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Integrated Sensing and Communication with Massive MIMO: A Unified Tensor Approach for Channel and Target Parameter Estimation

Ruoyu Zhang, Lei Cheng, Shuai Wang, Yi Lou, Yulong Gao, Wen Wu, Derrick Wing Kwan Ng

arXiv:2401.01738v1cs.ITeess.SP

TL;DR

Massive MIMO-ISAC must acquire accurate communication channels and target parameters despite its many antennas. The paper develops a unified tensor framework using physical channel parameters and shared training resources, with numerical results showing improvements in resolvable targets, sensing resolution, training overhead, and channel-estimation accuracy.

  • Problem

    Massive MIMO-ISAC creates challenges in acquiring accurate channel state information and target parameters, while beam squint makes steering vectors frequency-dependent in wideband systems.

  • Method

    The paper parameterizes communication channels by physical parameters, uses shared time-frequency training, and formulates channel and target estimation as structured tensor decomposition problems.

  • Results

    Numerical results verify the analysis and report superiority in resolvable targets, sensing resolution, training overhead reduction, and channel-estimation accuracy.

  • Takeaways & Limitations

    A unified CPD-based model connects channel and target estimation across angular, delay, and Doppler dimensions for massive MIMO-ISAC.

Abstract

from arXiv · show

Benefitting from the vast spatial degrees of freedom, the amalgamation of integrated sensing and communication (ISAC) and massive multiple-input multiple-output (MIMO) is expected to simultaneously improve spectral and energy efficiencies as well as the sensing capability. However, a large number of antennas deployed in massive MIMO-ISAC raises critical challenges in acquiring both accurate channel state information and target parameter information. To overcome these two challenges with a unified framework, we first analyze their underlying system models and then propose a novel tensor-based approach that addresses both the channel estimation and target sensing problems. Specifically, by parameterizing the high-dimensional communication channel exploiting a small number of physical parameters, we associate the channel state information with the sensing parameters of targets in terms of angular, delay, and Doppler dimensions. Then, we propose a shared training pattern adopting the same time-frequency resources such that both the channel estimation and target parameter estimation can be formulated as a canonical polyadic decomposition problem with a similar mathematical expression. On this basis, we first investigate the uniqueness condition of the tensor factorization and the maximum number of resolvable targets by utilizing the specific Vandermonde

I. INTRODUCTION

The paper frames massive MIMO-ISAC as a unified channel-estimation and target-sensing problem, exploiting shared physical parameters and tensor structure across communication and sensing. It proposes shared training, uniqueness analysis, structured estimation, and a beam-squint-aware extension.

  • Motivation: Massive MIMO-ISAC combines communication and sensing to pursue high-quality wireless communication and high-resolution, robust sensing.The motivation includes applications such as autonomous driving and unmanned aerial vehicle networks.
  • Unified framework: The paper unifies channel state information and target parameters by representing high-dimensional channels with physical parameters such as delay, Doppler, AoA, and AoD.These parameters connect communication-path characteristics with target range, velocity, and azimuth angle.
  • Tensor formulation: A shared training pattern uses the same time-frequency resources to formulate channel estimation and target estimation as structured tensor decomposition problems.The formulation captures angular, delay, and Doppler dimensions and supports analysis of resolvable-target limits through tensor uniqueness.
  • Estimation algorithm: The proposed algorithm exploits Vandermonde structure and spatial smoothing to separately estimate AoA, AoD, delay, Doppler shift, and coefficients.An iterative method addresses AoD–Doppler coupling and is reported to improve angle, velocity, and channel-estimation accuracy.
  • Beam-squint extension: For significant beam squint, the paper introduces segment-based training and a tensor estimation scheme that avoids angle–Doppler coupling while analyzing uniqueness.Beam squint makes steering vectors frequency-dependent and has been less studied for target parameter estimation than for channel estimation.
  • System model: The system model uses shared wireless signals for a communication UE and simultaneous sensing of surrounding targets, with separate channel and target estimation at the UE and BS.The communication model includes multipath delay, Doppler shift, and antenna-dependent delays associated with AoA and AoD.

B. Radar Sensing Model

The radar sensing model describes echoes from multiple moving point targets collected by separated receive antennas and converts them into an equivalent sensing-channel representation. Target reflection, round-trip delay, Doppler, and array-angle effects determine the received signal.

