APQ: Polar Quantization for RIS Beamforming

• APQ is a quantization framework for RIS beamforming that projects ideal continuous-phase configurations onto a discrete phase set, offering a low-complexity heuristic.
• It leverages the RIS coefficient model with phase-dependent amplitudes, using nearest-neighbor phase selection while ignoring amplitude attenuation in the decision rule.
• APQ provides analytical performance approximations relative to an ideal continuous-phase benchmark and serves as a basis for comparing amplitude-aware methods like EAPQ.

Searching arXiv for relevant papers on APQ, H-APQ, and polar quantization. arxiv_search(query="amplitude-introduced polar quantization OR hybrid amplitude-phase quantization OR Gaussian channel polar quantization", max_results=10, sort_by="relevance") Searching arXiv. Searching arXiv for “amplitude-introduced polar quantization”. to=arxiv_search {"query":"amplitude-introduced polar quantization", "max_results": 5, "sort_by": "relevance"} Amplitude-Introduced Polar Quantization (APQ) is a quantization framework for discrete beamforming in reconfigurable intelligent surfaces (RISs) with phase-dependent amplitude (PDA) and a finite phase set. In the formulation introduced for RISs with a limited phase range and amplitude attenuation, APQ is defined as the quantization of the ideal continuous-phase/no-attenuation solution: each ideal RIS phase is mapped to the nearest available discrete phase, yielding a low-complexity heuristic for received-power maximization under practical RIS constraints (Pekcan et al., 9 Jul 2025). Although the term includes “amplitude-introduced,” APQ does not use amplitude attenuation in its decision rule; rather, the amplitude dependence enters through the RIS coefficient model and through the approximation analysis built around that model. This distinguishes APQ both from amplitude-aware extensions such as EAPQ and from other amplitude-phase quantization methods developed in relay and receiver architectures.

1. Definition and problem setting

APQ arises in an RIS configuration problem where each element can realize only a discrete phase shift and where the reflection amplitude depends on the chosen phase. The underlying optimization seeks to maximize the received power at a user equipment through a direct path and an RIS-assisted path. For the $n$-th RIS element, the cascaded BS–RIS–UE channel is written as

$$h_n=\beta_n e^{j\alpha_n},\qquad \beta_n\ge 0,\ \alpha_n\in[-\pi,\pi),$$

while the direct BS–UE link is

$h_0=\beta_0 e^{j\alpha_0}.$

The selected RIS coefficient is

$w_n=\beta^r(\theta_n)e^{j\theta_n},$

where $\theta_n\in\Phi_K$ is a discrete phase and $\beta^r(\theta_n)\in[0,1]$ is the corresponding phase-dependent amplitude gain (Pekcan et al., 9 Jul 2025).

The received-power maximization problem is posed as

$$\max_{\boldsymbol{\theta}} f(\boldsymbol{\theta}) \quad\text{s.t.}\quad \theta_n\in\Phi_K,\ n=1,\dots,N,$$

with objective

$$f(\boldsymbol{\theta})= \left| \beta_0e^{j\alpha_0} +\sum_{n=1}^N \beta_n\beta^r(\theta_n)e^{j(\alpha_n+\theta_n)} \right|^2.$$

Within this setting, APQ serves two roles. First, it is a low-complexity baseline for discrete RIS beamforming. Second, it is an analytical device used to derive closed-form approximation ratios relative to an ideal continuous-phase, no-attenuation benchmark. The latter role is central to the framework’s significance, because the paper uses APQ to quantify how closely a practical PDA-constrained RIS can approach an idealized continuous, unit-gain surface (Pekcan et al., 9 Jul 2025).

