Diverging Polar-Domain Codebook (DPC)

• The paper introduces DPC as a hierarchical near-field beam-training framework that uses phase-only, diverging codewords with virtual focal points to efficiently localize users.
• It partitions the user half-plane into angular sectors and constructs a polar-domain codebook that replaces conventional focusing beams with diverging beams for wider coverage.
• The method achieves near-optimal beam-training accuracy in ELAA systems while dramatically reducing pilot overhead and operating under constant-envelope, single-RF-chain constraints.

Searching arXiv for the cited DPC and related near-field codebook papers. Diverging Polar-Domain Codebook (DPC) denotes a near-field beam-training codebook for extremely large-scale antenna arrays (ELAAs) that uses diverging beams, rather than conventional focusing beams, to perform angle–range localization under constant-envelope, single-RF-chain constraints. In the formulation introduced in “Near-Field Beam Training Through Beam Diverging” (Li et al., 19 Sep 2025), DPC is built from a new diverging codeword associated with a virtual focal point behind the array and organized as a hierarchical polar-domain structure parameterized by angle and range. The resulting framework supports angular-domain localization with only $2\log_2(N)$ pilots, followed by near-field refinement using a polar-domain focusing codebook, and is augmented by an angular range reduction strategy and a pilot set expansion method to improve robustness and accuracy (Li et al., 19 Sep 2025). In later work, the term has also been used more broadly or interpretively to describe polar-domain codebooks whose angular and range sampling patterns “diverge” according to geometry, user distribution, or propagation regime, but the explicit DPC construction is given in (Li et al., 19 Sep 2025).

1. Near-field beam training context

DPC arises from the beam-training problem in near-field ELAA systems, where the array aperture is sufficiently large that many users lie inside the Rayleigh distance $R_{\text{Rayleigh}} \sim \frac{2D^2}{\lambda}$, and the wavefront is spherical rather than planar (Li et al., 19 Sep 2025). In this regime, the channel depends jointly on angle and range, so beam training is naturally formulated in the polar domain.

For the uniform linear array model used in (Li et al., 19 Sep 2025), the base station has an $N$-element array with spacing $d=\lambda/2$, aperture $D=(N-1)d$, and element coordinates

$$\mathbf{p}_n = (0,\delta_n d),\quad \delta_n=\frac{2n-N-1}{2},\quad n\in\{1,\dots,N\}.$$

A single-antenna user equipment is located at $\mathbf{u}$, and the near-field steering vector for a point $\mathbf{s}$ is

$$\mathbf{b}(\mathbf{s}) \triangleq \left[ e^{-j\frac{2\pi}{\lambda}\|\overrightarrow{\mathbf{p}_1\mathbf{s}}\|},\; e^{-j\frac{2\pi}{\lambda}\|\overrightarrow{\mathbf{p}_2\mathbf{s}}\|},\; \ldots,\; e^{-j\frac{2\pi}{\lambda}\|\overrightarrow{\mathbf{p}_N\mathbf{s}}\|} \right]^T.$$

The downlink channel is modeled as Rician: $$\mathbf{h} = \sum_{l=0}^L g_l\,\mathbf{b}(\alpha_l,r_l),$$ where $R_{\text{Rayleigh}} \sim \frac{2D^2}{\lambda}$0 is the user’s angle–range pair (Li et al., 19 Sep 2025).

The conventional near-field beam-training baseline is the polar-domain codebook, which uses focusing beams

$R_{\text{Rayleigh}} \sim \frac{2D^2}{\lambda}$1

sampled over discrete angle–range pairs. Exhaustive training transmits all such beams and selects

$R_{\text{Rayleigh}} \sim \frac{2D^2}{\lambda}$2

According to (Li et al., 19 Sep 2025), this approach suffers from high pilot overhead, extremely narrow spatial coverage, high sensitivity to angle/range misalignment, and hardware difficulty in generating hierarchical wide beams under constant-envelope constraints. These limitations motivate the DPC paradigm.

2. Diverging-codeword principle

The defining concept of DPC is the diverging codeword, which reverses the usual focusing viewpoint. Instead of concentrating energy at a physical focal point in front of the array, the design uses a virtual focal point $R_{\text{Rayleigh}} \sim \frac{2D^2}{\lambda}$3 behind the array with $R_{\text{Rayleigh}} \sim \frac{2D^2}{\lambda}$4, creating a beam that diverges into a sector in front of the aperture (Li et al., 19 Sep 2025).

