Pinching Discretization Efficiency

• Pinching discretization efficiency is defined as the ratio of the ergodic rate of a discrete PAS to that of a continuous PAS, capturing performance loss due to finite pinching points.
• Analytical models using dielectric-waveguide PASs yield closed-form expressions for metrics like outage probability and ergodic rate, highlighting trade-offs between antenna density and throughput.
• Design guidelines derived from the analysis recommend optimal PA counts for different spatial dimensions to balance hardware simplicity with near-continuous performance in programmable wireless environments.

Pinching discretization efficiency quantifies the loss—or efficiency—incurred when a theoretically continuous actuator (such as a pinching antenna capable of arbitrary placement along a waveguide) is constrained to operate on a discrete, finite set of positions. In programmable wireless environments (PWEs), this metric rigorously characterizes how well a practical two-state pinching-antenna system (PAS) with a finite number of fixed pinching points can approximate the ideal continuous limit in terms of achievable throughput or ergodic data rate. Pinching discretization efficiency has become a central analytic tool in evaluating and optimizing the design of PASs and in engineering trade-offs between hardware simplicity and communication performance (Tyrovolas et al., 21 Dec 2025, Tyrovolas et al., 3 Nov 2025).

1. Definition and Formalism

Pinching discretization efficiency, denoted $\overline\eta_r$ or $\eta_d$ in the literature, is defined as the ratio of the ergodic achievable data rate of a discrete (two-state) PAS to that of the ideal continuous PAS: $\overline\eta_r = \frac{\overline R}{R_c}$ where

• $\overline R$: Ergodic rate with $M$ fixed pinching-antenna (PA) locations, activating exactly one per slot.
• $R_c$: Ergodic rate when PAs can be formed at any real-valued position along the waveguide.

This definition encapsulates, in a single normalized scalar, the degree to which the discrete hardware approaches the ideal propagation control promise of continuous reconfigurability. By construction, $0 < \overline\eta_r \leq 1$, and $\overline\eta_r \rightarrow 1$ as $M \to \infty$ (i.e., as the discretization becomes vanishingly fine) (Tyrovolas et al., 21 Dec 2025, Tyrovolas et al., 3 Nov 2025).

2. Analytical Framework and Key Assumptions

The canonical analysis is based on a dielectric-waveguide PAS, where radiation is produced by selectively exciting one of $M$ pre-placed PAs along a waveguide of length $\eta_d$0. The main modeling assumptions are:

• PAs are located at $\eta_d$1, with grid spacing $\eta_d$2.
• Each user is randomly positioned in the $\eta_d$3-plane, and the optimal PA (maximizing user SNR) is chosen in each access.
• The signal undergoes exponential attenuation along the waveguide ($\eta_d$4), with additional path-loss effects from the antenna to the user.

The received SNR for user $\eta_d$5 from the $\eta_d$6th PA is: $\eta_d$7 where $\eta_d$8 and $\eta_d$9 is the waveguide height.

With these assumptions, the ergodic rate $\overline\eta_r = \frac{\overline R}{R_c}$0 is computed by integrating the log-rate over the user distribution and the discrete PA serving regions partitioned by proximity (Tyrovolas et al., 21 Dec 2025, Tyrovolas et al., 3 Nov 2025).

3. Closed-form Expressions, Region Partitioning, and PDE Calculation

For two-state PASs, the discrete spatial structure allows the entire axis to be partitioned into $\overline\eta_r = \frac{\overline R}{R_c}$1 serving rectangles, each associated with a PA and width $\overline\eta_r = \frac{\overline R}{R_c}$2. With uniform user distribution, key outcomes include:

• Outage Probability:

$\overline\eta_r = \frac{\overline R}{R_c}$3

$\overline\eta_r = \frac{\overline R}{R_c}$4 is the 1D outage probability, detailed in five piecewise expressions parameterized by $\overline\eta_r = \frac{\overline R}{R_c}$5.

• Ergodic Rate:

$\overline\eta_r = \frac{\overline R}{R_c}$6

$\overline\eta_r = \frac{\overline R}{R_c}$7

$\overline\eta_r = \frac{\overline R}{R_c}$8 and $\overline\eta_r = \frac{\overline R}{R_c}$9 are computed in closed form, and $\overline R$0.

Pinching discretization efficiency is then evaluated as $\overline R$1, with $\overline R$2 determined by the same formalism but with $\overline R$3 (continuous limit) (Tyrovolas et al., 21 Dec 2025, Tyrovolas et al., 3 Nov 2025).