2. RIS coefficient model, PDA law, and ideal benchmark

The RIS coefficient set is defined as

$$\mathbf{W}_K= \left\{ \beta^r(\phi_1)e^{j\phi_1}, \beta^r(\phi_2)e^{j\phi_2}, \ldots, \beta^r(\phi_K)e^{j\phi_K} \right\}.$$

The PDA model adopted is

$$\beta^r(\theta_n)= (1-\beta^r_{\min}) \left(\frac{\sin(\theta_n-\phi^r)+1}{2}\right)^{\alpha^r} +\beta^r_{\min}.$$

Its parameters are $$h_n=\beta_n e^{j\alpha_n},\qquad \beta_n\ge 0,\ \alpha_n\in[-\pi,\pi),$$0, which controls the minimum amplitude; $$h_n=\beta_n e^{j\alpha_n},\qquad \beta_n\ge 0,\ \alpha_n\in[-\pi,\pi),$$1, a phase offset or rotation of the gain curve; and $$h_n=\beta_n e^{j\alpha_n},\qquad \beta_n\ge 0,\ \alpha_n\in[-\pi,\pi),$$2, a steepness parameter. The numerical studies later fix $$h_n=\beta_n e^{j\alpha_n},\qquad \beta_n\ge 0,\ \alpha_n\in[-\pi,\pi),$$3 and often use $$h_n=\beta_n e^{j\alpha_n},\qquad \beta_n\ge 0,\ \alpha_n\in[-\pi,\pi),$$4 (Pekcan et al., 9 Jul 2025).

APQ is defined by reference to an idealized benchmark in which phases are continuous and $$h_n=\beta_n e^{j\alpha_n},\qquad \beta_n\ge 0,\ \alpha_n\in[-\pi,\pi),$$5 for all $$h_n=\beta_n e^{j\alpha_n},\qquad \beta_n\ge 0,\ \alpha_n\in[-\pi,\pi),$$6. Under that benchmark, the optimal phase choice is pure alignment: $$h_n=\beta_n e^{j\alpha_n},\qquad \beta_n\ge 0,\ \alpha_n\in[-\pi,\pi),$$7 The corresponding ideal received power is

$$h_n=\beta_n e^{j\alpha_n},\qquad \beta_n\ge 0,\ \alpha_n\in[-\pi,\pi),$$8

This benchmark is not the solution to the PDA-constrained problem; it is the target that APQ quantizes. That distinction is important. The APQ construction begins from the phase-alignment rule that would be optimal only in the continuous, no-attenuation case, and then discretizes it. This suggests that APQ is best understood as a projection of an idealized design onto a constrained RIS codebook, rather than as a direct optimizer of the PDA-constrained objective.

3. Quantization rule and geometric interpretation

APQ quantizes each ideal phase $$h_n=\beta_n e^{j\alpha_n},\qquad \beta_n\ge 0,\ \alpha_n\in[-\pi,\pi),$$9 to the nearest discrete phase in $h_0=\beta_0 e^{j\alpha_0}.$0. The rule is

$h_0=\beta_0 e^{j\alpha_0}.$1

This is the standard nearest-neighbor phase quantization rule on the unit circle (Pekcan et al., 9 Jul 2025).

The geometric interpretation is explicit. APQ selects the discrete phase that maximizes

$h_0=\beta_0 e^{j\alpha_0}.$2

so each RIS term is aligned as closely as possible with the direct-link direction $h_0=\beta_0 e^{j\alpha_0}.$3. However, APQ does not account for the fact that $h_0=\beta_0 e^{j\alpha_0}.$4 varies with $h_0=\beta_0 e^{j\alpha_0}.$5. In the terminology of the source paper, APQ is a straightforward “polar” quantizer because it uses the ideal phase direction and snaps it to the nearest discrete angle, but it ignores amplitude attenuation of the RIS coefficients (Pekcan et al., 9 Jul 2025).

A common misconception follows from the name. “Amplitude-introduced” does not mean that APQ is amplitude-aware in the decision step. The amplitude enters through the RIS model and through the approximation ratio formulas, whereas the decision rule itself remains nearest-angle quantization of the ideal phase. This is one of the principal conceptual distinctions between APQ and its extension EAPQ.

