Extended Cluster-Based Channel Models

• Extended cluster-based channel modeling is a set of propagation techniques that refine traditional cluster abstractions by preserving spatial structure, geometry, and time evolution.
• It applies methodologies like reduced clustered delay lines, semi-deterministic clustering, and object-based ray tracing to capture realistic angular profiles, phase coherence, and eigenmode behavior.
• The approach enhances advanced MIMO, positioning, and ISAC performance by addressing gaps in legacy models and enabling accurate simulation of spatial correlation and non-stationary propagation effects.

Search arXiv for papers on "extended cluster-based channel modeling MIMO 3GPP rCDL ISAC THz" Extended cluster-based channel modeling denotes a family of propagation-modeling approaches in which multipath components are still organized into clusters and rays, but the baseline cluster abstraction is extended so that the model preserves spatial propagation characteristics, embeds geometry or object information, captures cross-link or cross-function correlation, or represents explicit time evolution. In recent work, these extensions appear as reduced clustered delay lines for realistic MIMO evaluation, semi-deterministic clusters for positioning, shared and newborn clusters for integrated sensing and communication, environment-aware terahertz models, and time-variant cluster processes for UAV, railway, massive-MIMO, and distributed-MIMO channels (Berrah et al., 2 Mar 2026, Alawieh et al., 2022, Zhang et al., 2024).

1. Rationale and departure from baseline statistical models

The immediate motivation for extended cluster-based modeling is the inadequacy of legacy statistical abstractions when the evaluation target depends on spatial structure rather than on delay statistics alone. In 3GPP-oriented MIMO testing, the tapped delay line model was shown to fail to capture the spatial propagation characteristics required for realistic MIMO evaluation, and the reduced clustered delay line model was introduced in 3GPP TR 38.753 as a more accurate alternative with manageable computational complexity (Berrah et al., 2 Mar 2026). In that comparison, the central issue was not merely delay spread, but the ability to reproduce angular profiles, phase coherence, channel rank, and eigenmode behavior.

Related limitations recur in other domains. For positioning, Alawieh et al. argue that ray-tracing simulations and statistical channel models do not fully capture important aspects applicable to positioning, especially the signal properties of the first arriving path and the spatial consistency of the propagation condition of multiple links (Alawieh et al., 2022). For ISAC, existing models often consider communication channels and sensing channels independently, thereby ignoring correlation under the consistent environment; this motivates cluster taxonomies in which communication and sensing clusters are jointly generated rather than superposed after the fact (Zhang et al., 2024). At THz, 3GPP-like defaults overstate richness: measurements at 100 GHz and 132 GHz yielded markedly fewer clusters and rays per cluster than standard defaults, making channel sparsity itself a modeling target (Chang et al., 2023).

A recurring misconception is that a model is adequate once it reproduces large-scale statistics such as delay spread or path loss. The literature surveyed here contradicts that view. The mismatch usually appears in spatial discrimination, persistence of directions of arrival, cluster visibility, cluster sharing across functions or links, or non-stationary evolution along routes and arrays. This suggests that the adjective “extended” is not primarily about adding parameters; it is about preserving structures that become operationally relevant once beamforming, CSI compression, sensing, positioning, or multi-link coordination are under study.

2. Mathematical structure and cluster semantics

Most extended models retain the standard double summation over clusters and rays. In the reduced clustered delay line formulation, the impulse response between a transmit array at positions $r_{\mathrm{TX}}$ and a receive array at $r_{\mathrm{RX}}$ is written as

$$h(t,r_{\mathrm{RX}},r_{\mathrm{TX}}) = \sum_{n=1}^{N}\sum_{m=1}^{M_n} \alpha_{n,m}\,e^{-j2\pi f_c \tau_{n,m}}\,\delta(t-\tau_{n,m})\, a_{\mathrm{RX}}(\phi^{\mathrm{AoA}}_{n,m},\theta^{\mathrm{ZoA}}_{n,m})\, a_{\mathrm{TX}}^{H}(\phi^{\mathrm{AoD}}_{n,m},\theta^{\mathrm{ZoD}}_{n,m}),$$

with cluster means, ray-level amplitudes, and steering vectors carrying the spatial information that a Kronecker-correlated TDL omits (Berrah et al., 2 Mar 2026). In this sense, the extension is not a replacement of cluster-based representation but a refinement of what a cluster is allowed to encode.