For a virtual point $R_{\text{Rayleigh}} \sim \frac{2D^2}{\lambda}$5 with polar coordinates $R_{\text{Rayleigh}} \sim \frac{2D^2}{\lambda}$6, the focusing codeword toward that point is

$R_{\text{Rayleigh}} \sim \frac{2D^2}{\lambda}$7

The diverging codeword is then defined as

$R_{\text{Rayleigh}} \sim \frac{2D^2}{\lambda}$8

Equivalently,

$R_{\text{Rayleigh}} \sim \frac{2D^2}{\lambda}$9

All entries have constant modulus, so the codeword is phase-only and compatible with a single RF chain (Li et al., 19 Sep 2025).

When $N$0 is transmitted, the received amplitude at location $N$1 is

$N$2

The paper identifies a beam diverging effect: if the user lies in a sector $N$3 determined by the rays from $N$4 through the edge antennas, the received power is significantly higher than for users outside that sector (Li et al., 19 Sep 2025). Formally,

$N$5

where the line $N$6 intersects the array segment at $N$7.

A corresponding twin effect is also reported: for fixed $N$8, the received power is much larger when the virtual point $N$9 lies in the sector defined by rays from $d=\lambda/2$0 to the edge antennas than when $d=\lambda/2$1 lies outside (Li et al., 19 Sep 2025). This makes sweeping over virtual focal points a mechanism for angular localization.

Theoretical support is provided through geometric expansion and a Fresnel-integral approximation of the received response. Under the approximations in (Li et al., 19 Sep 2025), the response is nearly flat and high inside the intended sector and near zero outside, confirming the sector-forming behavior of the diverging beam.

3. Codeword construction and hierarchical codebook structure

DPC is a hierarchical collection of diverging codewords indexed by angular sectors. The construction begins by partitioning the user half-plane $d=\lambda/2$2 into $d=\lambda/2$3 origin-centered angular sectors

$d=\lambda/2$4

with angles sampled according to

$d=\lambda/2$5

The corresponding origin-centered angular regions are

$d=\lambda/2$6

For each sector, a virtual focal point $d=\lambda/2$7 is chosen so that rays from $d=\lambda/2$8 through the two edge antennas have angles $d=\lambda/2$9 and $D=(N-1)d$0. The formula given in (Li et al., 19 Sep 2025) is

$D=(N-1)d$1

By construction, the diverging beam associated with $D=(N-1)d$2 covers the intended angular sector in front of the array (Li et al., 19 Sep 2025).

The level-$D=(N-1)d$3 DPC is then

$D=(N-1)d$4

This yields a multi-tier codebook in which low levels provide few wide beams and high levels provide many narrower beams. The hierarchy is nested: parent sectors equal the union of child sectors, enabling binary refinement (Li et al., 19 Sep 2025).

A central quantitative notion in the design is the diverging degree $D=(N-1)d$5, defined as a measure of how well the codeword separates its intended region $D=(N-1)d$6 from the rest of a bounding near-field region. The observations reported in (Li et al., 19 Sep 2025) are that $D=(N-1)d$7 increases with array size $D=(N-1)d$8, is independent of carrier frequency for fixed geometry, and is high when the virtual focal point is not too close to the array, with the design guideline $D=(N-1)d$9.

4. Beam-training algorithm and pilot complexity

The DPC beam-training procedure combines hierarchical angular localization by diverging beams with near-field refinement by focusing polar-domain beams. Let $$\mathbf{p}_n = (0,\delta_n d),\quad \delta_n=\frac{2n-N-1}{2},\quad n\in\{1,\dots,N\}.$$0. Stage 1 performs binary search over DPC levels:

1. Initialize $$\mathbf{p}_n = (0,\delta_n d),\quad \delta_n=\frac{2n-N-1}{2},\quad n\in\{1,\dots,N\}.$$1.
2. At level $$\mathbf{p}_n = (0,\delta_n d),\quad \delta_n=\frac{2n-N-1}{2},\quad n\in\{1,\dots,N\}.$$2, generate the two child codewords

$$\mathbf{p}_n = (0,\delta_n d),\quad \delta_n=\frac{2n-N-1}{2},\quad n\in\{1,\dots,N\}.$$3

1. Transmit both as pilots; the user compares the received amplitudes.
2. The index of the stronger branch becomes $$\mathbf{p}_n = (0,\delta_n d),\quad \delta_n=\frac{2n-N-1}{2},\quad n\in\{1,\dots,N\}.$$4.
3. Repeat for $$\mathbf{p}_n = (0,\delta_n d),\quad \delta_n=\frac{2n-N-1}{2},\quad n\in\{1,\dots,N\}.$$5.

