Localized Statistical Channel Modeling (LSCM)

• LSCM is a class of site-specific channel modeling techniques that harnesses localized statistical methods to estimate multipath behavior tailored to a deployment area.
• It utilizes approaches like APS recovery, link-state mapping, and hybrid deterministic-statistical modeling to capture channel characteristics conditioned on local geography.
• By solving sparse inverse problems and integrating multi-modal information, LSCM enhances prediction accuracy in dynamic networks while balancing efficiency with precision.

Localized Statistical Channel Modeling (LSCM) denotes a class of site-specific channel-modeling methods that estimate geographically conditioned channel representations for a particular deployment region rather than relying only on broad scenario classes such as urban, rural, or indoor. In the APS-centric formulation that dominates recent cellular work, the localized object is the grid-level channel angular power spectrum (APS), inferred from multi-beam reference signal receiving power (RSRP) and then reused to predict behavior under changed beam or array settings (Zhang et al., 2023). Adjacent works broaden the same localized principle to probabilistic link-state maps, site-trained shadowing or loss fields, deterministic location-to-channel maps, and hybrid geometry-conditioned stochastic models (Yang et al., 2024, Wang et al., 2023, Chatelier et al., 2023). Taken together, these works support an umbrella view in which LSCM is any geographically indexed channel representation whose statistics or effective parameters are conditioned on local environment structure rather than on a single wide-area stationary law.

1. Concept, scope, and boundaries

The defining motivation of LSCM is that conventional channel models are often too coarse for offline network optimization in a specific target region. APS-based LSCM papers make this explicit: geometry-based stochastic models are designed for broad scenario classes and do not capture the local geographic structure of a specific deployment region, whereas deterministic ray tracing can be too computationally expensive and may require accurate map information that is unavailable (Zhang et al., 2023). In that setting, the goal is not merely to reproduce average path loss, but to estimate a localized statistical descriptor of multipath that remains useful when antenna or beam configurations change.

Within this literature, the most explicit and standardized LSCM object is the localized APS. For a target grid or local region, the model estimates the average power associated with discrete angle-of-departure bins and treats that APS as the environmental invariant of the local propagation structure (Zhang et al., 2023). This invariant can then be pushed through a different beam measurement matrix to predict new beam-domain received powers without recollecting measurements. The same logic underlies recent digital-twin formulations, which describe LSCM as crucial for performance evaluation during parameter tuning because the environment-specific statistical multipath structure is more stable than any single beam configuration (Qin et al., 16 Sep 2025).

The scope of LSCM in the broader literature is wider than APS recovery alone. Some works model a binary spatial random field of line-of-sight probability, calling the result a link state map within the channel knowledge map framework; the object stored at each location is then a probability $\Pr(l(\mathbf{x})=1)$ rather than an APS (Yang et al., 2024). Other works learn a site-specific loss field that explains shadowing over an area and then aggregates that field along arbitrary transmitter–receiver links (Wang et al., 2023). A different but closely related line learns deterministic local channel maps $h(\mathbf{x})$ at wavelength-scale spatial resolution; this is not a full statistical channel model, but it still constructs a geographically conditioned channel field that can serve as a deterministic layer underneath a local statistical model (Chatelier et al., 2023). This suggests that LSCM is best understood as a family of localized channel representations rather than a single fixed model class.

A common simplification is to equate LSCM with a scalar radio map. The recent literature does not support so narrow a definition. APS-centric methods target localized directional power structure (Zhang et al., 2023), link-state methods target localized visibility statistics (Yang et al., 2024), loss-field methods target localized shadowing structure (Wang et al., 2023), and hybrid geometry-based methods target localized cluster or path structure tied to physical objects (Alawieh et al., 2022, Azim et al., 28 Nov 2025). The unifying feature is geographic conditioning, not any one choice of channel descriptor.