Across the literature, the same formal skeleton is preserved while the meaning of a cluster changes. Alawieh et al. divide the CIR into fixed-position SDCs, specular SDCs, and random clusters, thereby merging geometry-defined and statistically drawn components in a single 3GPP-style framework (Alawieh et al., 2022). In 28 GHz ISAC modeling, communication clusters coexist with shared sensing clusters and newborn sensing clusters, with an evolution probability controlling whether a communication cluster survives in the sensing channel (Zhang et al., 2024). In the extended GBSM for 3GPP ISAC standardization, the sensing channel is decomposed into target and background components, and the target part is built by concatenating a Tx→SP link and an SP→Rx link for each scattering point on the target (Zhang et al., 14 Apr 2025). In object-based ray-tracing fusion, a cluster is defined as all MPCs that interact with the same physical object, and the cluster state becomes a 4-vector of center coordinates and power over snapshots (Zhang et al., 2023).

Extension mechanism	Cluster semantics	Representative source
rCDL	Reduced set of spatially faithful clusters and rays	(Berrah et al., 2 Mar 2026)
SDC extension	Fixed, specular, and random clusters	(Alawieh et al., 2022)
ISAC shared/newborn modeling	Communication, shared sensing, and newborn sensing clusters	(Zhang et al., 2024)
Target/background E-GBSM	Target scattering points plus background clusters	(Zhang et al., 14 Apr 2025)
Object-based RT fusion	Clusters tied to physical objects across snapshots	(Zhang et al., 2023)

This semantic broadening is one of the defining features of extended cluster-based modeling. The cluster ceases to be only a statistical group in delay–angle space and becomes, depending on the application, an interacting object, a shared scatterer, a semi-deterministic reflector, a target facet, or a visibility-limited propagation entity.

3. Cluster extraction, metric learning, and tracking

Because the model is built on clusters, clustering itself becomes a first-class modeling problem. In the Mahalanobis-distance framework for MIMO channel analysis, each MPC is represented by a 7-dimensional real vector composed of delay and unit-sphere coordinates for transmit and receive angles, and the distance between two MPCs is defined by

$d_A(x_i,x_j)=\sqrt{(x_i-x_j)^T A (x_i-x_j)},$

with $A\succeq 0$ (Chen et al., 2021). The popular multipath component distance is proved to be a special case obtained by a particular diagonal matrix $A$, so learning $A$ generalizes MCD by adapting the metric to real cluster shapes. Two supervised metric-learning strategies are then introduced: weak-supervised Mahalanobis metric for clustering and fully supervised large margin nearest neighbor. On modified 3GPP SCM data, KPowerMeans F-measure improved from approximately $0.60\to0.40$ under standard MCD as the number of clusters increased from $10$ to $30$, to approximately $r_{\mathrm{RX}}$0 with diagonal $r_{\mathrm{RX}}$1 from MMC, and to approximately $r_{\mathrm{RX}}$2 with full $r_{\mathrm{RX}}$3 from LMNN; the label-effort study reported saturation at $r_{\mathrm{RX}}$4, $r_{\mathrm{RX}}$5, $r_{\mathrm{RX}}$6, i.e. 25 labelled MPCs (Chen et al., 2021).

