---
title: 'APQ: Polar Quantization for RIS Beamforming'
url: https://www.emergentmind.com/topics/amplitude-introduced-polar-quantization-apq
type: topic
---

# APQ: Polar Quantization for RIS Beamforming

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 [2507.07342]. 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 [2507.07342].

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 [2507.07342].

## 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 \(\beta^r_{\min}\), which controls the minimum amplitude; \(\phi^r\), a phase offset or rotation of the gain curve; and \(\alpha^r\), a steepness parameter. The numerical studies later fix \(\phi^r=\pi/2\) and often use \(\alpha^r=1.6\) [2507.07342].

APQ is defined by reference to an idealized benchmark in which phases are continuous and \(\beta^r(\theta)=1\) for all \(\theta\). Under that benchmark, the optimal phase choice is pure alignment:
\[
\theta_n^{\text{ideal}}=\alpha_0-\alpha_n,\qquad n=1,\ldots,N.
\]
The corresponding ideal received power is
\[
f(\boldsymbol{\theta}^{\text{ideal}})=\left(\sum_{n=0}^{N}\beta_n\right)^2.
\]

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 \(\theta_n^{\text{ideal}}\) to the nearest discrete phase in \(\Phi_K\). The rule is
\[
\theta_n^{\text{APQ}}=
\begin{cases}
\phi_1, & \frac{2\pi-\phi_K+\phi_1}{2}\le \theta_n^{\text{ideal}} < \frac{\phi_1+\phi_2}{2},\\
\phi_2, & \frac{\phi_1+\phi_2}{2}\le \theta_n^{\text{ideal}} < \frac{\phi_2+\phi_3}{2},\\
\vdots \\
\phi_{K-1}, & \frac{\phi_{K-2}+\phi_{K-1}}{2}\le \theta_n^{\text{ideal}} < \frac{\phi_{K-1}+\phi_K}{2},\\
\phi_K, & \text{otherwise}.
\end{cases}
\]
This is the standard nearest-neighbor phase quantization rule on the unit circle [2507.07342].

The geometric interpretation is explicit. APQ selects the discrete phase that maximizes
\[
\cos(\theta_n+\alpha_n-\alpha_0),
\]
so each RIS term is aligned as closely as possible with the direct-link direction \(\alpha_0\). However, APQ does not account for the fact that \(\beta^r(\theta_n)\) varies with \(\theta_n\). 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 [2507.07342].

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
\[
\theta_n^{\text{EAPQ}}
=
\arg\max_{\theta_n\in\Phi_K}
\beta^r(\theta_n)\cos(\theta_n+\alpha_n-\alpha_0).
\]
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” [2507.07342].

The paper also derives an exact discrete optimization criterion. For the globally optimal discrete phases \(\theta_n^*\), it is necessary and sufficient that
\[
\theta_n^*=
\arg\max_{\theta_n\in\Phi_K}
\beta^r(\theta_n)\cos(\theta_n+\alpha_n-\angle\mu),
\]
where
\[
g=h_0+\sum_{n=1}^N h_n\beta^r(\theta_n^*)e^{j\theta_n^*},
\qquad
\mu=\frac{g}{|g|}.
\]
The difference between the three procedures is therefore structural: the optimal algorithm uses the unknown optimal aggregate direction \(\angle\mu\), EAPQ approximates that direction by \(\alpha_0\), and APQ further removes the amplitude weighting.

| Method | Decision rule | Amplitude treatment |
|---|---|---|
| APQ | Nearest discrete phase to \(\theta_n^{\text{ideal}}=\alpha_0-\alpha_n\) | Ignores \(\beta^r(\theta_n)\) in the decision |
| EAPQ | \(\arg\max_{\theta_n\in\Phi_K}\beta^r(\theta_n)\cos(\theta_n+\alpha_n-\alpha_0)\) | Amplitude-aware heuristic |
| Optimal algorithm | \(\arg\max_{\theta_n\in\Phi_K}\beta^r(\theta_n)\cos(\theta_n+\alpha_n-\angle\mu)\) | Exact PDA-aware benchmark |

The optimal search algorithm is proven to converge to the global optimum in at most \(NK\) boundary steps, with \(\mathcal{O}(N(2K+1))\) 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 [2507.07342].

