Multi-path Conditional Power Profile (MCPP)

• MCPP is a conditional power representation that captures significant multipath components using Single-path Conditional Power values for RSRP prediction.
• It decouples physical propagation characteristics from fast fading, allowing analytical mapping to beamformed gains through closed-form expressions.
• Integrated in a Neural Beam Field framework, MCPP uses a Pretrain-and-Calibrate strategy to enhance convergence and adaptability in dense wireless networks.

Searching arXiv for the specified paper and related context. Multi-path Conditional Power Profile (MCPP) is a path-wise intermediate representation introduced in “Neural Beam Field for Spatial Beam RSRP Prediction” (Guo et al., 9 Aug 2025) for beam-level reference signal received power (RSRP) prediction in dense multi-user wireless networks. It is defined at a user equipment location $r \in \mathbb{R}^3$ as the collection of Single-path Conditional Power (SCP) values associated with the significant multipath components at that location, where each path is characterized by its direction of departure, direction of arrival, delay, and conditional power. Within the Neural Beam Field (NBF) framework, MCPP serves as the interface between site-specific propagation structure and antenna or beamforming configuration: a neural module infers MCPP from sparse measurements and positions, while a physics-inspired module maps the inferred profile to beam-level RSRP statistics through closed-form expressions (Guo et al., 9 Aug 2025).

1. Formal definition

In the formulation of NBF, a location $r$ is associated with $L$ significant multipath components indexed by $l \in \mathcal{L} \triangleq \{1,\dots,L\}$. For each path, the model specifies a unit direction of departure vector $u_{\mathrm{tx},l} \in \mathbb{R}^3$ with azimuth-elevation angles $$\Pi_{\mathrm{tx},l} \triangleq (\phi_{\mathrm{tx},l}, \theta_{\mathrm{tx},l})$$, a unit direction of arrival vector $u_{\mathrm{rx},l} \in \mathbb{R}^3$ with angles $$\Pi_{\mathrm{rx},l} \triangleq (\phi_{\mathrm{rx},l}, \theta_{\mathrm{rx},l})$$, a delay $\tau_l \ge 0$, a complex small-scale phase $\Phi_l \in [0,2\pi)$ modeled as independent and uniform, and an amplitude $r$0, treated as deterministic per location (Guo et al., 9 Aug 2025).

The Single-path Conditional Power for path $r$1 is defined as

$r$2

where $r$3 is the conditional power associated to the 7-D path condition $r$4. The Multi-path Conditional Power Profile at location $r$5 is then

$r$6

This representation is site-specific and explicitly decouples from the fast phases $r$7. The paper’s central claim is that, once MCPP is known, beam-level RSRP statistics can be obtained for a specified antenna panel and beamforming vector via closed-form analytical modeling (Guo et al., 9 Aug 2025). This suggests that MCPP is not merely a compressed channel descriptor, but a structural abstraction that isolates geometry- and environment-dependent propagation from small-scale stochastic variation.

2. Analytical structure and array-dependent mapping

The MCPP formalism is developed for a base station sector panel configured as a uniform planar array (UPA) with $r$8 elements along a local horizontal axis $r$9 and $L$0 elements along a local vertical axis $L$1, for a total of $L$2 elements. Inter-element spacings are $L$3 and $L$4, the carrier frequency is $L$5, and the element radiation pattern gain is $L$6 (Guo et al., 9 Aug 2025).

For a departure direction $L$7, the array response vector is constructed from the horizontal and vertical array response vectors and combined through a Kronecker product:

$L$8

Beamforming uses a DFT beamforming vector parameterized by spatial frequencies $L$9:

$l \in \mathcal{L} \triangleq \{1,\dots,L\}$0

For path $l \in \mathcal{L} \triangleq \{1,\dots,L\}$1, the effective scalar contribution after transmit beamforming is

$l \in \mathcal{L} \triangleq \{1,\dots,L\}$2

The normalized received power, with transmit symbol power $l \in \mathcal{L} \triangleq \{1,\dots,L\}$3 factored out, is

$l \in \mathcal{L} \triangleq \{1,\dots,L\}$4

A key analytical simplification is the separability of the array factor. Defining

$l \in \mathcal{L} \triangleq \{1,\dots,L\}$5

the beam-path coupling becomes

$l \in \mathcal{L} \triangleq \{1,\dots,L\}$6

where

$l \in \mathcal{L} \triangleq \{1,\dots,L\}$7

Hence,

$l \in \mathcal{L} \triangleq \{1,\dots,L\}$8

These expressions show that MCPP is not itself beam-specific. Rather, beam specificity enters only through the analytical map from path geometry and conditional powers to beamformed gain. A plausible implication is that the same learned MCPP can be reused across many beams in a codebook without relearning environment-dependent propagation.