A different line of work emphasizes physical interpretability. Wang and Han’s geography-inspired and self-adaptive clustering algorithm interprets the power–delay–angle profile as a topographic landscape, constructs contour lines, organizes them into a rooted tree, identifies characteristic points as ridge skeletons, and then fits either single-point or wide-spread reflector models by least-squares or RANSAC (Wang et al., 29 Apr 2025). The proposed power gradient consistency index is calculated as the weighted Spearman correlation coefficient between the power and the distance to the center. On a 300 GHz outdoor street-canyon measurement, the method achieved approximately $r_{\mathrm{RX}}$7 average Silhouette index, approximately $r_{\mathrm{RX}}$8 weighted Spearman correlation coefficient, and approximately $r_{\mathrm{RX}}$9 m average RMSE of estimated wall or single-point scatterer location (Wang et al., 29 Apr 2025).

At THz, high-resolution extraction is commonly performed before cluster synthesis. In the 300 GHz monostatic sensing study, a SAGE algorithm estimates amplitude, delay, and azimuth from directional CFR measurements, after which connected component labeling is applied to a thresholded PADP, following morphological closing, to obtain delay–angle consistent clusters (Lyu et al., 2 Sep 2025). Cluster-level parameters then include power, delay centroid, angle centroid, intra-cluster spreads, delay depth, and angular width. In distributed MIMO, clustering is instead performed jointly over multiple links by interacting-object centers and partial delays, and a Kalman filter tracks cluster state vectors containing object coordinates and delay components over snapshots (Xu et al., 16 Apr 2025).

These works show that clustering is not merely a pre-processing convenience. It determines whether cluster parameters remain physically interpretable, whether common clusters are identified across links or functions, and whether subsequent statistical fitting reflects actual propagation geometry or artifacts of the distance metric.

4. Spatial correlation, visibility, and non-stationary cluster evolution

One of the clearest distinctions between conventional and extended models is the treatment of spatial correlation. In TDL, correlation is commonly imposed through a Kronecker form,

$$h(t,r_{\mathrm{RX}},r_{\mathrm{TX}}) = \sum_{n=1}^{N}\sum_{m=1}^{M_n} \alpha_{n,m}\,e^{-j2\pi f_c \tau_{n,m}}\,\delta(t-\tau_{n,m})\, a_{\mathrm{RX}}(\phi^{\mathrm{AoA}}_{n,m},\theta^{\mathrm{ZoA}}_{n,m})\, a_{\mathrm{TX}}^{H}(\phi^{\mathrm{AoD}}_{n,m},\theta^{\mathrm{ZoD}}_{n,m}),$$0

with exponential entries and no phase information. In rCDL, by contrast, correlation is inherited directly from the geometric ray model through sums of steering-vector outer products, which produce realistic angular power spectra and phase coherence (Berrah et al., 2 Mar 2026). This difference explains why TDL can produce either excessive randomness or rank deficiency, whereas rCDL can sustain multiple stable directions of arrival and realistic eigenmode separation.

Near-field and array-variant effects extend this idea further. In the 2D non-stationary wideband massive-MIMO GBSM, spherical wavefronts are approximated by second-order parabolic wavefronts, which induce linear drifts of effective angles along the array (Lopez et al., 2016). Cluster evolution is modeled by a two-state Markov process for visibility versus invisibility and a spatial lognormal process for smooth power variation. The resulting channel is wide-sense stationary in time but non-WSS in space, and quantities such as local Rician factor and spatial CCF vary over the array (Lopez et al., 2016).

Route-dependent non-stationarity appears in aerial and railway channels. For UAV air-to-ground channels, SAGE-based MPC extraction is followed by KPowerMeans clustering and a clustering-based tracking method that links clusters over time by a weighted distance in link distance, centroid delay, and centroid power (Cui et al., 2021). The survival length of cluster trajectories is modeled as $$h(t,r_{\mathrm{RX}},r_{\mathrm{TX}}) = \sum_{n=1}^{N}\sum_{m=1}^{M_n} \alpha_{n,m}\,e^{-j2\pi f_c \tau_{n,m}}\,\delta(t-\tau_{n,m})\, a_{\mathrm{RX}}(\phi^{\mathrm{AoA}}_{n,m},\theta^{\mathrm{ZoA}}_{n,m})\, a_{\mathrm{TX}}^{H}(\phi^{\mathrm{AoD}}_{n,m},\theta^{\mathrm{ZoD}}_{n,m}),$$1, and the final CIR explicitly separates centroid delays and intra-cluster delay offsets (Cui et al., 2021). In 5G-R, cluster birth and death are represented both by a Poisson birth–death description and by a four-state discrete-time Markov chain with states corresponding to no births or deaths, births only, deaths only, and both births and deaths (Zhang et al., 2024). That framework preserves the 3GPP CDL procedure for large-scale and small-scale parameter generation but adds time evolution of delays, powers, angles, and cluster count, with validation showing less than 5 % mismatch in key percentiles and correct reproduction of stationarity distance distribution (Zhang et al., 2024).