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

The RIS phase range is denoted \(R\in[0,2\pi]\), and allowed phases lie in \([-\tfrac{R}{2},\tfrac{R}{2}]\) without loss of generality. The discrete phase set is
\[
\Phi_K=
\begin{cases}
\left\{-\frac{R}{2},\ \frac{R}{K-1}-\frac{R}{2},\ \ldots,\ \frac{R}{2}\right\}, & R<2\pi\frac{K-1}{K},\\[1ex]
\{0,\omega',\ldots,(K-1)\omega'\}-\frac{(K-1)\omega'}{2}, & R\ge 2\pi\frac{K-1}{K},
\end{cases}
\]
where
\[
\omega'=\frac{2\pi}{K}.
\]
If the phase range is sufficient, the \(K\) phases are uniformly spaced around the unit circle; if the range is limited, they are uniformly spaced only within the available interval [2507.07342].

The APQ approximation ratio against the ideal continuous, no-attenuation benchmark is defined by
\[
E_{PDA}
=
\lim_{N\to\infty}
\frac{\mathbb{E}[f_{\text{rx}}(\boldsymbol{\theta}^{\text{APQ}})]}
{\mathbb{E}\left[\left(\sum_{n=0}^N \beta_n\right)^2\right]}.
\]
After derivation, the paper obtains
\[
E_{PDA}
=
\left(\mathbb{E}\big[\beta^r(\theta_n^{\text{APQ}})\cos(\delta_n)\big]\right)^2
+
\left(\mathbb{E}\big[\beta^r(\theta_n^{\text{APQ}})\sin(\delta_n)\big]\right)^2,
\]
where
\[
\delta_n=\theta_n^{\text{APQ}}-\theta_n^{\text{ideal}}.
\]
By symmetry, the sine term vanishes.

For uniformly spaced phases and sufficiently large phase range,
\[
\mathbb{E}\left[\beta^r(\theta_n^{\text{APQ}})\cos(\delta_n)\right]
=
\frac{\sinc(1/K)}{K}\sum_{k=1}^K \beta^r(\phi_k),
\]
hence
\[
E_{PDA}(K)
=
\left(
\frac{\sinc(1/K)}{K}\sum_{k=1}^K \beta^r(\phi_k)
\right)^2,
\]
with
\[
\sinc(x)=\frac{\sin(\pi x)}{\pi x}.
\]
The source explicitly interprets this expression as separating two losses in the uniform case: a quantization loss \(\sinc(1/K)\) and an average attenuation loss \(\frac{1}{K}\sum_k \beta^r(\phi_k)\) [2507.07342].

When \(R<2\pi\frac{K-1}{K}\), the phase set is nonuniform over the full circle. The selection probabilities become
\[
p(\phi_k)=
\begin{cases}
\frac{A}{2\pi}, & k=1,\\
\frac{R}{2\pi(K-1)}, & 2\le k\le K-1,\\
\frac{A}{2\pi}, & k=K,
\end{cases}
\qquad
A=\pi-\frac{R}{2}+\frac{R}{2(K-1)}.
\]
For \(K>2\),
\[
E_{PDA}(R,K)
=
\frac{1}{\pi^2}
\Bigg(
\left[\sum_{k=2}^{K-1}\beta^r(\phi_k)\right]
\sin\!\left(\frac{R}{2(K-1)}\right)
+
\frac{\beta^r(\phi_1)+\beta^r(\phi_K)}{2}
\left[
\sin\!\left(\frac{R}{2(K-1)}\right)+\sin\!\left(\frac{R}{2}\right)
\right]
\Bigg)^2,
\]
and for \(K=2\),
\[
E_{PDA}(R,2)
=
\frac{\sin^2(R/2)}{\pi^2}
\left(\beta^r(\phi_1)+\beta^r(\phi_K)\right)^2.
\]

The analytical and empirical conclusions are correspondingly specific. Increasing the number of discrete phases beyond \(K=4\) yields only marginal gains when the RIS has a sufficiently wide phase range \(R\). When \(R\) is limited, performance is sensitive to attenuation for larger \(R\), and sensitive to \(R\) when there is less attenuation. Smaller \(\beta^r_{\min}\) implies stronger attenuation and larger performance loss, and EAPQ tends to outperform APQ more noticeably in the high-attenuation regime [2507.07342].

## 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 [2507.07342]. 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 [2502.12592]. 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 \(b_1\)-bit phase quantizer and an optimized single-bit magnitude quantizer [2205.05850]. 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 [2507.07342], [2502.12592], [2205.05850].

Source: https://www.emergentmind.com/topics/amplitude-introduced-polar-quantization-apq