4. EAPQ and relation to the globally optimal discrete algorithm

The amplitude-aware extension is Extended Amplitude-Introduced Polar Quantization (EAPQ). Its decision rule is

$h_0=\beta_0 e^{j\alpha_0}.$6

Relative to APQ, EAPQ replaces pure phase proximity with an amplitude-weighted projection criterion. In interpretive terms given by the source, APQ “quantize[s] the ideal phase,” whereas EAPQ “choose[s] the discrete phase giving the largest projection onto the direct-link direction, while accounting for the RIS amplitude loss” (Pekcan et al., 9 Jul 2025).

The paper also derives an exact discrete optimization criterion. For the globally optimal discrete phases $h_0=\beta_0 e^{j\alpha_0}.$7, it is necessary and sufficient that

$h_0=\beta_0 e^{j\alpha_0}.$8

where

$h_0=\beta_0 e^{j\alpha_0}.$9

The difference between the three procedures is therefore structural: the optimal algorithm uses the unknown optimal aggregate direction $w_n=\beta^r(\theta_n)e^{j\theta_n},$0, EAPQ approximates that direction by $w_n=\beta^r(\theta_n)e^{j\theta_n},$1, and APQ further removes the amplitude weighting.

Method	Decision rule	Amplitude treatment
APQ	Nearest discrete phase to $w_n=\beta^r(\theta_n)e^{j\theta_n},$2	Ignores $w_n=\beta^r(\theta_n)e^{j\theta_n},$3 in the decision
EAPQ	$w_n=\beta^r(\theta_n)e^{j\theta_n},$4	Amplitude-aware heuristic
Optimal algorithm	$w_n=\beta^r(\theta_n)e^{j\theta_n},$5	Exact PDA-aware benchmark

The optimal search algorithm is proven to converge to the global optimum in at most $w_n=\beta^r(\theta_n)e^{j\theta_n},$6 boundary steps, with $w_n=\beta^r(\theta_n)e^{j\theta_n},$7 vector-addition complexity in the paper’s accounting. The same work presents this exact algorithm as a generic upper bound for discrete beamforming with amplitude constraints, while APQ and EAPQ remain lower-complexity alternatives (Pekcan et al., 9 Jul 2025).

5. Discrete phase sets, approximation ratios, and parameter sensitivity

The RIS phase range is denoted $w_n=\beta^r(\theta_n)e^{j\theta_n},$8, and allowed phases lie in $w_n=\beta^r(\theta_n)e^{j\theta_n},$9 without loss of generality. The discrete phase set is

$\theta_n\in\Phi_K$0

where

$\theta_n\in\Phi_K$1

If the phase range is sufficient, the $\theta_n\in\Phi_K$2 phases are uniformly spaced around the unit circle; if the range is limited, they are uniformly spaced only within the available interval (Pekcan et al., 9 Jul 2025).

The APQ approximation ratio against the ideal continuous, no-attenuation benchmark is defined by

$\theta_n\in\Phi_K$3

After derivation, the paper obtains

$\theta_n\in\Phi_K$4

where

$\theta_n\in\Phi_K$5

By symmetry, the sine term vanishes.

For uniformly spaced phases and sufficiently large phase range,

$\theta_n\in\Phi_K$6

hence

$\theta_n\in\Phi_K$7

with

$\theta_n\in\Phi_K$8

The source explicitly interprets this expression as separating two losses in the uniform case: a quantization loss $\theta_n\in\Phi_K$9 and an average attenuation loss $\beta^r(\theta_n)\in[0,1]$0 (Pekcan et al., 9 Jul 2025).