Visibility regions provide a complementary description in distributed MIMO. There, BS-side and UE-side visibility-region lengths are modeled as exponential random variables, with maximum-likelihood estimates $$h(t,r_{\mathrm{RX}},r_{\mathrm{TX}}) = \sum_{n=1}^{N}\sum_{m=1}^{M_n} \alpha_{n,m}\,e^{-j2\pi f_c \tau_{n,m}}\,\delta(t-\tau_{n,m})\, a_{\mathrm{RX}}(\phi^{\mathrm{AoA}}_{n,m},\theta^{\mathrm{ZoA}}_{n,m})\, a_{\mathrm{TX}}^{H}(\phi^{\mathrm{AoD}}_{n,m},\theta^{\mathrm{ZoD}}_{n,m}),$$2 m and $$h(t,r_{\mathrm{RX}},r_{\mathrm{TX}}) = \sum_{n=1}^{N}\sum_{m=1}^{M_n} \alpha_{n,m}\,e^{-j2\pi f_c \tau_{n,m}}\,\delta(t-\tau_{n,m})\, a_{\mathrm{RX}}(\phi^{\mathrm{AoA}}_{n,m},\theta^{\mathrm{ZoA}}_{n,m})\, a_{\mathrm{TX}}^{H}(\phi^{\mathrm{AoD}}_{n,m},\theta^{\mathrm{ZoD}}_{n,m}),$$3 m, corresponding to radii $$h(t,r_{\mathrm{RX}},r_{\mathrm{TX}}) = \sum_{n=1}^{N}\sum_{m=1}^{M_n} \alpha_{n,m}\,e^{-j2\pi f_c \tau_{n,m}}\,\delta(t-\tau_{n,m})\, a_{\mathrm{RX}}(\phi^{\mathrm{AoA}}_{n,m},\theta^{\mathrm{ZoA}}_{n,m})\, a_{\mathrm{TX}}^{H}(\phi^{\mathrm{AoD}}_{n,m},\theta^{\mathrm{ZoD}}_{n,m}),$$4 m and $$h(t,r_{\mathrm{RX}},r_{\mathrm{TX}}) = \sum_{n=1}^{N}\sum_{m=1}^{M_n} \alpha_{n,m}\,e^{-j2\pi f_c \tau_{n,m}}\,\delta(t-\tau_{n,m})\, a_{\mathrm{RX}}(\phi^{\mathrm{AoA}}_{n,m},\theta^{\mathrm{ZoA}}_{n,m})\, a_{\mathrm{TX}}^{H}(\phi^{\mathrm{AoD}}_{n,m},\theta^{\mathrm{ZoD}}_{n,m}),$$5 m (Xu et al., 16 Apr 2025). This formulation makes the common-cluster ratio across panels an explicit statistical object rather than an incidental property of independently generated links.

5. Measurement validation across domains

Extended cluster-based models are strongly measurement-driven, and the validation targets are usually application-specific rather than limited to generic CDF matching.