3. Beam-level RSRP statistics

The analytical map from MCPP to beam-level statistics assumes independent and uniformly distributed small-scale phases $l \in \mathcal{L} \triangleq \{1,\dots,L\}$9, while treating per-path geometry and conditional power as deterministic (Guo et al., 9 Aug 2025). Under this model, the per-path average gain is defined as

$u_{\mathrm{tx},l} \in \mathbb{R}^3$0

Proposition 1 in the paper gives the mean and variance of the normalized RSRP:

$u_{\mathrm{tx},l} \in \mathbb{R}^3$1

$u_{\mathrm{tx},l} \in \mathbb{R}^3$2

These moment expressions can be evaluated in $u_{\mathrm{tx},l} \in \mathbb{R}^3$3 per beam once the per-path geometry, conditional powers, and array parameters are given (Guo et al., 9 Aug 2025).

The importance of this result is methodological as well as computational. Because the distribution is not explicitly given, the framework works directly with analytically tractable moments derived from the sum of phasors with independent uniform phases. This permits training and inference against measured RSRP statistics without requiring explicit sampling of the fast-fading process. The paper further reports that the analytical $u_{\mathrm{tx},l} \in \mathbb{R}^3$4 and $u_{\mathrm{tx},l} \in \mathbb{R}^3$5 were validated by Monte Carlo simulations under QuaDRiGa channels with $u_{\mathrm{tx},l} \in \mathbb{R}^3$6 (Guo et al., 9 Aug 2025).

A common misconception would be to regard MCPP as equivalent to a conventional channel state representation. In the NBF formulation, MCPP does not retain the instantaneous phase realization; instead, it is a conditional power profile over path geometry and delays. Its intended output is therefore beam-level statistical prediction rather than instantaneous coherent channel reconstruction.

4. Role within the decoupled blackbox–whitebox architecture

NBF adopts a decoupled “blackbox–whitebox” design in which the blackbox is a Transformer-based deep neural network $u_{\mathrm{tx},l} \in \mathbb{R}^3$7 that learns MCPP from sparse spatial measurements and positions, and the whitebox is a physics-inspired module that maps MCPP to beam RSRP statistics through the closed-form formulas above (Guo et al., 9 Aug 2025).

The blackbox takes as input a UE location $u_{\mathrm{tx},l} \in \mathbb{R}^3$8 or $u_{\mathrm{tx},l} \in \mathbb{R}^3$9, embedded using random Fourier position embedding into a 256-D token $$\Pi_{\mathrm{tx},l} \triangleq (\phi_{\mathrm{tx},l}, \theta_{\mathrm{tx},l})$$0, together with $$\Pi_{\mathrm{tx},l} \triangleq (\phi_{\mathrm{tx},l}, \theta_{\mathrm{tx},l})$$1 learnable target tokens $$\Pi_{\mathrm{tx},l} \triangleq (\phi_{\mathrm{tx},l}, \theta_{\mathrm{tx},l})$$2. These tokens are concatenated and passed through $$\Pi_{\mathrm{tx},l} \triangleq (\phi_{\mathrm{tx},l}, \theta_{\mathrm{tx},l})$$3 Transformer encoder blocks with multi-head self-attention, MLP, layer normalization, and residual connections. Each target token attends to the location token to extract location-conditioned multipath features (Guo et al., 9 Aug 2025).

For each path $$\Pi_{\mathrm{tx},l} \triangleq (\phi_{\mathrm{tx},l}, \theta_{\mathrm{tx},l})$$4, a two-layer regression head outputs 8 features corresponding to one SCP and its conditions:

• $$\Pi_{\mathrm{tx},l} \triangleq (\phi_{\mathrm{tx},l}, \theta_{\mathrm{tx},l})$$5 (3D),
• $$\Pi_{\mathrm{tx},l} \triangleq (\phi_{\mathrm{tx},l}, \theta_{\mathrm{tx},l})$$6 (3D),
• $$\Pi_{\mathrm{tx},l} \triangleq (\phi_{\mathrm{tx},l}, \theta_{\mathrm{tx},l})$$7 (1D),
• $$\Pi_{\mathrm{tx},l} \triangleq (\phi_{\mathrm{tx},l}, \theta_{\mathrm{tx},l})$$8 (1D).