  • Target and array model: The model considers Q point targets characterized by distance R_q and radial velocity V_q, whose echoes are collected by the BS receive array.The target reflection coefficient is proportional to radar cross section, and antenna geometry introduces angle-dependent delays.
  • Model assumption: Assuming target velocity is much smaller than the speed of light, the model treats the propagation delay as constant during the coherent processing interval.This approximation enables the stated equivalent sensing-channel model.
  • Sensing-channel representation: The equivalent sensing channel is expressed using a linear time-varying filter, and Fourier transformation followed by antenna stacking yields the multi-antenna sensing-channel representation.The representation incorporates array steering vectors for the target angles.

C. Unified Model for Massive MIMO-ISAC

Communication and sensing channels share angular, delay, and Doppler structure, enabling common training resources and analogous tensor formulations for estimating channel and target parameters.

  • Shared channel structure: The communication and sensing channels have similar mathematical expressions, linking channel physical parameters with target parameters across angular, delay, and Doppler dimensions.Channel parameters describe propagation paths, whereas target parameters describe target scattering; one-way and round-trip delays have different physical meanings.
  • Shared training: The shared intrinsic structure supports using the same signals for channel and target estimation, potentially reducing training overhead and improving spectral efficiency.
  • Beam squint: Beam squint makes array steering vectors frequency-dependent because propagation delay across a wideband massive-MIMO aperture cannot be ignored.For 64 BS antennas, 1 GHz bandwidth, 28 GHz carrier frequency, and θ = 60°, the aperture delay is 0.9743T_s.
  • Signal model: The OFDM cyclic prefix suppresses intersymbol interference from multipath and maximum detectable-range delays before CP removal and frequency transformation.
  • Shared training: The proposed training pattern uses s_n,k = p_nx_k, with symbol-varying precoders and unchanged unit-modulus pilot symbols across training subcarriers.This precoded structure facilitates a third-order tensor representation of the echo signals.
  • Tensor formulation: Collecting echoes over training symbols and subcarriers forms a third-order tensor, whose factor matrices encode observations across space, time, and frequency.The UE training signals are modeled similarly as a third-order tensor for channel estimation.

B. Uniqueness Property Analysis for Unified Tensor Formulation

The paper analyzes CPD identifiability for the unified tensor model, first using Kruskal’s condition and then exploiting a Vandermonde factor structure to obtain stronger recovery implications.

  • General CPD uniqueness: The unified channel and target estimation problems require CPD uniqueness so decomposed factor matrices retain the unknown-parameter information.
  • General CPD uniqueness: Kruskal’s condition k_B(1) + k_B(2) + k_B(3) ≥ 2Q + 2 provides a sufficient condition for unique CPD.In the generic case, this becomes min(I_1,Q) + min(I_2,Q) + min(I_3,Q) ≥ 2Q + 2.
  • Implications: Table I compares the maximum number of resolvable targets under the uniqueness conditions of Lemma 1 and Lemma 2 for K = 16 and M_re = 8.
  • Vandermonde-constrained uniqueness: The Vandermonde structure of B(3) yields a relaxed uniqueness condition beyond the generic CPD condition.The condition applies when B(3) has distinct generators.
  • Implications: The Vandermonde-constrained result permits reliable parameter estimation even when factor-matrix columns are dependent, including closely spaced target directions.It therefore indicates higher spatial resolution for target sensing within the stated identifiability setting.

IV. UNIFIED CHANNEL AND TARGET PARAMETER ESTIMATION ALGORITHM

The proposed algorithm estimates tensor factor matrices first, using Vandermonde structure and spatial smoothing, then uses those factors to recover the underlying channel and target parameters.

  • Algorithm structure: The algorithm has two phases: factor-matrix estimation followed by extraction of the unknown physical parameters.It also discusses practical implementation issues.
  • Factor-matrix estimation: Instead of unconstrained ALS, the method exploits the Vandermonde structure in B(3) to improve parameter estimation and avoid limiting resolvable targets.
  • Factor-matrix estimation: Spatial smoothing expands the tensor-derived observation dimension by exploiting the Vandermonde structure of B(3).The unfolded model includes a corresponding noise matrix.
  • Factor-matrix estimation: An ESPRIT-like procedure estimates factor matrices from signal subspaces obtained through singular value decomposition and an eigenvalue decomposition.
  • Factor ambiguity: The estimated factor matrices match the true factors up to diagonal scaling, permutation, and tensor-factorization error.The scaling matrices satisfy Δ_1Δ_2Δ_3 = I_Q.