Domain	Measurement setting	Key validation result
3GPP-oriented MIMO	Commercial 4×4 sector, $$h(t,r_{\mathrm{RX}},r_{\mathrm{TX}}) = \sum_{n=1}^{N}\sum_{m=1}^{M_n} \alpha_{n,m}\,e^{-j2\pi f_c \tau_{n,m}}\,\delta(t-\tau_{n,m})\, a_{\mathrm{RX}}(\phi^{\mathrm{AoA}}_{n,m},\theta^{\mathrm{ZoA}}_{n,m})\, a_{\mathrm{TX}}^{H}(\phi^{\mathrm{AoD}}_{n,m},\theta^{\mathrm{ZoD}}_{n,m}),$$6 GHz, 15 MHz BW	rCDL-C reveals 3–4 distinct DoAs over ms timescales; TDL fails (Berrah et al., 2 Mar 2026)
Positioning with SDCs	FR1, 3.75 GHz, 100 MHz BW, moving blocker, six TRPs	FAP and diffraction tails reproduced within $$h(t,r_{\mathrm{RX}},r_{\mathrm{TX}}) = \sum_{n=1}^{N}\sum_{m=1}^{M_n} \alpha_{n,m}\,e^{-j2\pi f_c \tau_{n,m}}\,\delta(t-\tau_{n,m})\, a_{\mathrm{RX}}(\phi^{\mathrm{AoA}}_{n,m},\theta^{\mathrm{ZoA}}_{n,m})\, a_{\mathrm{TX}}^{H}(\phi^{\mathrm{AoD}}_{n,m},\theta^{\mathrm{ZoD}}_{n,m}),$$7 dB (Alawieh et al., 2022)
28 GHz ISAC	Outdoor measurements	90 % CDF relative errors: 2.74 %, 9.84 %, 4.68 % for $$h(t,r_{\mathrm{RX}},r_{\mathrm{TX}}) = \sum_{n=1}^{N}\sum_{m=1}^{M_n} \alpha_{n,m}\,e^{-j2\pi f_c \tau_{n,m}}\,\delta(t-\tau_{n,m})\, a_{\mathrm{RX}}(\phi^{\mathrm{AoA}}_{n,m},\theta^{\mathrm{ZoA}}_{n,m})\, a_{\mathrm{TX}}^{H}(\phi^{\mathrm{AoD}}_{n,m},\theta^{\mathrm{ZoD}}_{n,m}),$$8, $$h(t,r_{\mathrm{RX}},r_{\mathrm{TX}}) = \sum_{n=1}^{N}\sum_{m=1}^{M_n} \alpha_{n,m}\,e^{-j2\pi f_c \tau_{n,m}}\,\delta(t-\tau_{n,m})\, a_{\mathrm{RX}}(\phi^{\mathrm{AoA}}_{n,m},\theta^{\mathrm{ZoA}}_{n,m})\, a_{\mathrm{TX}}^{H}(\phi^{\mathrm{AoD}}_{n,m},\theta^{\mathrm{ZoD}}_{n,m}),$$9, $d_A(x_i,x_j)=\sqrt{(x_i-x_j)^T A (x_i-x_j)},$0 (Zhang et al., 2024)
THz GBSM	100 GHz office and 132 GHz UMi	Gini-factor maximum deviation of approximately 0.04 (Chang et al., 2023)
mmWave RT-ICM	60 GHz classroom	Max cluster AoA error 1°, mean angle spread error 9°, peak-power RMS error 2.2 dB (Yaman et al., 2019)

The MIMO evidence is especially explicit. In the 4×4 commercial deployment, post-SVD SINR histograms spanned approximately 28 dB between strongest and weakest eigenmodes, and Bartlett DoA profiles showed multiple stable directions over tens of milliseconds. rCDL-C reproduced these behaviors, whereas low- or fixed-correlation TDL did not (Berrah et al., 2 Mar 2026). For positioning, the SDC-augmented model mapped measured MPCs in the 40–120 ns range one-to-one to fixed room SDCs and reproduced blockage and diffraction effects that a random-only TR 38.901 model missed (Alawieh et al., 2022).

