Sparse Target Parameter and Channel Estimation in MIMO-OOFDM-Aided Visible Light Integrated Sensing and Communication Systems

I. INTRODUCTION

• Practical scenes are modeled as sparse because only a few dominant targets and propagation paths occupy the larger delay-Doppler search space; off-grid estimation addresses mismatch when parameters fall between predefined grid points.DCO-OFDM uses Hermitian symmetry and a DC bias to meet the real-valued, non-negative IM/DD waveform constraint.Source: §I · p. 1, A. MIMO DCO-OFDM VL-ISAC system model · pp. 3–4

A. Literature review

• Existing VL-ISAC work largely emphasizes waveform or resource allocation, while a unified MIMO DCO-OFDM framework incorporating multipath, target reflections, Doppler, clutter, channel estimation, and sensing parameter estimation remains largely unexplored.The introduction motivates this gap by noting optical IM/DD constraints and the limited attention to signal-level estimation in MIMO DCO-OFDM VL-ISAC.Source: A. Literature review · pp. 2–3

B. Contributions

• The proposed framework formulates joint delay-Doppler sparse sensing models and off-grid Bayesian learning methods, alongside sparse channel estimation, range and velocity recovery, and closed-form BCRB benchmarks.1D-OffGBL uses a joint DD dictionary; 2D-OffGBL separates Doppler and delay estimation to reduce complexity.Source: B. Contributions · p. 3

II. RADAR AND COMMUNICATION VL SYSTEM MODELS

• The monostatic system uses a shared DCO-OFDM waveform for sensing and downlink communication, with the model accounting for target reflections, clutter, Doppler, and the communication channel.The system model specifies multiple LEDs and sensing photodetectors, target reflectors, and dominant communication scatterers.Source: B. Contributions · p. 3, A. MIMO DCO-OFDM VL-ISAC system model · pp. 3–4

III. 1D OFF-GRID SPARSE MODELING AND BL-BASED PE

• A first-order Taylor approximation represents continuous delay and Doppler values as deviations from nearby virtual-grid points, reducing basis mismatch in the 1D sparse sensing model.The model combines the on-grid dictionary with delay- and Doppler-derivative dictionaries weighted by the respective deviations.Source: B. 1D off-grid sparse representation… · p. 5
• The 1D-OffGBL procedure iteratively updates sparsity-controlling hyperparameters and off-grid delay-Doppler deviations to improve the estimated RCS matrix.The BL model assigns Gaussian priors to coefficient rows and uses hyperparameters to promote row sparsity.Source: C. 1D off-grid BL (1D-OffGBL)-based… · p. 5

H MIMOyr. (25)

• After convergence, rows associated with large precision values can be pruned using a chosen threshold to refine the RCS estimate.The coefficient rows encode candidate components in the sparse representation.Source: H MIMOyr. (25) · p. 6
• The high-dimensional joint dictionary makes 1D-OffGBL computationally costly, motivating sequential 2D-OffGBL as a complexity-saving performance-complexity trade-off.The 2D formulation processes Doppler and delay in separate stages.Source: H MIMOyr. (25) · p. 6

IV. 2D OFF-GRID SPARSE MODELING AND BL-BASED PE

• Separable Doppler and delay steering matrices form a vectorized MIMO observation model that underpins on-grid and off-grid 2D sparse sensing PE.The model stacks contributions across transmit LEDs and receive PDs.Source: A. 2D on-grid sparse representation… · pp. 6–7
• The off-grid formulation models true delay and Doppler as offsets from nearby grid points and uses first-order Taylor corrections to capture basis mismatch.The on-grid support gives coarse DD locations, while the deviations refine delay and Doppler estimates.Source: B. 2D off-grid sparse representation… · p. 7
• The two-stage 2D-OffGBL procedure estimates sparse coefficients using joint sparsity across receive PDs, then recovers scattering matrices and refines DD estimates with reduced computational complexity.The final estimate also prunes rows associated with large precision values.Source: C. 2D off-grid BL (2D-OffGBL)-based… · p. 7, H MIMOY, (46) · p. 8