When $\beta^r(\theta_n)\in[0,1]$1, the phase set is nonuniform over the full circle. The selection probabilities become

$\beta^r(\theta_n)\in[0,1]$2

For $\beta^r(\theta_n)\in[0,1]$3,

$\beta^r(\theta_n)\in[0,1]$4

and for $\beta^r(\theta_n)\in[0,1]$5,

$\beta^r(\theta_n)\in[0,1]$6

The analytical and empirical conclusions are correspondingly specific. Increasing the number of discrete phases beyond $\beta^r(\theta_n)\in[0,1]$7 yields only marginal gains when the RIS has a sufficiently wide phase range $\beta^r(\theta_n)\in[0,1]$8. When $\beta^r(\theta_n)\in[0,1]$9 is limited, performance is sensitive to attenuation for larger $$\max_{\boldsymbol{\theta}} f(\boldsymbol{\theta}) \quad\text{s.t.}\quad \theta_n\in\Phi_K,\ n=1,\dots,N,$$0, and sensitive to $$\max_{\boldsymbol{\theta}} f(\boldsymbol{\theta}) \quad\text{s.t.}\quad \theta_n\in\Phi_K,\ n=1,\dots,N,$$1 when there is less attenuation. Smaller $$\max_{\boldsymbol{\theta}} f(\boldsymbol{\theta}) \quad\text{s.t.}\quad \theta_n\in\Phi_K,\ n=1,\dots,N,$$2 implies stronger attenuation and larger performance loss, and EAPQ tends to outperform APQ more noticeably in the high-attenuation regime (Pekcan et al., 9 Jul 2025).

6. Relation to broader polar and amplitude-phase quantization literature

APQ belongs to a broader family of methods that operate in polar coordinates, but its technical role differs sharply across application domains. In the RIS setting, APQ is a beamforming heuristic based on nearest-angle quantization of an ideal phase-alignment solution under PDA constraints (Pekcan et al., 9 Jul 2025). In contrast, the relay quantization method termed hybrid amplitude-phase quantization (H-APQ) was proposed for autoencoder-based MIMO quantize-forward relay systems and combines uniform phase quantization with ordered amplitude quantization based on the relative order of amplitudes rather than absolute amplitude bins (Kim et al., 18 Feb 2025). H-APQ therefore introduces amplitude information into the relay representation through rank-based grouping, reduces relay memory, and is explicitly described as an APQ-like hybrid method rather than standard uniform APQ.

A second, more theoretical line of work studies polar quantization at the receiver for a complex AWGN channel. There, the receiver converts the observation into amplitude and phase, quantizes them separately, and the main result is that the capacity-achieving input has an amplitude phase shift keying (APSK) structure. The optimization can be reduced to the amplitude probability mass function, with special emphasis on a $$\max_{\boldsymbol{\theta}} f(\boldsymbol{\theta}) \quad\text{s.t.}\quad \theta_n\in\Phi_K,\ n=1,\dots,N,$$3-bit phase quantizer and an optimized single-bit magnitude quantizer (Bernardo et al., 2022). That usage of polar quantization is information-theoretic and receiver-centric, rather than a discrete beamforming rule for RIS configuration.

These comparisons clarify what APQ is not. It is not standard uniform amplitude-phase quantization in the relay sense, because its decision rule does not quantize amplitudes jointly with phases. It is not the same as the capacity-oriented polar-quantized AWGN model, where amplitude quantization is an explicit receiver operation and APSK emerges as the optimal signaling structure. APQ is instead a nearest-phase quantizer built around an RIS coefficient model with phase-dependent amplitude, and its main analytical value lies in providing closed-form performance approximations for practical discrete RISs.

A plausible implication is that the shared “polar” terminology across these works reflects a common decomposition of complex variables into amplitude and phase, while the actual optimization target differs: beamforming under PDA-constrained RIS hardware in APQ, memory-efficient relay forwarding in H-APQ, and mutual-information maximization under finite-precision polar reception in the AWGN setting (Pekcan et al., 9 Jul 2025, Kim et al., 18 Feb 2025, Bernardo et al., 2022).