EAPQ: Amplitude-Aware Polar Quantization

• EAPQ is an amplitude-aware polar quantization framework that integrates discrete beamforming, capacity optimization in AWGN channels, LLM weight quantization, and VLC modulation.
• It employs separate amplitude and phase quantization with techniques like Lloyd–Arimoto optimization and geometric projection to enhance reliability and mitigate hardware constraints.
• The approach leverages tailored amplitude adjustments—via radial bins, rings, and optimized thresholds—to counteract phase limitations, improving metrics such as beamforming gain and distortion.

Searching arXiv for the specified papers to ground the article and citations. arxiv_search(query="(Bernardo et al., 2022)", max_results=5, sort_by="relevance") arxiv_search(query="(Pekcan et al., 9 Jul 2025)", max_results=5, sort_by="relevance") arxiv_search(query="(Sharma et al., 25 May 2026)", max_results=5, sort_by="relevance") arxiv_search(query="(Abumarshoud et al., 2021)", max_results=5, sort_by="relevance") Extended Amplitude-Introduced Polar Quantization (EAPQ) designates a family of amplitude-aware constructions built on polar representations, that is, representations separating magnitude and angle or closely related geometric variables. In the literature, the term is used most explicitly for discrete beamforming of reconfigurable intelligent surfaces (RISs) with phase-dependent amplitude and limited phase range, while closely related amplitude-extended polar-quantization principles also appear in information-theoretic analyses of polar-quantized AWGN channels, in constructive extensions of amplitude–phase–quadrant modulation for visible light communications, and as an inferred polar variant within LLM weight quantization studies (Pekcan et al., 9 Jul 2025, Bernardo et al., 2022, Abumarshoud et al., 2021, Sharma et al., 25 May 2026). Across these settings, the common technical motif is that amplitude is not treated as a passive by-product of phase quantization; instead, it is explicitly modeled, quantized, optimized, or projected in order to improve rate, reliability, beamforming gain, or distortion under finite-precision or hardware constraints.

1. Terminological scope and common structure

EAPQ is not a single standardized algorithm across all fields. The term appears explicitly in RIS beamforming work, where it denotes an extension of amplitude-introduced polar quantization using geometric projections under phase-dependent amplitude (PDA) and limited phase range. In the AWGN polar-receiver literature, the same expression is used in the supplied description as a design paradigm derived from the capacity-achieving structure of polar-quantized channels. In the LLM and VLC materials, EAPQ is presented as an inferred or constructive extension rather than as the formal name of the method in the original paper (Pekcan et al., 9 Jul 2025, Bernardo et al., 2022, Sharma et al., 25 May 2026, Abumarshoud et al., 2021).

A concise way to organize the term is to distinguish the underlying domains and the status of the name.

Domain	Meaning of EAPQ	Status
Polar-quantized AWGN channel	Extend amplitude resolution within polar quantization to approach capacity	Design paradigm in supplied summary
RIS beamforming with PDA	Geometric PDA-aware projection over discrete polar coefficients	Explicitly named
LLM weight quantization	Amplitude-aware polar variant contrasted with joint 2D codebooks	Inferred interpretation
VLC under IM/DD	Extension of APQ via optimized quantization, power allocation, and SIC	Constructive extension

Despite this heterogeneity, the constructions share four recurring ingredients. First, the signal or coefficient is represented in a polar or quasi-polar form. Second, finite-resolution constraints act differently on amplitude and angle. Third, amplitude-aware design is introduced to compensate for coarse phase resolution, amplitude attenuation, or product-cell distortion. Fourth, optimization is performed over discrete geometric objects: rings, radial bins, phase sectors, discrete reflection coefficients, or amplitude/phase codebooks. This suggests a unifying viewpoint in which EAPQ denotes amplitude-explicit design within a polar discretization framework, although the exact optimization target differs by application.