After $$\mathbf{p}_n = (0,\delta_n d),\quad \delta_n=\frac{2n-N-1}{2},\quad n\in\{1,\dots,N\}.$$6 iterations, the user is localized to one of $$\mathbf{p}_n = (0,\delta_n d),\quad \delta_n=\frac{2n-N-1}{2},\quad n\in\{1,\dots,N\}.$$7 angular sectors. Because each level uses 2 pilots, the total pilot count for the angular stage is

$$\mathbf{p}_n = (0,\delta_n d),\quad \delta_n=\frac{2n-N-1}{2},\quad n\in\{1,\dots,N\}.$$8

(Li et al., 19 Sep 2025).

Stage 2 restricts attention to the identified angular region and performs near-field refinement using focusing polar-domain beams whose angle–range samples lie in that region. The user selects the codeword with maximum received power, obtaining fine angle–range alignment (Li et al., 19 Sep 2025).

The paper reports that, for $$\mathbf{p}_n = (0,\delta_n d),\quad \delta_n=\frac{2n-N-1}{2},\quad n\in\{1,\dots,N\}.$$9, coarse localization uses $\mathbf{u}$0 pilots and the total pilot count is in the order of 70–80, compared with $\mathbf{u}$1 for exhaustive polar-domain search when $\mathbf{u}$2 rings (Li et al., 19 Sep 2025). The same section also states that the method requires only one RF chain and obeys constant-envelope constraints, whereas the cited far-field hierarchical baseline also uses $\mathbf{u}$3 pilots but needs $\mathbf{u}$4 RF chains and amplitude tapering in practice (Li et al., 19 Sep 2025).

A closely related but distinct near-field training strategy is given in “Near-field Beam Training with Sparse DFT Codebook” (Zhou et al., 2024), which uses sparse DFT sweeping for angle estimation and a conventional polar-domain codebook for range refinement. That work reports a three-phase overhead

$\mathbf{u}$5

with $\mathbf{u}$6 scaling and a 98.67% reduction relative to exhaustive search in the reported setting (Zhou et al., 2024). This provides context for DPC: both seek to reduce 2D near-field search overhead, but DPC does so by replacing coarse focusing or DFT-style angle search with hierarchical diverging beams (Li et al., 19 Sep 2025).

5. Enhancement techniques

Two refinements are integral to the DPC method in (Li et al., 19 Sep 2025): DPC angular range reduction and pilot set expansion.

The first addresses a pathology at extreme angles. When $\mathbf{u}$7 approaches $\mathbf{u}$8, the virtual focal points produced by the basic formula can lie on or near the array, yielding low diverging degree $\mathbf{u}$9. To avoid this, the angular range is reduced to $\mathbf{s}$0, motivated by the near-field boundary approximation

$\mathbf{s}$1

and the coverage metric

$\mathbf{s}$2

The paper reports $\mathbf{s}$3, supporting the claim that most near-field users lie within that angular interval (Li et al., 19 Sep 2025).

The clamped angle samples are

$\mathbf{s}$4

and the virtual points are recomputed as

$\mathbf{s}$5

These modified virtual points all lie in a region where simulations show $\mathbf{s}$6, so all DPC beams exhibit strong diverging contrast (Li et al., 19 Sep 2025).

The second refinement, pilot set expansion, addresses limited angular resolution in the refinement stage. The concern is that the standard polar-domain codebook uses only $\mathbf{s}$7 angular samples, so sweeping only the beams inside the selected sector may miss the truly optimal focusing beam. Observation 4 in (Li et al., 19 Sep 2025) characterizes how the optimal focusing angle can deviate from the sector label as range changes, using the polar rings

$\mathbf{s}$8

The refinement set is therefore expanded ring-by-ring: farther ranges use a narrower angular interval around the detected sector, while nearer rings use broader intervals. This is reported to maintain reasonable pilot overhead while markedly improving accuracy (Li et al., 19 Sep 2025).

