin(\frac{\pi}{2^b}) \right)^2$$

yielding, for example, a $-3.9$ dB penalty at $1$-bit, $-0.9$ dB at $2$-bit, and $-0.2$ dB at $3$-bit (Wu et al., 2018, Wu et al., 2019, Pekcan et al., 2023, Sang et al., 2023). These results are robust across SISO and MISO architectural variants, holding for both single-user and multi-user optimization.

Angular and distance scaling laws remain consistent with continuous-phase counterparts, and the dominant loss is constant across elements. Overhead-aware energy efficiency and rate optimization frameworks demonstrate similar loss ratios up to 99% reduction in signaling overhead at $b=5$ bits with only sub-dB performance loss (Shekhar et al., 2022).

5. Experimental Validation and Application Domains

Validation of discrete phase shift optimization has been performed at gigahertz and millimeter-wave frequencies. Field trials exhibit, for instance, up to $27.2$ dB gain of DTPQ-configured 1-bit RIS over fixed-threshold schemes at $35$ GHz (Sang et al., 2023).

Key application domains include:

• Uplink and downlink multiuser MIMO (sum rate, MMSE, SINR, and linear precoder beamforming) (Wu et al., 2019, Ramezani et al., 2023)
• Joint radar–communication integrated sensing (Liu et al., 2022)
• mmWave position sensing, with combined discrete-phase RIS activation and sequential beamforming for minimizing PEB (Cheng et al., 9 Aug 2025)
• Secure UAV communications via directional modulation under discrete IRS constraints (Li et al., 2024)
• Codebook design for beam squint and beamwidth control (Ghanem et al., 2022)
• Large-scale PIN-diode arrays for 3D far-field beamforming using QUBO (Kim et al., 2023)

Practicalities include constructing optimal training matrices for channel estimation (DFT/Hadamard) under phase quantization (You et al., 2019, Sun et al., 2021), and establishing that $b\sim2-4$ bits suffices for negligible MSE penalty.

6. Complexity, Scalability, and Practical Guidelines

Table: Complexity Summary of Main Algorithms

Method	Complexity (per iteration)	Optimality
Brute-force	$O(2^{bN})$	Global
DTPQ	$O(MN)$	Global (single-user)
EIPQ	$O(K)$	Near-optimal
Coord.-descent	$O(N^2)$	Local optimum
Sphere decoding	$O(L^\delta)$, $\delta<N$ (avg)	Global (LS)
Probabilistic	$O(MN^2)$ (samples)	Opt. in expectation
Sim. annealing	$O(\#\,bit\text{-}flips)$, scalable	Global (QUBO)
DDQN + GA	$O(\sqrt{N})$ actions, $O(N)$ greedy	Near-optimal

DTPQ/EIPQ, sphere decoding, and QUBO transformation enable real-time RIS optimization for hundreds to thousands of elements (Sang et al., 2023, Pekcan et al., 2023, Ramezani et al., 2023, Kim et al., 2023).

A practical guideline is to quantize continuous-phase solutions for initialization, iterate coordinate-wise updates for refinement, and resort to global combinatorial or metaheuristic solvers for critical applications with strong nonconvexity, many users, or non-uniform phase codebooks. Hadamard-based training patterns are optimal at 1-bit for LS channel estimation (Sun et al., 2021).

7. Generalizations, Limitations, and Outlook

Modern discrete phase optimization frameworks generalize to arbitrary phase codebooks (non-uniformly spaced, elements allowed to be off) and hybrid amplitude–phase designs (Hashemi et al., 2023). The switching between transmission/reflection states in STAR-RIS architectures introduces additional combinatorial layers (Alishahi et al., 20 Oct 2025). Multiuser/multicast scalability is addressed via smart gradient or probabilistic search over codebooks or categorical policy spaces.

Limitations arise in strongly coupled multiuser MIMO, fully nonconvex combinatorial optimization, and real-time adaptation under fast-fading. For very large arrays, physics-inspired solvers (QUBO + simulated annealing) enable tractable near-optimal solutions (Kim et al., 2023).

Current research directions involve integration with reinforcement learning for online control in dynamic environments (Wang et al., 7 May 2025), adaptive codebook design for channel estimation and feedback reduction (Ghanem et al., 2022), and joint control with power/amplitude, analog–digital hybrid architectures, or quantum-inspired hardware accelerators.

Discrete phase shift optimization frameworks thus represent a mature, multifaceted research area with rigorous theoretical foundation, efficient algorithms, and validated impact on both classic and emerging RIS applications.