2. Information-theoretic foundation in polar-quantized AWGN channels

The most rigorous foundation for amplitude-extended polar design is the analysis of the additive white Gaussian noise channel with polar quantization at the receiver (Bernardo et al., 2022). A polar receiver measures the envelope and phase of the complex baseband output,

$R = |Y|,\qquad \Theta = \arg(Y),$

rather than the in-phase and quadrature components. The channel input satisfies $\mathbb{E}[|X|^2]\le P$, and the output is

$Y = X + Z,\qquad Z\sim \mathcal{CN}(0,N_0),$

with per-real-dimension variance $\sigma^2=N_0/2$.

Polar quantization applies separate quantizers to $R$ and $\Theta$. For a $b_1$-bit phase quantizer, the phase interval $[-\pi,\pi)$ is partitioned into $B=2^{b_1}$ equal sectors of width $\Delta=2\pi/B$, with sector centers

$\mathbb{E}[|X|^2]\le P$0

Amplitude is quantized by radial thresholds $\mathbb{E}[|X|^2]\le P$1, with the single-bit magnitude case using one optimized threshold $\mathbb{E}[|X|^2]\le P$2 and bins $\mathbb{E}[|X|^2]\le P$3 and $\mathbb{E}[|X|^2]\le P$4.

The polar-form conditional density for input $\mathbb{E}[|X|^2]\le P$5 is

$\mathbb{E}[|X|^2]\le P$6

and the induced transition probability to quantizer output $\mathbb{E}[|X|^2]\le P$7 is obtained by integrating this density over the corresponding radial and angular bin. The capacity under polar quantization is

$\mathbb{E}[|X|^2]\le P$8

The central theorem is structural: the capacity-achieving input has amplitude phase shift keying (APSK) form. Writing

$\mathbb{E}[|X|^2]\le P$9

the optimal phase pmf is uniform over the $Y = X + Z,\qquad Z\sim \mathcal{CN}(0,N_0),$0 sector centers because the AWGN channel and uniform phase quantizer are jointly rotationally invariant. Consequently, the capacity optimization reduces to the amplitude distribution: the ring radii $Y = X + Z,\qquad Z\sim \mathcal{CN}(0,N_0),$1 and the amplitude probabilities $Y = X + Z,\qquad Z\sim \mathcal{CN}(0,N_0),$2.

This result is the main information-theoretic basis for amplitude-extended polar design. EAPQ, in this setting, means exploiting that reduction by extending amplitude resolution through more radial bins, more rings, or both. The optimization problem is to maximize $Y = X + Z,\qquad Z\sim \mathcal{CN}(0,N_0),$3 over the discrete radii and their pmf subject to

$Y = X + Z,\qquad Z\sim \mathcal{CN}(0,N_0),$4

For a fixed candidate set of radii, the channel becomes a discrete-input discrete-output memoryless channel, and a Blahut–Arimoto procedure can optimize the pmf; the radii can then be updated by coordinate search or other gradient-free methods.

The numerical findings expose an important non-smooth phenomenon: SNR thresholds at which the number of amplitude levels and their probabilities change abruptly. At low SNR, the optimal structure is one ring with uniform phases; as SNR increases, the optimum transitions to two rings and possibly more, with probability mass shifting toward outer rings. The location of these thresholds depends on the number of phase bits. More phase bits generally delay the need for additional rings, whereas fewer phase bits induce earlier amplitude-structure transitions to compensate for limited phase distinguishability. In the single-bit magnitude case, the optimized threshold $Y = X + Z,\qquad Z\sim \mathcal{CN}(0,N_0),$5 depends on both SNR and $Y = X + Z,\qquad Z\sim \mathcal{CN}(0,N_0),$6; increasing $Y = X + Z,\qquad Z\sim \mathcal{CN}(0,N_0),$7 improves capacity at all SNRs, and optimizing $Y = X + Z,\qquad Z\sim \mathcal{CN}(0,N_0),$8 yields nontrivial gains over naïve thresholds. The paper leaves open the joint optimization of multi-bit amplitude thresholds and input rings, analytical characterization of the SNR transitions, and extension beyond AWGN to fading or nonuniform-sector quantizers.