THz studies emphasize sparsity and environmental semantics. The 3GPP-like THz GBSM reported measured cluster numbers of 4 versus default 15 for indoor office LoS and 3 versus default 12 for UMi LoS, together with much smaller intra- and inter-cluster spreads (Chang et al., 2023). The environment-aware 300 GHz monostatic sensing framework went further by fitting specular reflection loss through Fresnel coefficients, diffuse scattering through a Lambertian law, and mappings from roughness, geometry, and material permittivity to cluster parameters, with the stated objective of extracting structural and material information from observed channel characteristics (Lyu et al., 2 Sep 2025).

What emerges from these validations is not a single universal extension but a repeated methodological pattern: the model is extended only insofar as the target domain reveals a measurable deficiency in a simpler abstraction.

6. Performance implications, standardization, and points of debate

The most direct performance consequence appears in advanced MIMO evaluation. In the 8×8 single-user case study with Type-I and eType-II codebooks, 40 MHz bandwidth, and 30 kHz SCS, TDL under ideal estimation made Type-I and eType-II perform almost identically and close to a random-PMI lower bound; rCDL-C instead yielded approximately 2 dB gain for eType-II at 70 % throughput, nearly reaching eigen-beamforming (Berrah et al., 2 Mar 2026). Under practical CSI-RS/DMRS estimation, the gap reduced to approximately 1.5 dB but remained visible, and increasing angular spread to CDL-901 values widened the gap to approximately 5 dB (Berrah et al., 2 Mar 2026). The implication is straightforward: if the model suppresses spatial richness, it also suppresses discrimination among feedback, beamforming, and multiplexing schemes.

For positioning and ISAC, the central issue is correlation and physical consistency. Alawieh et al. report that machine-learning-based ToA and fingerprinting algorithms trained on classic TR 38.901 data severely underfit OLOS and early MPCs, whereas SDC-enhanced simulated training data can support sub-meter and even centimeter-level positioning in challenging environments (Alawieh et al., 2022). In the shared-cluster ISAC model, the channel Sharing Degree increases with the number of shared clusters, and measured indoor examples yielded sensing-channel Sharing Degree values of approximately 0.65, 0.64, and 0.54 for two LOS and one NLOS case, respectively (Liu et al., 2022). The extended GBSM proposed for 3GPP ISAC standardization formalizes a related idea by separating target and background channels, introducing a power control factor $d_A(x_i,x_j)=\sqrt{(x_i-x_j)^T A (x_i-x_j)},$1, and supporting both mono-static and bi-static sensing modes; in the reported measurements, $d_A(x_i,x_j)=\sqrt{(x_i-x_j)^T A (x_i-x_j)},$2 lay in the range 0.67–0.95 and the sharing degree between target and background channels reached up to approximately 30 % (Zhang et al., 14 Apr 2025).

Another implication concerns scalability of channel generation. In the ray-tracing and deep-learning fusion model, high-resolution RT for $d_A(x_i,x_j)=\sqrt{(x_i-x_j)^T A (x_i-x_j)},$3 snapshots could take several hours per scenario, whereas the proposed fusion required only $d_A(x_i,x_j)=\sqrt{(x_i-x_j)^T A (x_i-x_j)},$4–4 coarse RT snapshots plus a single forward pass of the multi-layer learning network of approximately 5 ms on RTX 3090, for an overall acceleration of 30×–100× while preserving cluster-level and CIR fidelity (Zhang et al., 2023). This suggests that extension does not always mean higher end-to-end cost; in some workflows, a richer cluster representation enables lower simulation cost than brute-force deterministic propagation.

A final point of debate concerns complexity versus necessity. The literature does not support the blanket claim that every use case requires an extended model. Rather, it shows that extensions are necessary when the target observable depends on structures erased by simpler models: stable DoAs, eigenmode spread, shared scatterers, target/background coupling, visibility regions, or route- and array-dependent non-stationarity. Conversely, when those structures are not operationally relevant, simpler models may remain adequate. Extended cluster-based channel modeling is therefore best understood not as a single standardized model family, but as a methodological program for reconciling cluster abstractions with the physical and algorithmic requirements of contemporary 5G and emerging 6G evaluations.