2. Canonical APS-centric formulation

In the APS-centric cellular formulation, the base station uses a uniform rectangular array with $N_T=N_x\times N_y$ antennas, and the downlink channel is expanded on a discretized angular grid with $N_V$ vertical and $N_H$ horizontal angles (Zhang et al., 2023). The channel coefficient at element $(x,y)$ is modeled as

$$h_{x, y} (t) = \sum_{i = 1}^{N_V} \sum_{j = 1}^{N_H} \sqrt{\alpha_{i, j}(t) } \times g_{i, j} \times e^{-j2\pi\frac{d_x x}{\lambda}\cos \theta_i\sin \varphi_j} \times e^{-j2\pi\frac{d_y y }{\lambda}\sin \theta_i } \times e^{-j\omega_{i,j} (t)-j\omega_{x,y} (t)}.$$

Here $\alpha_{i,j}(t)$ is the path power in angular bin $(\theta_i,\varphi_j)$, $g_{i,j}$ is antenna gain, and the remaining factors encode array manifold and random phase terms (Zhang et al., 2023).

For the $h(\mathbf{x})$0-th transmit beam, the beam-level RSRP is

$h(\mathbf{x})$1

The key first-order relationship used by LSCM is

$h(\mathbf{x})$2

where $h(\mathbf{x})$3 is the mean angular power and $h(\mathbf{x})$4 (Zhang et al., 2023). After vectorization,

$h(\mathbf{x})$5

with $h(\mathbf{x})$6 the nonnegative APS vector and $h(\mathbf{x})$7 the expected multi-beam RSRP (Zhang et al., 2023). Localized statistical channel modeling then becomes a sparse inverse problem,

$h(\mathbf{x})$8

because only a few angular bins are expected to carry significant power (Zhang et al., 2023).

A practical difficulty is that $h(\mathbf{x})$9 is not column-normalized and can be highly nonuniform. The weighted non-negative orthogonal matching pursuit (WNOMP) method addresses this by combining a normalized correlation term with a magnitude prior. With

$N_T=N_x\times N_y$0

the support selection step becomes

$N_T=N_x\times N_y$1

followed by a nonnegative least-squares update on the active support (Zhang et al., 2023). In this formulation, LSCM is explicitly a localized sparse-recovery problem over angle.

Recent MR-driven work keeps the same APS object but elevates the formulation from sample level to grid level. For grid $N_T=N_x\times N_y$2, with sample set $N_T=N_x\times N_y$3, centroid $N_T=N_x\times N_y$4, and grid APS $N_T=N_x\times N_y$5, the objective jointly penalizes RSRP inconsistency and geographic spread,

$N_T=N_x\times N_y$6

subject to a partition constraint and sparsity/nonnegativity of $N_T=N_x\times N_y$7 (Qin et al., 16 Sep 2025). This makes explicit that localization of regions and localization of channel statistics are coupled.

3. Spatial representation, grids, and local fields

LSCM requires not only a local channel descriptor but also a spatial organization scheme. In APS-centric cellular formulations, this is usually a target grid or a collection of local regions, each assigned its own APS (Zhang et al., 2023). The grid-based viewpoint is computationally attractive, but it immediately raises a consistency problem: geographically uniform cells need not be channel-homogeneous when measurements are non-uniform and local blockage is strong.

The MR-driven framework addresses this by jointly optimizing region construction and APS estimation rather than fixing grids a priori (Qin et al., 16 Sep 2025). This is a direct response to two facts stated in the literature: first, MR data are spatially non-uniform and incomplete; second, geographical proximity alone does not guarantee channel homogeneity in complex environments (Qin et al., 16 Sep 2025). A plausible implication is that “locality” in LSCM should be interpreted as joint spatial-and-channel coherence, not merely Euclidean closeness.

Other LSCM-adjacent works use explicit spatial fields instead of APS-per-grid parameterization. The link state map for cellular-connected UAVs models the environment as a binary spatial random field

$N_T=N_x\times N_y$8

with prior

$N_T=N_x\times N_y$9

and recursively updated posterior log-odds

$N_V$0

using distance-domain and angular-domain spatial correlation models (Yang et al., 2024). Here the localized object is not multipath power over angle but LoS probability over space.