3. Explicit EAPQ in RIS beamforming with phase-dependent amplitude

The most explicit usage of the term EAPQ appears in RIS beamforming with limited phase range and phase-dependent amplitude attenuation (Pekcan et al., 9 Jul 2025). Each RIS element has reflection coefficient

$Y = X + Z,\qquad Z\sim \mathcal{CN}(0,N_0),$9

with $\sigma^2=N_0/2$0 and $\sigma^2=N_0/2$1. The phase range is $\sigma^2=N_0/2$2, and the feasible discrete phase set $\sigma^2=N_0/2$3 depends on both $\sigma^2=N_0/2$4 and the number of discrete phases $\sigma^2=N_0/2$5. For sufficiently wide range, the $\sigma^2=N_0/2$6 phases are uniformly spaced over the unit circle; for limited range, they are uniformly spaced within $\sigma^2=N_0/2$7.

For a canonical single-user narrowband RIS-assisted link, with cascaded channel coefficients $\sigma^2=N_0/2$8 and direct link $\sigma^2=N_0/2$9, the received baseband sum is

$R$0

and the design objective is to maximize

$R$1

The paper derives a necessary and sufficient optimality condition. If

$R$2

then each element must choose

$R$3

This converts the global beamforming problem into a geometric projection rule onto feasible PDA-aware polar coefficients. The paper then constructs a linear-time optimal search algorithm, under local convexity of the sampled coefficient set, by moving $R$4 along the unit circle and updating only those elements whose decision boundaries are crossed. The resulting complexity is linear in $R$5 and $R$6, with at most $R$7 update steps, in contrast to exhaustive search of order $R$8.

Within this framework, APQ and EAPQ are distinct. APQ quantizes the ideal continuous phase alignment

$R$9

to the nearest discrete phase and ignores amplitude variation, effectively treating $\Theta$0. EAPQ instead incorporates PDA directly through the projection

$\Theta$1

for a desired unconstrained coefficient $\Theta$2. When $\Theta$3, this is equivalent to the PDA-aware rule

$\Theta$4

EAPQ therefore remains low complexity, requiring only $\Theta$5 candidate evaluations per element, yet directly accounts for amplitude–phase coupling.

The paper also provides closed-form performance characterizations. For sufficiently wide phase range, the large-$\Theta$6 normalized performance relative to the ideal continuous-phase no-attenuation case is

$\Theta$7

For limited range, $\Theta$8 becomes a nonuniform expression depending on endpoint amplitudes and the interior sum. These formulas separate amplitude loss from phase quantization loss and explain the principal sensitivity results: increasing the number of discrete phases beyond $\Theta$9 yields only marginal gains when the phase range is sufficiently wide; when the phase range is limited, performance is sensitive to attenuation for larger $b_1$0, and sensitive to $b_1$1 when attenuation is milder.

Validation results support the analytical picture. For $b_1$2, $b_1$3, and $b_1$4, the proposed optimal algorithm matches exhaustive search in CDFs of SNR boost, establishing global optimality under the stated conditions. EAPQ significantly outperforms APQ under high attenuation, and under high attenuation the first-percentile performance of EAPQ with $b_1$5 is approximately that of APQ with $b_1$6. The optimal algorithm remains the absolute benchmark, but EAPQ closes much of the gap at $b_1$7 complexity.

4. Polar EAPQ as an inferred variant in LLM weight quantization

In LLM weight quantization, the paper introducing QAM-W does not formally define an algorithm called EAPQ; rather, it analyzes a polar baseline and then describes what an “Extended Amplitude-Introduced Polar Quantization” would amount to within that framework (Sharma et al., 25 May 2026). The setting is row-wise post-training quantization of weight matrices. Each row is $b_1$8-normalized, block-Hadamard rotated, paired into 2D coordinates, and either quantized by a joint 2D Lloyd–Max codebook or by a polar amplitude–phase factorization.