6. Performance characteristics and comparisons

The numerical results reported in (Li et al., 19 Sep 2025) compare the DPC-based hierarchical method against exhaustive polar-domain search, far-field hierarchical methods, a near-field hierarchical focusing-codeword method, a far-field sweeping method with near-field refinement, and a spatial-chirp hierarchical method. The simulation setting includes $\mathbf{s}$9, $$\mathbf{b}(\mathbf{s}) \triangleq \left[ e^{-j\frac{2\pi}{\lambda}\|\overrightarrow{\mathbf{p}_1\mathbf{s}}\|},\; e^{-j\frac{2\pi}{\lambda}\|\overrightarrow{\mathbf{p}_2\mathbf{s}}\|},\; \ldots,\; e^{-j\frac{2\pi}{\lambda}\|\overrightarrow{\mathbf{p}_N\mathbf{s}}\|} \right]^T.$$0 GHz, Rician factor 13 dB, and $$\mathbf{b}(\mathbf{s}) \triangleq \left[ e^{-j\frac{2\pi}{\lambda}\|\overrightarrow{\mathbf{p}_1\mathbf{s}}\|},\; e^{-j\frac{2\pi}{\lambda}\|\overrightarrow{\mathbf{p}_2\mathbf{s}}\|},\; \ldots,\; e^{-j\frac{2\pi}{\lambda}\|\overrightarrow{\mathbf{p}_N\mathbf{s}}\|} \right]^T.$$1 rings.

The principal findings are fourfold. First, the method offers low pilot overhead together with single-RF-chain, constant-envelope implementation. Second, the beam patterns of DPC concentrate energy within intended angular sectors with low side-lobes, in contrast to the strong side-lobes or overlapping angular patterns reported for some comparison methods. Third, the accuracy is described as virtually identical to exhaustive polar-domain search across user ranges, angles within the near-field region, SNR from approximately 10–40 dB, and antenna counts $$\mathbf{b}(\mathbf{s}) \triangleq \left[ e^{-j\frac{2\pi}{\lambda}\|\overrightarrow{\mathbf{p}_1\mathbf{s}}\|},\; e^{-j\frac{2\pi}{\lambda}\|\overrightarrow{\mathbf{p}_2\mathbf{s}}\|},\; \ldots,\; e^{-j\frac{2\pi}{\lambda}\|\overrightarrow{\mathbf{p}_N\mathbf{s}}\|} \right]^T.$$2, with only slight degradation at low $$\mathbf{b}(\mathbf{s}) \triangleq \left[ e^{-j\frac{2\pi}{\lambda}\|\overrightarrow{\mathbf{p}_1\mathbf{s}}\|},\; e^{-j\frac{2\pi}{\lambda}\|\overrightarrow{\mathbf{p}_2\mathbf{s}}\|},\; \ldots,\; e^{-j\frac{2\pi}{\lambda}\|\overrightarrow{\mathbf{p}_N\mathbf{s}}\|} \right]^T.$$3 due to weaker divergence. Fourth, the DPC method is reported to outperform the comparison methods in robustness, especially under low SNR or near-field misalignment-sensitive conditions (Li et al., 19 Sep 2025).

The paper’s summary phrase is that the method achieves near-optimal beam-training accuracy with dramatically reduced pilot overhead, while remaining phase-only and single-RF-chain (Li et al., 19 Sep 2025). Because these claims are tied to the paper’s simulations and specific baselines, they should be read as properties of the evaluated setting rather than universal guarantees.

A broader codebook-design perspective appears in “Codebook Design for Limited Feedback in Near-Field XL-MIMO Systems” (Yao et al., 15 Jan 2026). That paper does not explicitly define DPC in the sense of (Li et al., 19 Sep 2025), but it studies polar-domain codebook design under limited feedback and user distribution, showing that uniform angle sampling is optimal for received-power maximization under uniform angle distribution and that geometric range sampling is a high-quality suboptimal solution for range (Yao et al., 15 Jan 2026). This suggests a different sense in which a codebook may be “diverging”: not via virtual focal points behind the array, but via range samples that spread geometrically and via adaptive allocation of resolution between angle and range. The paper further reports that as array size increases, the optimal allocation increasingly favors range bits over angle bits (Yao et al., 15 Jan 2026).

7. Related interpretations and extensions

The explicit DPC construction belongs to planar ELAA near-field beam training (Li et al., 19 Sep 2025), but several related papers provide frameworks that can be interpreted as adjacent or extended forms of diverging polar-domain design.