A related field-based formulation appears in site-trained shadowing prediction. CELF represents the latent environmental attenuation as a loss field $N_V$1 and uses the linear model

$N_V$2

where $N_V$3 contains link fading losses, $N_V$4 is a link-to-pixel weight matrix, and $N_V$5 has a Gaussian prior with exponentially decaying covariance (Wang et al., 2023). Inference then yields a site-specific field that can be reused for arbitrary transmitter–receiver pairs. This is localized statistical channel modeling in the sense of a latent spatial random field rather than an angle-domain inverse problem.

An earlier location-based channel database adopts an explicitly gridded map of channel gain and fills missing entries through a two-step interpolation procedure: K-nearest-neighbor coarse completion followed by a mask-aware convolutional neural network refinement (Deng et al., 2018). Although that work models scalar channel gain rather than APS, it already exhibits a basic LSCM pattern: channel behavior is indexed by location, nearby points are statistically related, and local spatial structure is learned from historical records.

4. Methodological families

One major methodological family is physics-based sparse APS recovery from RSRP. In this family, the measurement model $N_V$6 is explicit, the APS is sparse and nonnegative, and inference is framed as structured sparse recovery (Zhang et al., 2023). WNOMP belongs to this class, as does the later MR-oriented GM-NNOMP routine, which is designed to remain robust under ill-conditioned $N_V$7 and incomplete beam observations while operating at grid level (Qin et al., 16 Sep 2025). These methods are close to classical inverse problems: the environment is summarized by a low-dimensional localized APS, and physical array structure is embedded directly in the sensing matrix.

A second family is multi-modal neural radio fields and radiance-field formulations. MM-LSCM introduces a self-supervised multi-modal neural radio radiance field in which LiDAR point clouds and RSRP are fused through a dual-branch architecture and volume-rendering-based multi-modal synthesis; rays emitted from the base station are integrated to render both APS and obstacle depth, with training driven by radio consistency and LiDAR-derived depth supervision (Wang et al., 8 Aug 2025). RF-LSCM extends this idea to multi-cell, multi-grid, and multi-frequency settings by representing attenuation and radiance within a shared radiance field, adding a physics-informed frequency-dependent attenuation model, point-cloud-aided environment enhancement, and a low-rank tensor representation with Hierarchical Tensor Angular Modeling (Peng et al., 17 Sep 2025). PC-TGS pushes the same line toward APS extrapolation at unmeasured outdoor grids by representing environmental scatterers as anisotropic 3D Gaussians, projecting them into local angular domains via tangent-plane projection, and using a closed-form Gaussian-weighted average for APS bin integration (Xue et al., 17 Jun 2026). In all three cases, the defining shift is from local inverse recovery at measured positions to a scene-conditioned generator of localized APS across space.

A third family is hybrid deterministic–statistical localized modeling. One branch learns deterministic local channel maps. The location-to-channel mapping work treats the target as a function

$N_V$8

uses a physically derived planar-wave dictionary

$N_V$9

and learns sparse coefficient fields with a hypernetwork, thereby separating slowly varying coefficients from wavelength-scale oscillations (Chatelier et al., 2023). This is not a full statistical LSCM, but it is directly relevant as a geographically conditioned channel field. Another branch keeps geometry deterministic but randomizes localized object responses. The quasi-deterministic ray-tracing model for mmWave street canyons preserves deterministic scene geometry and specular paths while replacing exact bistatic RCS of irregular objects by samples from fitted object-class PDFs (Ebrahimizadeh et al., 2023). The low-THz indoor hybrid model keeps ray-traced LoS and wall-reflection clusters deterministic and adds statistically generated intra-cluster and non-ray-traced components around them (Chen et al., 2021). Semi-deterministic-cluster extensions for positioning define fixed or specular clusters through explicit $N_H$0 coordinates instead of only angle-delay tuples, making channels depend on actual environmental objects and blocker states (Alawieh et al., 2022). For ISAC, a dual-component model decomposes the channel into target and background components and injects deterministic target scattering points and deterministic clusters into a TR 38.901-compatible stochastic framework (Azim et al., 28 Nov 2025). These methods suggest that one persistent interpretation of LSCM is hybridization: dominant local geometry is modeled explicitly, residual structure statistically.