For a 2D pair $b_1$9, the polar mapping is

$[-\pi,\pi)$0

with reconstruction $[-\pi,\pi)$1. The polar baseline quantizes amplitude with a scalar Lloyd–Max quantizer trained on the unit Rayleigh distribution and phase with uniform bins on $[-\pi,\pi)$2. The exact distortion identity for $[-\pi,\pi)$3 and $[-\pi,\pi)$4 is

$[-\pi,\pi)$5

which decomposes total distortion into an amplitude term and a phase term weighted by amplitude.

The proposed QAM-W codec replaces this product quantization by a single 2D Lloyd–Max codebook trained on the unit circular Gaussian. Empirically, the joint 2D codebook is decisively stronger. At approximately $[-\pi,\pi)$6 bpw, joint 2D coding outperforms the polar baseline by $[-\pi,\pi)$7–$[-\pi,\pi)$8 percentage points in $[-\pi,\pi)$9PPL, with reported examples of $B=2^{b_1}$0 pp on TinyLlama, $B=2^{b_1}$1 pp on Mistral, and $B=2^{b_1}$2 pp on Qwen. At approximately $B=2^{b_1}$3 bpw, the activation-aware joint 2D variant remains within $B=2^{b_1}$4 of BF16 WikiText-2 perplexity across TinyLlama, Qwen, and Mistral, matching the SmoothQuant W8A8 quality envelope with approximately $B=2^{b_1}$5 fewer weight bits. Paired KL against BF16 tracks $B=2^{b_1}$6PPL\% with Spearman $B=2^{b_1}$7 across $B=2^{b_1}$8 method–model rows.

Within this paper, EAPQ is explicitly presented as an interpretation rather than a named contribution. The described extensions are amplitude-aware refinements of the polar baseline: tighter pair calibration through per-pair $B=2^{b_1}$9, activation-aware per-channel scaling $\Delta=2\pi/B$0, amplitude-centric bit allocation with $\Delta=2\pi/B$1, and weak coupling between $\Delta=2\pi/B$2 and $\Delta=2\pi/B$3 by allocating finer phase resolution to larger amplitudes. These modifications aim to reduce the amplitude distortion term and the amplitude-weighted phase penalty, especially on high-RMS activation channels.

The paper’s diagnostic conclusion is precise. Such EAPQ-style polar refinements can narrow the gap to joint 2D coding, particularly on quantization-tolerant architectures and at modest bitrates, but they do not eliminate the core limitation of separate amplitude and phase quantization. Joint 2D Lloyd coding adapts its Voronoi geometry directly to the rotated pair distribution, whereas polar coding uses rectangular product cells in $\Delta=2\pi/B$4 space. A plausible implication is that, in this setting, EAPQ is best understood as a structured low-complexity compromise rather than as a replacement for learned 2D vector quantization.

5. Constructive EAPQ extension of APQ modulation for visible light communications

In visible light communications, the base object is APQ modulation, not EAPQ, and the supplied material develops EAPQ as an extension of APQ under IM/DD constraints (Abumarshoud et al., 2021). The optical signal must be real and nonnegative, so a complex symbol

$\Delta=2\pi/B$5

is decomposed into three unipolar components conveying amplitude, phase, and quadrant information. The quadrant variable $\Delta=2\pi/B$6 contributes two bits, while amplitude and intra-quadrant phase are mapped to PAM levels. The transmitted optical intensity is the superposition

$\Delta=2\pi/B$7

where each component is formed from a DC bias and a weighted unipolar PAM level.

The indoor VLC channel is modeled in line-of-sight by the Lambertian DC gain, and the received baseband signal is

$\Delta=2\pi/B$8

with AWGN variance $\Delta=2\pi/B$9. The receiver uses successive interference cancellation: with fixed power allocation satisfying $\mathbb{E}[|X|^2]\le P$00, the amplitude component is decoded first, then phase, then quadrant. The paper derives exact SER expressions for the $\mathbb{E}[|X|^2]\le P$01-ary case $\mathbb{E}[|X|^2]\le P$02, $\mathbb{E}[|X|^2]\le P$03, $\mathbb{E}[|X|^2]\le P$04, including error-propagation terms through the SIC chain, and reports that the analytical SER closely matches Monte Carlo simulation.