A one-layer classification head outputs $$\Pi_{\mathrm{tx},l} \triangleq (\phi_{\mathrm{tx},l}, \theta_{\mathrm{tx},l})$$9 indicating whether path $u_{\mathrm{rx},l} \in \mathbb{R}^3$0 exists at that location, allowing variable numbers of significant paths to be represented (Guo et al., 9 Aug 2025).

The whitebox then computes $u_{\mathrm{rx},l} \in \mathbb{R}^3$1 and $u_{\mathrm{rx},l} \in \mathbb{R}^3$2 from $u_{\mathrm{rx},l} \in \mathbb{R}^3$3 and the panel geometry, forms $u_{\mathrm{rx},l} \in \mathbb{R}^3$4 and $u_{\mathrm{rx},l} \in \mathbb{R}^3$5, and evaluates $u_{\mathrm{rx},l} \in \mathbb{R}^3$6 and $u_{\mathrm{rx},l} \in \mathbb{R}^3$7. For one beam at one location, the complexity is $u_{\mathrm{rx},l} \in \mathbb{R}^3$8; for $u_{\mathrm{rx},l} \in \mathbb{R}^3$9 beams, it is $$\Pi_{\mathrm{rx},l} \triangleq (\phi_{\mathrm{rx},l}, \theta_{\mathrm{rx},l})$$0, which the paper identifies as scalable to dense beam codebooks (Guo et al., 9 Aug 2025).

This partition is central to the interpretation of MCPP. The learned component is responsible for inferring a latent propagation profile, whereas the deterministic component ensures consistency with antenna geometry and beam design. This suggests that MCPP functions as a physically constrained latent variable rather than an unconstrained feature embedding.

5. Pretrain-and-Calibrate strategy

The paper introduces a Pretrain-and-Calibrate (PaC) strategy to improve convergence and adaptivity (Guo et al., 9 Aug 2025). In pretraining, when ray-tracing or site-survey provides prior MCPP labels $$\Pi_{\mathrm{rx},l} \triangleq (\phi_{\mathrm{rx},l}, \theta_{\mathrm{rx},l})$$1, the network is trained to regress MCPP directly. Because path ordering is permutation-invariant, the matching between predicted $$\Pi_{\mathrm{rx},l} \triangleq (\phi_{\mathrm{rx},l}, \theta_{\mathrm{rx},l})$$2 and ground truth $$\Pi_{\mathrm{rx},l} \triangleq (\phi_{\mathrm{rx},l}, \theta_{\mathrm{rx},l})$$3 is handled with the Hungarian algorithm. Variable path count is handled by the path-existence classifier $$\Pi_{\mathrm{rx},l} \triangleq (\phi_{\mathrm{rx},l}, \theta_{\mathrm{rx},l})$$4 and a binary cross-entropy loss, while matched path parameters and powers are trained with a regression loss such as MAE or smooth L1. The overall pretraining loss is a weighted sum of MCPP regression loss and BCE loss (Guo et al., 9 Aug 2025).

The stated benefit is that pretraining positions the model near a physically plausible optimum and reduces nonconvex training difficulty. Calibration then fine-tunes $$\Pi_{\mathrm{rx},l} \triangleq (\phi_{\mathrm{rx},l}, \theta_{\mathrm{rx},l})$$5 using real RSRP measurements through an end-to-end loss on $$\Pi_{\mathrm{rx},l} \triangleq (\phi_{\mathrm{rx},l}, \theta_{\mathrm{rx},l})$$6 and optionally $$\Pi_{\mathrm{rx},l} \triangleq (\phi_{\mathrm{rx},l}, \theta_{\mathrm{rx},l})$$7 as predicted by the whitebox from the DNN output (Guo et al., 9 Aug 2025). The calibration stage is intended to adapt to unmodeled environmental factors and measurement noise.

The paper reports that PaC notably improves convergence and generalization. In the reported results, NBF trained end-to-end on RSRP achieves an MAE of $$\Pi_{\mathrm{rx},l} \triangleq (\phi_{\mathrm{rx},l}, \theta_{\mathrm{rx},l})$$8 dB with a model size of $$\Pi_{\mathrm{rx},l} \triangleq (\phi_{\mathrm{rx},l}, \theta_{\mathrm{rx},l})$$9 MB, while NBF with PaC achieves an MAE of $\tau_l \ge 0$0 dB (Guo et al., 9 Aug 2025). This performance difference is attributed in the paper to the use of MCPP-prior pretraining combined with on-site calibration, rather than to changes in the analytical mapping itself.