5. Empirical behavior and application domains

The empirical literature shows that localized representations materially improve prediction in settings where global models are too coarse. In APS recovery from real 5G measurements in Chengdu, the WNOMP-based framework predicts post-rotation beam RSRP with mean absolute error $N_H$1 dB over all grids for SSB beams and $N_H$2 dB for CSI-RS beams, outperforming LASSO and usually outperforming NNOMP; the same work reports that performance degrades as the angular dictionary grows and improves as the number of measured beams increases (Zhang et al., 2023). This directly supports the APS-centric claim that localized angular statistics inferred from low-dimensional beam measurements can be predictive under changed antenna settings.

The radiance-field line reports substantially lower errors when geometry is explicitly fused. MM-LSCM achieves MAE $N_H$3 dB for rotated-RSRP prediction in the explored region, $N_H$4 dB for RSRP prediction in the unexplored region, and $N_H$5 dB for rotated-RSRP prediction in the unexplored region; it remains strongest under added $N_H$6 dB noise and improves markedly over the single-modal radio-only variant (Wang et al., 8 Aug 2025). RF-LSCM reports SSB-RSRP MAE $N_H$7 dB and CSI-RSRP MAE $N_H$8 dB in a real-world multi-cell setting, improves 2.1 GHz prediction from $N_H$9 dB to $(x,y)$0 dB at $(x,y)$1 training ratio by using full 3.5 GHz augmentation, and converges in about $(x,y)$2 minutes on a single NVIDIA 4090 GPU (Peng et al., 17 Sep 2025). PC-TGS, evaluated on a LiDAR-scanned city-scale dataset with over $(x,y)$3 million LiDAR points and $(x,y)$4 RSRP samples, reports $(x,y)$5 dB on rotated RSRP in measured regions, $(x,y)$6 dB on RSRP prediction in unmeasured regions, and $(x,y)$7 dB on rotated RSRP prediction in unmeasured regions, while also reducing per-query inference time relative to neural radiance-field baselines (Xue et al., 17 Jun 2026). Across these results, the consistent message is that geometry-aware localization of channel statistics improves extrapolation to unmeasured space.