H MIMOY, (46)

• The Bayesian formulation assigns row-wise Gaussian priors with sparsity-controlling precision hyperparameters and a Gamma prior to those hyperparameters.The hyperparameters and off-grid deviations are updated iteratively.Source: C. 2D off-grid BL (2D-OffGBL)-based… · p. 7
• The update procedure refines the Doppler deviation using the effective dictionary derivative and prunes rows associated with large precision values.The update derivations follow the same steps as those for 1D-OffGBL.Source: H MIMOY, (46) · p. 8

A. 1D sparse representation and CE

• The 1D communication channel coefficients are sparse when dominant scatterers are few, so the sensing 1D on-grid and off-grid recovery frameworks and 1D-OffGBL can be adapted for channel estimation.The model uses a communication sensing dictionary with stacked received signals and noise.Source: A. 1D sparse representation and CE · p. 8

B. 2D sparse representation and CE

• The 2D channel-estimation model assumes pilots separable across Doppler and delay, then uses the resulting model as the basis for on-grid, off-grid, and 2D-OffGBL recovery.The vectorized channel model contains sparse channel coefficients.Source: B. 2D sparse representation and CE · pp. 8–9

1D range and velocity estimation

• The 1D model detects active DD bins from averaged RCS-coefficient magnitudes, applies a threshold, and uses estimated off-grid deviations to refine range and velocity.The range and velocity estimates are first obtained from the detected on-grid DD indices.Source: 1) 1D range and velocity estimation · p. 9

2D range and velocity estimation

• The 2D model detects DD bins by averaging coefficient magnitudes across transmit LEDs and receive PDs, then refines velocity with Doppler deviations and range with delay deviations.The deviations are applied after the 2D off-grid BL algorithm converges.Source: 2) 2D range and velocity estimation · p. 9

• BCRBs are provided for parameter, channel, range, and velocity estimation as theoretical lower bounds for the discussed signal models.Source: §VI · p. 9

A. BCRB for sensing PE

• For each receive photodiode, the received signal is modeled as a linear function of its RCS coefficient vector plus independent complex Gaussian noise.The coefficient vector contains the target RCS coefficients for that photodiode.Source: A. BCRB for sensing PE · p. 9
• A complex Gaussian prior yields a Bayesian Fisher information matrix formed from measurement and prior information.The likelihood and prior log densities are specified for the Gaussian model.Source: A. BCRB for sensing PE · p. 9
• The MSE lower bound for each coefficient vector is the trace of the inverse Bayesian Fisher information matrix, and independent photodiode errors give a matrix-level BCRB for the full coefficient matrix.The per-column bound is extended across receive photodiodes because their estimation errors are independent.Source: A. BCRB for sensing PE · pp. 9–10

B. BCRB for CE

• The downlink signal is modeled as the product of a sensing matrix and a sparse channel coefficient matrix, plus additive noise.The channel matrix is organized into vectors for the receive channels.Source: B. BCRB for CE · p. 10
• Following the sensing-coefficient derivation, the MSE lower bound for channel estimation is obtained from the channel Bayesian Fisher information matrix.The relevant information matrix is associated with each receive channel-coefficient vector.Source: B. BCRB for CE · p. 10

C. BCRB for range and velocity estimation

• For each receive photodiode, the observation is modeled using target steering atoms parameterized by delay and Doppler.Each atom also depends on the transmitted LED and target.Source: C. BCRB for range and velocity… · p. 10
• The unknown parameter vector stacks the target delays and Doppler frequencies, and its Bayesian Fisher information matrix is partitioned into blocks.The delay and Doppler bounds follow from the inverse of this joint information matrix.Source: C. BCRB for range and velocity… · p. 10
• The delay and Doppler estimation errors are bounded using the corresponding blocks of the inverse joint information matrix.The resulting lower bounds are expressed as traces of the relevant covariance blocks.Source: C. BCRB for range and velocity… · p. 10
• Covariance transformation maps the delay-Doppler bounds to physical range and velocity bounds.The resulting range and velocity BCRBs are the respective diagonal blocks of the transformed covariance bound.Source: C. BCRB for range and velocity… · p. 10