6. Evaluation context and empirical characteristics

The reported evaluation uses a three-sector macrocell with base station height $\tau_l \ge 0$1 m, UE height $\tau_l \ge 0$2 m, a UPA panel with $\tau_l \ge 0$3 and $\tau_l \ge 0$4 for $\tau_l \ge 0$5, half-wavelength spacing $\tau_l \ge 0$6, carrier frequency $\tau_l \ge 0$7 GHz, panel down-tilt $\tau_l \ge 0$8, peak element gain $\tau_l \ge 0$9 dBi, and ERP $\Phi_l \in [0,2\pi)$0 consistent with 3GPP TR 38.901. The map area is a $\Phi_l \in [0,2\pi)$1 grid (Guo et al., 9 Aug 2025).

Ray-traced channels via SionnaRT provide deterministic MCPP priors, while QuaDRiGa injects spatially consistent random perturbations to mimic unknown environmental factors. RSRP measurements are aggregated at grid anchors using inverse-distance weighting and temporal decay to handle sparse, noisy, time-staggered measurements. The train/validation split is $\Phi_l \in [0,2\pi)$2 (Guo et al., 9 Aug 2025).

The baselines and reported metrics are summarized below.

Method	Reported result	Additional note
NBF (end-to-end on RSRP)	MAE = 3.547 dB	Model size 8.21 MB
MLP baseline	MAE = 4.105 dB	Model size 10.57 MB
NBF with PaC	MAE = 2.416 dB	MCPP-prior pretraining + calibration
IDW baseline (with whitebox mapping)	MAE = 3.328 dB	Storage 27.86 MB

The metric is mean absolute error in dB of predicted mean RSRP versus ground truth, normalized to $\Phi_l \in [0,2\pi)$3, and the comparison also considers model size, training efficiency, and generalization (Guo et al., 9 Aug 2025). The paper states that NBF significantly outperforms conventional table-based channel knowledge maps and pure blackbox DNNs in prediction accuracy, training efficiency, and generalization, while maintaining a compact model size. It further states that NBF captures fine spatial details missed by the pure blackbox MLP.

Within this empirical setting, MCPP’s significance lies in enabling the comparison between a pure interpolation-based estimate of path descriptors, a pure end-to-end regressor, and a hybrid model in which learned path structure is analytically propagated through the antenna model. A plausible implication is that the primary empirical advantage comes from constraining the regression target to a physically meaningful object rather than directly regressing beam-level power.

7. Interpretability, scalability, limitations, and extensions

The paper explicitly identifies interpretability as a property of MCPP because it encodes per-path geometry and power through $\Phi_l \in [0,2\pi)$4 while separating fast phases. As a result, beam RSRP statistics are traceable to physical multipath and array geometry through closed-form $\Phi_l \in [0,2\pi)$5 sums (Guo et al., 9 Aug 2025). This differs from a pure blackbox RSRP regressor, where the relation between location and predicted beam power is not decomposed into interpretable propagation components.

Scalability follows from the whitebox complexity. Since the mapping is $\Phi_l \in [0,2\pi)$6 per beam and $\Phi_l \in [0,2\pi)$7 for $\Phi_l \in [0,2\pi)$8 beams, the computation scales linearly with both the number of paths and the beam count, and is parallelizable across beams and users (Guo et al., 9 Aug 2025). The representation is also described as robust to fast fading because the random phase $\Phi_l \in [0,2\pi)$9 is integrated out in the statistics.

The paper also lists several limitations. The current formulas assume a narrowband setting at a single carrier frequency $r$00, so wideband frequency selectivity and delay spread are not fully exploited. Accuracy depends on the quality of MCPP priors from ray tracing and on panel calibration, including ERP $r$01, rotation, and spacing. The formulation assumes analog DFT beamforming and ignores polarization, so hybrid or digital beamforming would require extended formulas. Measurement errors and severe nonstationarity can degrade learning if pretraining priors are poor (Guo et al., 9 Aug 2025).

Future-oriented variants are noted. A wideband extension is sketched by defining frequency-dependent array factors $r$02 for subcarriers with $r$03, and corresponding per-subcarrier moments

$r$04

The paper also mentions a conceptual temporal MCPP indexed by time $r$05 to handle slow environmental changes, with the possibility that $r$06 could incorporate temporal tokens or recurrent modules to track drifts (Guo et al., 9 Aug 2025).

Taken together, these points situate MCPP as a structured conditional representation for spatial beam prediction rather than a general-purpose channel model. Its current scope is defined by the narrowband, analog-beamforming setting, but its formulation around path geometry, delay, and conditional power makes extension to wider-bandwidth and temporally varying scenarios a natural direction.