Deterministic or hybrid localized models show the same pattern in different descriptors. The model-based location-to-channel map uses about $(x,y)$8M parameters, versus $(x,y)$9M for a baseline MLP and $$h_{x, y} (t) = \sum_{i = 1}^{N_V} \sum_{j = 1}^{N_H} \sqrt{\alpha_{i, j}(t) } \times g_{i, j} \times e^{-j2\pi\frac{d_x x}{\lambda}\cos \theta_i\sin \varphi_j} \times e^{-j2\pi\frac{d_y y }{\lambda}\sin \theta_i } \times e^{-j\omega_{i,j} (t)-j\omega_{x,y} (t)}.$$0M for a nonlinear random Fourier feature model, and achieves test-grid NMSE $$h_{x, y} (t) = \sum_{i = 1}^{N_V} \sum_{j = 1}^{N_H} \sqrt{\alpha_{i, j}(t) } \times g_{i, j} \times e^{-j2\pi\frac{d_x x}{\lambda}\cos \theta_i\sin \varphi_j} \times e^{-j2\pi\frac{d_y y }{\lambda}\sin \theta_i } \times e^{-j\omega_{i,j} (t)-j\omega_{x,y} (t)}.$$1 dB on a synthetic six-path scenario and $$h_{x, y} (t) = \sum_{i = 1}^{N_V} \sum_{j = 1}^{N_H} \sqrt{\alpha_{i, j}(t) } \times g_{i, j} \times e^{-j2\pi\frac{d_x x}{\lambda}\cos \theta_i\sin \varphi_j} \times e^{-j2\pi\frac{d_y y }{\lambda}\sin \theta_i } \times e^{-j\omega_{i,j} (t)-j\omega_{x,y} (t)}.$$2 dB on ray-traced channels, far better than the tested baselines (Chatelier et al., 2023). The quasi-deterministic street-canyon ray tracer matches deterministic ray-tracing path-loss and excess-delay distributions according to the two-sample Cramér-von Mises test, with reported $$h_{x, y} (t) = \sum_{i = 1}^{N_V} \sum_{j = 1}^{N_H} \sqrt{\alpha_{i, j}(t) } \times g_{i, j} \times e^{-j2\pi\frac{d_x x}{\lambda}\cos \theta_i\sin \varphi_j} \times e^{-j2\pi\frac{d_y y }{\lambda}\sin \theta_i } \times e^{-j\omega_{i,j} (t)-j\omega_{x,y} (t)}.$$3-values $$h_{x, y} (t) = \sum_{i = 1}^{N_V} \sum_{j = 1}^{N_H} \sqrt{\alpha_{i, j}(t) } \times g_{i, j} \times e^{-j2\pi\frac{d_x x}{\lambda}\cos \theta_i\sin \varphi_j} \times e^{-j2\pi\frac{d_y y }{\lambda}\sin \theta_i } \times e^{-j\omega_{i,j} (t)-j\omega_{x,y} (t)}.$$4, $$h_{x, y} (t) = \sum_{i = 1}^{N_V} \sum_{j = 1}^{N_H} \sqrt{\alpha_{i, j}(t) } \times g_{i, j} \times e^{-j2\pi\frac{d_x x}{\lambda}\cos \theta_i\sin \varphi_j} \times e^{-j2\pi\frac{d_y y }{\lambda}\sin \theta_i } \times e^{-j\omega_{i,j} (t)-j\omega_{x,y} (t)}.$$5, $$h_{x, y} (t) = \sum_{i = 1}^{N_V} \sum_{j = 1}^{N_H} \sqrt{\alpha_{i, j}(t) } \times g_{i, j} \times e^{-j2\pi\frac{d_x x}{\lambda}\cos \theta_i\sin \varphi_j} \times e^{-j2\pi\frac{d_y y }{\lambda}\sin \theta_i } \times e^{-j\omega_{i,j} (t)-j\omega_{x,y} (t)}.$$6, and $$h_{x, y} (t) = \sum_{i = 1}^{N_V} \sum_{j = 1}^{N_H} \sqrt{\alpha_{i, j}(t) } \times g_{i, j} \times e^{-j2\pi\frac{d_x x}{\lambda}\cos \theta_i\sin \varphi_j} \times e^{-j2\pi\frac{d_y y }{\lambda}\sin \theta_i } \times e^{-j\omega_{i,j} (t)-j\omega_{x,y} (t)}.$$7 for pedestrian and parked-car path loss and excess delay at significance level $$h_{x, y} (t) = \sum_{i = 1}^{N_V} \sum_{j = 1}^{N_H} \sqrt{\alpha_{i, j}(t) } \times g_{i, j} \times e^{-j2\pi\frac{d_x x}{\lambda}\cos \theta_i\sin \varphi_j} \times e^{-j2\pi\frac{d_y y }{\lambda}\sin \theta_i } \times e^{-j\omega_{i,j} (t)-j\omega_{x,y} (t)}.$$8 (Ebrahimizadeh et al., 2023). The low-THz indoor hybrid model yields mean PDAP RMSE $$h_{x, y} (t) = \sum_{i = 1}^{N_V} \sum_{j = 1}^{N_H} \sqrt{\alpha_{i, j}(t) } \times g_{i, j} \times e^{-j2\pi\frac{d_x x}{\lambda}\cos \theta_i\sin \varphi_j} \times e^{-j2\pi\frac{d_y y }{\lambda}\sin \theta_i } \times e^{-j\omega_{i,j} (t)-j\omega_{x,y} (t)}.$$9 dB and SSIM $\alpha_{i,j}(t)$0, outperforming both a conventional statistical model and a reparameterized 3GPP GSCM baseline in preservation of local temporal–angular structure (Chen et al., 2021). These results reinforce a broader LSCM lesson: preserving local structure in the channel descriptor matters more than matching only coarse marginal statistics.