B. Extraction of Unknown Parameters

After estimating the factor matrices, the method separately extracts angles, delays, Doppler shifts, and reflection coefficients, iteratively resolving the coupled AoD–Doppler parameters.

  • Parameter extraction: The extracted target parameters are θ_q, ϕ_q, τ_q, β_q, and ν_q, while the UE estimates the analogous five-tuple of channel parameters.
  • Direct extraction: AoA is obtained separately from each B(1) column using a correlation-based method, and time delay is directly recovered from B(3) generators.
  • Coupled estimation: AoD and Doppler shift are coupled in B(2), while Doppler-induced phase offsets make conventional correlation methods severely degrade.
  • Coupled estimation: The algorithm alternately refines AoD and Doppler estimates until the iterative updates converge to a stationary point of the joint objective.
  • Closed-form estimation: Polynomial methods exploit Vandermonde structure to replace one-dimensional searches for AoD and Doppler generators with closed-form estimation.
  • Coefficient and channel recovery: Reflection coefficients are estimated by least squares after the delay and other factor-related parameters have been obtained.The reconstructed multipath parameters then produce the full channel matrix at the UE.

C. Discussions

The discussion addresses practical implementation issues, including interference, unknown target count, pilot design, and the tradeoff created by sharing resources between sensing and communication.

  • Practical issues: Interference from non-target objects or passive links can either increase effective noise or be modeled as virtual targets.The first treatment lowers the effective SNR, while the second preserves noise power but adds estimated non-target objects.
  • Practical issues: Distinguishing true targets from non-target objects remains an open target-classification and identification problem.The discussion explicitly leaves this problem for future research.
  • Practical issues: The algorithm assumes that the tensor rank, equivalently the number of targets Q, is known during factor-matrix estimation.When Q is unavailable, the paper suggests sparsity-promoting tensor methods or a preliminary rank-determination step.
  • Pilot design: Random-phase pilot signals make the training beams isotropic, so overhead depends on targets or channel multipaths rather than antenna count.The required channel-estimation time is also described as proportional to the number of targets or channel multipaths.
  • Resource sharing: Sharing time-frequency resources creates a performance tradeoff because communication multipaths and sensing targets need not have equal counts.Fewer channel multipaths favor channel estimation, whereas fewer targets favor target-parameter estimation; signal energy can be optimized around the weaker task.

V. PARAMETER ESTIMATION WITH BEAM SQUINT EFFECTS

Beam squint makes steering vectors frequency-dependent and prevents direct multidimensional decomposition across subcarriers, so the paper designs segment-based training and a multi-time-slot CPD formulation.

  • Problem formulation: Beam squint prevents independently decomposing multi-subcarrier observations into multiple dimensions under the original tensor model.The resulting frequency dependence motivates a new formulation using signals from multiple time slots.
  • Training design: A segment-based training precoder is designed to enable CPD-based parameter estimation under beam squint effects.The new training pattern facilitates construction of a tensor from received signals collected over multiple time slots.
  • Training design: The time-domain training frame is divided into L segments of Nd OFDM symbols, satisfying NdL = N.The precoder is updated Nd times within one time slot and repeated L times per frame.
  • Tensor formulation: Within each segment, the model uses a quasi-static Doppler approximation across consecutive Nd OFDM symbols.The approximation is based on the target speed being far below the speed of light, so Doppler-induced phase changes little within a segment.
  • Tensor formulation: Collecting the segment echoes over multiple time slots forms a received-signal tensor Yk ∈ C^Mre×Nd×L for each subcarrier.The segment observations are assembled after pilot removal and concatenation of the corresponding training-symbol echoes.

B. Uniqueness Condition

The beam-squint formulation establishes CPD uniqueness through the Vandermonde structure of a factor matrix, with a sufficient condition that applies separately at every subcarrier.

  • Uniqueness condition: The Doppler-related factor matrix C(3)k has Vandermonde structure with distinct generators.This structure is the basis for the uniqueness analysis under beam squint effects.
  • Algorithm: Algorithm 2 processes received echoes Yk,l across segments and subcarriers to estimate the model factors and target parameters.Its listed output includes θ̂q, τ̂q, ϕ̂q, ν̂q, and β̂q.
  • Uniqueness condition: The CPD is unique when the stated k-rank condition satisfies K3 + L3 = L + 1.The passage identifies this as the uniqueness condition before giving its generic-case form.
  • Uniqueness condition: The beam-squint uniqueness proof follows the earlier lemma after replacing the original factor matrices with the beam-squint factors.The proof therefore transfers the earlier structural argument to the frequency-dependent model.
  • Implication: The sufficient uniqueness condition holds for arbitrary subcarrier k, allowing factor matrices to be recovered from individual subcarriers.This avoids potential performance degradation attributed to beam squint effects.