In (Yao et al., 15 Jan 2026), the authors discuss how user-distribution-aware polar-domain codebooks can be designed by solving an expected beamforming-gain maximization problem. Their results include the optimal uniform angle samples

$$\mathbf{b}(\mathbf{s}) \triangleq \left[ e^{-j\frac{2\pi}{\lambda}\|\overrightarrow{\mathbf{p}_1\mathbf{s}}\|},\; e^{-j\frac{2\pi}{\lambda}\|\overrightarrow{\mathbf{p}_2\mathbf{s}}\|},\; \ldots,\; e^{-j\frac{2\pi}{\lambda}\|\overrightarrow{\mathbf{p}_N\mathbf{s}}\|} \right]^T.$$4

and the geometric range samples

$$\mathbf{b}(\mathbf{s}) \triangleq \left[ e^{-j\frac{2\pi}{\lambda}\|\overrightarrow{\mathbf{p}_1\mathbf{s}}\|},\; e^{-j\frac{2\pi}{\lambda}\|\overrightarrow{\mathbf{p}_2\mathbf{s}}\|},\; \ldots,\; e^{-j\frac{2\pi}{\lambda}\|\overrightarrow{\mathbf{p}_N\mathbf{s}}\|} \right]^T.$$5

This line of work does not use diverging codewords, but it interprets “divergence” as controlled spreading of polar samples according to user distribution and near-field sensitivity (Yao et al., 15 Jan 2026). A plausible implication is that DPC-style beam training and distribution-aware sampling could be combined in future designs.

In “A Unified Codebook Design for Curvature-Reconfigurable Apertures: Seamless Near to Far Field Coverage” (You et al., 28 Mar 2026), the proposed hierarchical codebook for curvature-reconfigurable apertures is described in the supplied material as “almost exactly what you would call a DPC,” because it uses a polar angular domain, hierarchical structure, and ERD-guided reciprocal-range sampling that transitions smoothly from dense near-field focusing to sparse far-field steering (You et al., 28 Mar 2026). The paper defines a direction-dependent effective Rayleigh distance

$$\mathbf{b}(\mathbf{s}) \triangleq \left[ e^{-j\frac{2\pi}{\lambda}\|\overrightarrow{\mathbf{p}_1\mathbf{s}}\|},\; e^{-j\frac{2\pi}{\lambda}\|\overrightarrow{\mathbf{p}_2\mathbf{s}}\|},\; \ldots,\; e^{-j\frac{2\pi}{\lambda}\|\overrightarrow{\mathbf{p}_N\mathbf{s}}\|} \right]^T.$$6

for 1-D curvature-reconfigurable apertures, and samples uniformly in reciprocal range $$\mathbf{b}(\mathbf{s}) \triangleq \left[ e^{-j\frac{2\pi}{\lambda}\|\overrightarrow{\mathbf{p}_1\mathbf{s}}\|},\; e^{-j\frac{2\pi}{\lambda}\|\overrightarrow{\mathbf{p}_2\mathbf{s}}\|},\; \ldots,\; e^{-j\frac{2\pi}{\lambda}\|\overrightarrow{\mathbf{p}_N\mathbf{s}}\|} \right]^T.$$7. In that framework, the “diverging” property refers to the way sampling density thins out with distance and the beam family degenerates from 3-D focal beams to angle-only steering beyond the ERD (You et al., 28 Mar 2026). This is conceptually related to DPC, but it is not the same construction as the diverging-codeword hierarchy of (Li et al., 19 Sep 2025).

A different adjacent framework is (Zhou et al., 2024), which uses sparse DFT sweeping plus conventional polar-domain refinement. The supplied details explicitly note that DPC is not named there, but that the paper provides a complete polar-domain beam-training framework from which “diverging” polar beams could be generalized. This suggests that DPC can be situated within a larger family of hierarchical near-field codebooks that reduce 2D beam-search complexity by separating coarse and fine localization stages (Zhou et al., 2024).

Overall, the strict definition of DPC is the diverging-beam hierarchical codebook introduced for near-field ELAA beam training in (Li et al., 19 Sep 2025). Related literature extends the idea in two directions: first, toward distribution-aware polar sampling for limited feedback (Yao et al., 15 Jan 2026); second, toward distance-adaptive unified codebooks that transition continuously between near and far field under geometry-dependent correlation rules (You et al., 28 Mar 2026). These connections suggest that DPC is both a specific codebook architecture and a point of reference within a broader research program on near-field polar-domain beam design.