Bayesian Predictive Beamforming for Vehicular Networks: A Low-overhead Joint Radar-Communication Approach

• DFRC jointly performs radar sensing and communication with one transmission, reducing hardware cost and increasing overall system throughput.Source: §I · p. 1
• High-mobility V2X scenarios require fast beam tracking, but feedback-based alignment sends pilots and returns estimated relative angles to the roadside unit.Source: §I · p. 2
• The proposed V2I scheme estimates vehicle beam directions and motion parameters from radar echoes, avoiding vehicle feedback and using the whole downlink block as data symbols and sensing pilots.Source: §I · p. 2
• The scheme predicts vehicle angles for beamformer design, reducing latency while achieving better estimation performance and higher spectral efficiency than conventional feedback-based schemes.Source: §I · p. 2
• A factor-graph framework and message passing algorithm track and predict vehicle motion parameters, with closed-form messages derived through mean-field message passing and Taylor expansion.Source: §I · pp. 2–3
• The system model uses a roadside unit with separate transmit and receive arrays to receive vehicle echoes while maintaining uninterrupted downlink transmission.Source: §II · p. 3

• Radar matched filtering estimates delay and Doppler, which correspond to vehicle range and speed.The measurement model assumes Gaussian observation noise.Source: B. Observation Model · p. 4
• Single-time-slot observations make vehicle-speed estimation challenging because the Doppler phase shift is not significant.The paper suggests accumulating Doppler shifts over multiple time slots.Source: B. Observation Model · p. 4
• The received signal samples across receive antennas support estimation of the vehicle angle and reflection coefficient after delay and Doppler filtering.The model includes receive-antenna noise samples and a transmit beamformer based on the predicted angle.Source: B. Observation Model · p. 4

• Vehicle range, angle, speed, and reflection coefficient evolve according to kinematic equations based on consecutive states.The transition noises are modeled as zero-mean Gaussian variables.Source: C. State Evolution Model · p. 5
• For short intervals with small positional variation, the angular transition is approximated using sin(∆θk,n) ≈ ∆θk,n.This approximation converts the nonlinear geometric relation into a tractable expression.Source: C. State Evolution Model · p. 5
• The range transition is simplified by treating consecutive ranges as approximately equal and omitting a negligible quadratic variation term.Source: C. State Evolution Model · p. 5
• The model further uses the reflection-coefficient model and assumes the vehicle speed is approximately constant.Source: C. State Evolution Model · p. 5
• Estimated previous states replace coupled variables so each motion-parameter evolution depends only on its own previous state.Source: C. State Evolution Model · p. 5

• Vehicle k receives the RSU transmission through a receive beamformer, with the received signal determined by the vehicle channel and steering vector.Source: D. Communication Model · pp. 5–6
• The receive beamformer is designed from the vehicle’s predicted angle relative to the RSU at time n.Source: D. Communication Model · p. 5
• The RSU’s beamforming update at time n uses an estimate from time n−2 conveyed in the DFRC signal at time n−1.Source: D. Communication Model · p. 5
• Achievable sum-rate depends on the transmit and receive beamformers, and perfect angle prediction yields maximum received SNR and maximum achievable sum-rate.Source: D. Communication Model · pp. 5–6
• The channel pathloss coefficient can be estimated from the vehicle range.Source: D. Communication Model · pp. 5–6

• The goal is to estimate vehicle range, speed, angle, and path loss using the state-evolution and observation models through MAP estimation.Source: §III · p. 6
• Direct marginalization of the joint posterior has prohibitively high complexity because it requires multidimensional integrations.Source: §III · p. 6
• The joint posterior is factorized into likelihood and prior terms using Bayes’ theorem and conditional independence.Source: §III · p. 6
• Initial vehicle parameters are obtained with an omnidirectional probing waveform and modeled with Gaussian initial distributions.Source: §III · p. 6
• An auxiliary variable is introduced because each received-sample factor contains Nt components, making inference difficult.Source: §III · pp. 6–7
• The factor graph separates state-evolution and observation-model factors, allowing message passing to compute approximate marginals, or beliefs, of unknown variables.Source: §III · p. 7, Figure 3 · p. 7

• Factor-to-variable messages are denoted by −→ µ f(x), while variable-to-factor messages are denoted by ←−ν f(x).Source: §IV · p. 7
• Belief propagation (BP) passes messages in both directions between factor vertices and variable vertices.The paper distinguishes factor-to-variable messages from variable-to-factor messages.Source: A. Conventional Message Passing… · p. 7, §IV · p. 7
• The message-update rules use the variables in a function and the factor vertices connected to a variable.Source: A. Conventional Message Passing… · p. 7
• After receiving messages from connected factors, the belief of a variable is computed as its approximate marginal.Source: A. Conventional Message Passing… · p. 7