C. Estimation of Unkown Parameters

After tensor factorization, the method extracts angular, Doppler, delay, and reflection parameters from estimated factor matrices and combines subcarrier estimates to reconstruct the communication channel.

  • Parameter extraction: The estimated factor matrices are used to recover the unknown parameters despite their nonlinear coupling.The procedure constructs auxiliary factors before estimating the remaining coefficients and delays.
  • Angular parameters: AoA is estimated from the columns of factor matrix C(1)k, while the corresponding factor columns encode the angular information.The paper explicitly associates target AoA with each column of C(1)k.
  • Doppler parameters: Doppler shift is directly estimated from the Vandermonde generator associated with each column of C(3)k.The generator ẑν,q,k is used for the q-th target at subcarrier k.
  • Delay and coefficient estimation: Reflection coefficients and time delays are obtained by combining coefficient estimates across multiple subcarriers.The delay factor matrix is constructed from the delay steering vectors before estimating τq and βq.
  • Communication channel: Under the same process, the communication receiver can estimate multipath parameters and reconstruct the entire channel matrix.The listed multipath parameters are θ̂p, τ̂p, ϕ̂p, ν̂p, and α̂p.

VI. SIMULATION RESULTS

Simulations evaluate the unified estimation schemes across target parameters, channel estimation, computational runtime, and significant beam squint. The proposed algorithms generally improve estimation accuracy and target distinguishability, while some settings impose higher runtime or performance gaps.

  • Target parameter estimation: At SNR = 10 dB with N = 16 and K = 16, Algorithm 1 accurately estimates target angles, ranges, and velocities using few time-frequency training resources.The benchmark recovers only part of the target parameters; limited receive-array aperture prevents distinguishing adjacent angles, while short observation duration reduces Doppler resolution.
  • Angle estimation: The proposed algorithm's AoD and AoA RMSE improves with SNR and approaches the corresponding CRBs, outperforming ALS and avoiding MUSIC's limited-training performance bottleneck.The tensor formulation exploits the Vandermonde structure of the factor matrix and addresses coupling between AoD and Doppler shift.
  • Range and velocity estimation: The proposed algorithm improves range and velocity estimation with SNR by independently estimating delays and using Doppler compensation, while ALS suffers from coupled angle-Doppler recovery.Range and velocity RMSE are relatively higher than angle RMSE because the parameters occupy different frequency-, time-, and space-domain resources.
  • Channel estimation: With K = 16 training subcarriers versus K = 32 for benchmarks, the proposed algorithm achieves superior channel-estimation performance as SNR increases.Joint Doppler-shift and AoD estimation enables effective Doppler compensation; ALS encounters an SNR bottleneck and BOMP is unsatisfactory in this scenario.
  • Computational complexity: The proposed algorithm has higher CPU runtime for AoD and velocity estimation and more than double the channel-estimation runtime of BOMP and ALS.The added runtime is associated with iterative estimation for AoD-Doppler coupling, alongside reported estimation-performance improvements.
  • Beam squint effects: Under significant beam squint, Algorithm 2 improves angle and channel estimation by decoupling parameter estimation across individual subcarriers, although a Doppler approximation error creates a performance gap from the ideal case.Algorithm 2 uses segment-based training for angle estimation and is evaluated across 600 MHz and 1 GHz bandwidths for channel estimation.

VII. CONCLUSION

The paper develops a unified tensor framework for massive MIMO-ISAC channel and target parameter estimation under negligible and severe beam squint. It establishes tensor-factorization conditions and reports numerical superiority in resolvable targets, sensing resolution, and training overhead reduction.

  • The proposed framework handles channel and target parameter estimation under both negligible and severe beam squint effects.
  • A shared time-frequency training pattern casts channel and target estimation as structured tensor decomposition problems following a unified CPD model.
  • The analysis derives uniqueness conditions for the tensor decomposition and exploits Vandermonde factor matrices to enhance parameter estimation.
  • Numerical results verify the theoretical analysis and the proposed approaches' superiority in resolvable targets, sensing resolution, and training overhead reduction.
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