4. Intuitive Interpretation and Limiting Behavior

The PDE quantifies how closely a discrete, grid-constrained PAS emulates the ideal PAS with continuous pinching. At small $\overline R$4 (larger $\overline R$5), the closest available PA may be far from the user-optimal $\overline R$6, incurring SNR and rate loss. As $\overline R$7 increases, the discretization penalty diminishes due to finer spatial granularity.

However, a notable artifact is that, under exponential waveguide attenuation, further densification of PAs may eventually overshoot the true optimum, so $\overline R$8 plateaus below unity rather than increasing indefinitely. This characterizes a fundamental performance saturation dictated by combined grid alignment and physical channel effects (Tyrovolas et al., 21 Dec 2025).

5. Design Criteria and Performance Guidelines

A key utility of pinching discretization efficiency lies in enabling principled design of PAS topologies. Analytical forms for $\overline R$9 provide explicit trade-offs between the number of PAs ($M$0), the physical environment dimensions ($M$1), and the desired proximity to the continuous optimum.

Key design recommendations, validated both numerically and by closed-form analysis, are:

• For room-scale environments ($M$2), $M$3 suffices for $M$4.
• For $M$5, $M$6.
• For $M$7, $M$8.
• For $M$9, $R_c$0–$R_c$1 achieves $R_c$2.
• Further increments in $R_c$3 yield rapidly diminishing returns, so hardware cost and complexity can be minimized by targeting $R_c$4 thresholds (e.g., $R_c$5 or $R_c$6) (Tyrovolas et al., 21 Dec 2025, Tyrovolas et al., 3 Nov 2025).

The penalty scales as $R_c$7, motivating design of $R_c$8 (and thus $R_c$9) such that the expected PA-user misalignment is below the characteristic vertical dimension.

6. Relationship with Energy Efficiency and Dual-Scale Resolution

Beyond raw ergodic rate, discretization efficiency interacts with broader system energy efficiency, particularly under dual-scale deployment (DSD) strategies. Here, coarse and fine grid resolutions are jointly optimized against actuation power cost, RF combining gain, and deployment protocol.

• Coarse resolution $0 < \overline\eta_r \leq 1$0 impacts amplitude alignment and transmission efficiency. Excessive coarseness ($0 < \overline\eta_r \leq 1$1) can cause substantial efficiency loss—up to $0 < \overline\eta_r \leq 1$2 in multi-PA systems.
• Fine resolution $0 < \overline\eta_r \leq 1$3 controls phase alignment and can induce a $0 < \overline\eta_r \leq 1$4 drop if too coarse.
• Practical energy-efficient PAS designs emerge by tuning $0 < \overline\eta_r \leq 1$5 and $0 < \overline\eta_r \leq 1$6 to achieve less than $0 < \overline\eta_r \leq 1$7 amplitude quantization and $0 < \overline\eta_r \leq 1$8 phase error, respectively (Gan et al., 31 Oct 2025).
• Overall system energy efficiency (spectral efficiency per total consumed power) is maximized by balancing RF, positioning, and mechanical costs with discretization-induced rate penalties using PDE as a guiding metric.

7. Numerical Validation and Practical Implications

Simulations consistently confirm the analytic framework, indicating:

• For typical PWEs ($0 < \overline\eta_r \leq 1$9, $\overline\eta_r \rightarrow 1$0, $\overline\eta_r \rightarrow 1$1, $\overline\eta_r \rightarrow 1$2), $\overline\eta_r \rightarrow 1$3 saturates quickly with moderate $\overline\eta_r \rightarrow 1$4.
• Even with moderate $\overline\eta_r \rightarrow 1$5, near-optimal performance is accessible; e.g., for $\overline\eta_r \rightarrow 1$6, $\overline\eta_r \rightarrow 1$7 at $\overline\eta_r \rightarrow 1$8, and $\overline\eta_r \rightarrow 1$9 at $M \to \infty$0.
• The impact of increased waveguide attenuation ($M \to \infty$1) mildly depresses the plateau in $M \to \infty$2 but does not shift the saturation point (Tyrovolas et al., 21 Dec 2025).

The analytic approach, validated by Monte Carlo, supports hardware-efficient deployment of PAS in practical indoor scenarios, with the ability to a priori size the number of discrete antennas for a target throughput or energy efficiency constraint.

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References:

(Tyrovolas et al., 21 Dec 2025, Tyrovolas et al., 3 Nov 2025, Gan et al., 31 Oct 2025)