VII. SIMULATION RESULTS

• The simulations compare 1D and 2D on-grid and off-grid Bayesian-learning estimators against LMMSE and OMP using NMSE for channel and RCS accuracy.The evaluation includes 1D and 2D MIMO configurations, with a stopping tolerance of 10^-6 and a maximum of 100 iterations.Source: §VII · p. 10
• 1D-OffGBL achieves the lowest channel-estimation NMSE across the considered SNR range and the most accurate RCS estimates among the proposed methods.For channel estimation, the 2D methods generally have lower accuracy than their 1D counterparts; for RCS, 2D performance is slightly degraded by inter-dimensional error propagation.Source: §VII · pp. 10–11, Figure 1 · p. 11
• As target density increases, 1D-OffGBL maintains low RCS NMSE by exploiting coupled delay-Doppler structure, while the 2D methods can suffer inter-stage error propagation.The performance gap among estimators becomes more pronounced at higher target density.Source: Figure 1 · p. 11, §VII · p. 11
• 1D-OffGBL achieves the lowest NMSE among the proposed methods for both range and velocity estimation, with performance near the corresponding BCRB.The range and velocity comparisons use a detection threshold of η = 0.01.Source: Figure 2 · p. 11, §VII · p. 11
• NMSE increases as virtual Doppler resolution becomes coarser, and 1D-OffGBL consistently achieves the lowest NMSE across the considered methods.Coarser Doppler grids introduce greater quantization error and basis mismatch; 2D-OffGBL outperforms 2D-OnGBL.Source: Figure 2 · p. 11, §VII · p. 11
• The RCS heat maps show 1D-OffGBL localizing dominant targets more sharply and with less energy leakage than OMP or 1D-OnGBL.OMP displays substantial energy dispersion and spurious components, while 1D-OnGBL retains some low-magnitude artifacts.Source: Figure 3 · p. 12, §VII · p. 12

VIII. CONCLUSIONS

• Both off-grid algorithms refine continuous-valued sensing parameters to improve their estimation.The 1D method jointly refines delay-Doppler grid mismatches, whereas the 2D method uses sequential Doppler and delay stages.Source: §VIII · p. 12
• 1D-OffGBL provides superior estimation accuracy, while 2D-OffGBL offers a favorable balance between computational complexity and estimation performance.The 1D method uses a joint delay-Doppler dictionary; the 2D method estimates Doppler and delay sequentially, avoiding a high-dimensional joint dictionary at the cost of moderate performance loss.Source: §VIII · p. 12
• The framework develops sparse channel representations and channel-estimation procedures for both formulations, with methods to recover target range and velocity.Source: §VIII · p. 12
• Derived BCRBs provide theoretical benchmarks for parameter estimation, channel estimation, range, and velocity estimation.Source: §VIII · p. 12

APPENDIX

• The sparse-indicator update is obtained by differentiating the surrogate objective with respect to z_g and setting the result to zero.The objective discards terms independent of z; its expressions use G = DτDνNt and posterior mean and covariance information.Source: APPENDIX · p. 12
• The delay-deviation update uses an effective dictionary with a delay-dependent diagonal deviation term and follows by differentiating the expected surrogate objective.The dictionary is expressed using Ψ_MIMO,τ, Φ_MIMO,τ, and diag(ετ).Source: APPENDIX · pp. 12–13
• The Doppler-deviation update uses the corresponding Doppler-dependent effective dictionary and surrogate objective.The appendix identifies the z_g, ετ, and εν update rules with equations (27), (28), and (29), respectively.Source: APPENDIX · p. 13