Within the constructive EAPQ extension, the basic APQ architecture is retained but several optimizations are introduced. The amplitude quantizer can be redesigned by a Lloyd–Max procedure minimizing

$\mathbb{E}[|X|^2]\le P$05

while the intra-quadrant phase quantizer can be optimized over $\mathbb{E}[|X|^2]\le P$06 using a circular Lloyd–Max criterion. Power allocation across amplitude, phase, and quadrant streams can be adapted to minimize overall SER under average optical power, peak optical power, and LED-linearity constraints. The SIC order can be made dynamic through instantaneous reliability scores, instead of remaining fixed as amplitude $\mathbb{E}[|X|^2]\le P$07 phase $\mathbb{E}[|X|^2]\le P$08 quadrant. Cross-component coding and Gray mapping can further reduce catastrophic propagation of early SIC errors.

These constructions preserve the original APQ motivation. APQ realizes high-order signaling with a single LED and a single photodiode, using three low-order PAM streams and scalar thresholding at the receiver. The paper reports that APQ offers higher reliability than generalized spatial shift keying across realistic indoor deployments while requiring lower hardware complexity. EAPQ, in this context, is therefore a refinement layer over APQ: it preserves the amplitude–phase–quadrant factorization but optimizes quantization, power allocation, and SIC to improve robustness under IM/DD and LED constraints.

6. Cross-domain principles, limitations, and open directions

Across AWGN polar receivers, RIS beamforming, LLM weight coding, and VLC modulation, EAPQ-like methods share a consistent design logic. Amplitude is elevated from a secondary coordinate to an optimization variable. In the AWGN setting this appears as optimal ring radii and amplitude pmf over APSK inputs; in RIS beamforming it appears as projection onto PDA-aware feasible coefficients; in LLM quantization it appears as amplitude calibration, activation-aware scaling, and amplitude-heavy bit allocation; in VLC it appears as explicit amplitude signaling and optimized amplitude quantization. The recurring benefit is compensation for phase coarsening, attenuation, or independent scalar quantization.

The domain-specific limitations are equally consistent. In polar-quantized AWGN channels, the reported numerics focus on single-bit magnitude and few-bit phase, and the continuous optimization of ring radii and thresholds is purely numerical; closed-form optimal radii and thresholds are unavailable (Bernardo et al., 2022). In RIS beamforming, global optimality of the linear-time search is guaranteed only under local convexity of the sampled RIS coefficient set; multi-user, wideband, and joint active/passive optimization are outside the scope (Pekcan et al., 9 Jul 2025). In LLM quantization, the EAPQ notion is interpretive rather than canonical, and the polar family remains inferior to joint 2D coding at equal bitrate in the reported study; checkpoint-conditioned codebooks and fused dequantization–matmul kernels remain open problems (Sharma et al., 25 May 2026). In VLC, the EAPQ extension is constructive rather than experimentally standardized, and practical deployment remains constrained by LED linearity, eye-safety, dimming, and SIC error propagation (Abumarshoud et al., 2021).

A plausible synthesis is that EAPQ is best viewed not as a single method but as a reusable geometric principle. Whenever a system admits a polar or angle-aware representation and hardware or algorithmic constraints make pure phase quantization inadequate, amplitude can be introduced in a structured way: through rings, thresholds, radial bins, PDA-aware projections, or amplitude-prioritized codebooks. The literature surveyed here shows that this principle can be capacity-oriented, beamforming-oriented, distortion-oriented, or reliability-oriented, depending on the objective function. The principal open challenge is not whether amplitude should be introduced, but how tightly amplitude and angle should be coupled for a given hardware budget: separate scalar quantization, partially coupled polar rules, or fully joint multidimensional codebooks.