• The RSU predicts angle θn from motion-parameter estimates at time n−1 using the state-evolution model.Source: B. Vehicle State Prediction · p. 7
• The predicted angle is sent to vehicles in the DFRC signal for receive-beam steering at the next time instant.Source: B. Vehicle State Prediction · pp. 7–8
• Accurate prediction aligns RSU transmit beams with vehicle receive beams, improving communication performance.Source: B. Vehicle State Prediction · p. 7
• Gaussian state-evolution messages provide prediction means and variances based on the previous marginal distribution.For vehicle speed, the predicted mean equals the previous mean and the variance adds process uncertainty.Source: B. Vehicle State Prediction · pp. 7–8
• The RSU further predicts the relative angle at time n+1 from the estimated angle and predicted motion parameters.Source: B. Vehicle State Prediction · p. 8

• Vehicles decode data and receive next-epoch angle predictions, while the RSU refines current motion estimates from vehicle echoes.Source: C. Vehicle State Tracking · p. 8
• The causal model lets messages depend only on past states and propagate forward along time.Source: 1) Messages related to dn · p. 8
• Range beliefs are Gaussian, so the range estimate is the belief mean and feeds the next state-evolution model.Source: 1) Messages related to dn · p. 8
• The nonlinear cosine relation is reconstructed with an auxiliary cosine variable because direct message computation lacks a closed-form Gaussian representation.Source: 2) Messages related to γn · p. 8
• Trigonometric nonlinearities are handled through Gaussian moment calculations and second-order Taylor approximations of inverse functions.The inverse-cosine approximation is arccos ϑ ≈π/2 −ϑ −ϑ^3/6.Source: 2) Messages related to γn · p. 9
• Observation messages update angular and motion-variable beliefs, including contributions from the receive antennas.Source: 2) Messages related to γn · p. 9
• The cyclic κq subgraph requires several message-passing iterations, with inverse trigonometric transformations reconstructed through auxiliary processing.Source: 4) Messages related to κq · pp. 9–10
• The resulting angle belief supplies the next prediction and the state-transition model, while estimated angles support reliable data links.Source: 4) Messages related to κq · p. 10

• The vehicle and RSU use predictions from different time instants, with Gaussian uncertainty associated with the vehicle’s next-epoch steering direction.Source: D. Beam Misalignment Analysis · p. 10
• Beam alignment requires the vehicle’s predicted direction to remain within δ of the actual angle, where δ = π/Nantenna.The beamwidth is defined as ∆ = 2δ.Source: D. Beam Misalignment Analysis · p. 10
• The misalignment probability pmis is obtained from cumulative distributions of the predicted-angle variables.Source: D. Beam Misalignment Analysis · pp. 10–11
• Misalignment probability depends on beamwidth, motivating future beamwidth optimization to minimize pmis.Source: D. Beam Misalignment Analysis · p. 11

• O(4K) complexity results from closed-form message calculations using appropriate approximations and message-updating rules.The calculations require only addition and multiplication operations.Source: E. Computational Complexity and… · p. 11
• The particle-filter-based algorithm has O(4KR) complexity because it represents nonlinear functions with R particles.A sufficiently large R is generally needed for reasonably good estimation performance.Source: E. Computational Complexity and… · p. 11
• The extended Kalman filter has O((4K)3) complexity because it requires matrix inversion.Source: E. Computational Complexity and… · p. 11
• The proposed approach provides zero signaling overhead for beam tracking, unlike feedback-based approaches that usually use one or two pilot symbols.Source: E. Computational Complexity and… · p. 11

Tracking performance

• The proposed approach tracks range and angle parameters across four antenna deployments and four vehicles in the simulation setup.The setup uses vehicles with randomly generated initial speeds and evaluates averaged tracking results.Source: §V · p. 11
• 10−2 rad is the reported error level for angle estimation, with larger antenna arrays providing higher array gain and more observations.Source: §V · pp. 11–12

Comparison with benchmarks

• The proposed parametric message passing method attains the performance of the PF-based method with significantly reduced complexity.The comparison uses the velocity-estimation error at the last time instant with M = Nt = Nr = 64 antennas.Source: §V · p. 12
• The proposed approach significantly outperforms feedback-based schemes using 1 and 2 pilots because of its higher SNR gain.Source: §V · p. 12

Communication performance

• The feedback scheme degrades with more antennas because one pilot cannot sufficiently track angular variation in narrower beams.The degradation is more significant with 128 antennas.Source: §V · pp. 12–13
• The proposed scheme achieves rates higher than 4 bps/Hz at SNR 10 dB, whereas the feedback-based scheme achieves much lower rates.The rate degradation is attributed to angle-estimation error causing beam misalignment and reduced received SNR.Source: Figure 11 · p. 12, §V · p. 13

Beam misalignment

• Low-speed vehicles have lower misalignment probability than high-speed vehicles, while high-speed vehicles are more sensitive to beamwidth.The high-speed case shows a significant rise in misalignment probability at δ = π/128.Source: §V · p. 13

• The scheme estimates vehicle motion parameters from RSU echo signals using a factor graph representing a factorized joint a posteriori distribution.Source: §VI · p. 13
• Second-order Taylor expansion produces closed-form factor-graph messages and a low-complexity beam-tracking solution.Source: §VI · p. 13
• A two-step angle prediction is transmitted through DFRC signals to reduce beam-alignment latency.Source: §VI · p. 13
• The proposed approach provides zero signaling overhead for beam tracking and outperforms the conventional feedback-based scheme in simulations.Source: §VI · p. 13