Application domains are correspondingly diverse. APS-centric formulations target offline 5G and cellular network optimization, including beam planning, antenna tilt and azimuth changes, and digital twins (Zhang et al., 2023, Qin et al., 16 Sep 2025). Link-state maps support UAV communication by learning a spatial probability field of LoS connectivity, which is especially relevant when blockage dominates (Yang et al., 2024). Loss-field models support repeated path-loss estimation for many transmitter–receiver pairs in dynamic networks without explicit site maps (Wang et al., 2023). Hybrid localized models support positioning and ISAC by preserving absolute delay alignment, target-dependent paths, and multi-link geometric consistency (Alawieh et al., 2022, Azim et al., 28 Nov 2025). This breadth indicates that LSCM is increasingly defined by locality of representation rather than by any single downstream task.

6. Limitations, misconceptions, and open directions

The first limitation, stated in multiple papers, is that many current LSCM methods do not constitute a full statistical channel model of every relevant variable. APS-centric methods recover first-order angular power statistics but not full higher-order dynamics (Zhang et al., 2023). Deterministic local channel-map learners estimate instantaneous complex coefficients $\alpha_{i,j}(t)$1, not local covariance, delay-angle power spectra, or uncertainty bounds (Chatelier et al., 2023). Link-state maps model LoS probability but not small-scale fading or delay–angle structure (Yang et al., 2024). A common misconception is therefore to treat any localized representation as a complete channel emulator; the literature supports a more modular view.

A second limitation is dependence on auxiliary information. RSRP-only APS recovery is attractive because it uses operational measurements, but it is fundamentally limited at unmeasured locations. Recent work makes this explicit by recasting the problem as APS extrapolation using dense LiDAR geometry, thereby acknowledging that sparse radio observations alone do not determine a full localized field (Xue et al., 17 Jun 2026). Conversely, multi-modal neural methods require scene geometry and coordinate alignment, which may be burdensome in practice (Wang et al., 8 Aug 2025, Peng et al., 17 Sep 2025). MR-driven formulations avoid dedicated drive tests but must first solve a location-inference problem because MR data often lack reliable positions (Qin et al., 16 Sep 2025). This suggests that future LSCM systems will continue to balance measurement convenience against side-information requirements.

A third limitation concerns spatial and temporal nonstationarity. Several adjacent works note that local consistency is not equivalent to full stationarity. The THz hybrid model states that its deterministic part presents spatial consistency in nature, but it does not provide explicit spatial consistency mechanisms such as correlated random fields or cluster birth/death across trajectories (Chen et al., 2021). The XL-MIMO mismatch analysis shows that spatial non-stationarity, spherical wave propagation, and beam squint make global stationary narrowband far-field models inadequate, implying that realistic localized models should be indexed by user position, subarray or visibility region, and frequency (Chen et al., 2022). The ISAC hybrid model similarly points to missing cluster birth/death dynamics as future work (Azim et al., 28 Nov 2025). A plausible implication is that a mature LSCM for 6G systems must be localized simultaneously over space, array aperture, and frequency.

The final open direction is joint localized inference across regions, modalities, and tasks. APS work already identifies joint estimation across multiple grids using spatial consistency as a next step (Zhang et al., 2023). MR-LSCM couples localization and APS estimation at grid level (Qin et al., 16 Sep 2025). Neural radiance-field methods incorporate LiDAR and radio jointly (Wang et al., 8 Aug 2025, Peng et al., 17 Sep 2025). Link-state mapping highlights the importance of more accurate spatial correlation models and fusion with physical sensing (Yang et al., 2024). Taken together, these works suggest that LSCM is evolving from single-site, single-descriptor estimation toward multi-modal, multi-domain localized channel modeling in which local geometry, local statistics, and local measurement processes are inferred jointly.