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A Novel Angle-Delay-Doppler Estimation Scheme for AFDM-ISAC System in Mixed Near-field and Far-field Scenarios
Yirui Luo, Yong Liang Guan, Yao Ge, David González G., Chau Yuen
TL;DR
Accurate AFDM-ISAC sensing is challenging with fractional delay and Doppler and coexisting near-field and far-field targets, while prior mixed-field work does not address high mobility. The paper models received signals as a low-rank tensor and develops structured decomposition and parameter estimators. Simulations report better performance than benchmark schemes and results relatively close to the corresponding CRLB.
Problem
Accurate sensing in AFDM-ISAC is challenged by fractional delay and Doppler and mixed near-field and far-field targets, while prior mixed-field studies do not consider high mobility.
Source: §I · pp. 1–2Method
The method models received signals as a third-order low-rank CP tensor and uses structured decomposition and low-complexity estimators for angle, delay, and Doppler parameters.
Source: §I · p. 2Results
Simulations report lower NMSE than benchmark schemes and performance relatively close to the corresponding CRLB for AoA, AoD, delay, and Doppler estimates.
Source: §I · p. 3Takeaways & Limitations
The proposed scheme estimates AoA, AoD, delay, and Doppler for mixed near-field and far-field targets, including fractional parameter values.
Source: §I · pp. 2–3
Abstract
from arXiv · showhide
The recently proposed multi-chirp waveform, affine frequency division multiplexing (AFDM), is considered as a potential candidate for integrated sensing and communication (ISAC). However, acquiring accurate target sensing parameter information becomes challenging due to fractional delay and Doppler shift occurrence, as well as effects introduced by the coexistence of near-field (NF) and far-field (FF) targets associated with large-scale antenna systems. In this paper, we propose a novel angle-delay-Doppler estimation scheme for AFDM-ISAC system in mixed NF and FF scenarios. Specifically, we model the received ISAC signals as a third-order tensor that admits a low-rank CANDECOMP/PARAFAC (CP) format. By employing the Vandermonde nature of the factor matrix and the spatial smoothing technique, we develop a structured CP decomposition method that guarantees the condition for uniqueness. We further propose a low-complexity estimation scheme to acquire target sensing parameters with fractional values, including angle of arrival/departure (AoA/AoD), delay and Doppler shift accurately. We also derive the Cram\'er-Rao Lower Bound (CRLB) as a benchmark and analyze the complexity of our proposed scheme. Finally, simulation results are provided to demonstrate the effectiveness and superiority of our proposed scheme.
I. INTRODUCTION
AFDM-ISAC sensing must handle high mobility, fractional delay and Doppler, and mixed near-field and far-field targets. The paper proposes a joint angle-delay-Doppler estimation scheme and describes the system and transmitted AFDM signal.
I. INTRODUCTION
- Existing AFDM-ISAC methods leave gaps in velocity sensing, fractional delay-Doppler estimation, and low-complexity processing, while prior mixed near-field/far-field studies do not address high mobility.AFDM's DAF-domain representation motivates its use for high-mobility sensing, but earlier approaches have limitations including grid assumptions, costly searches, or large matrix inversions.Source: §I · p. 2
- The proposed scheme estimates AoA, AoD, delay, and Doppler jointly for mixed near-field and far-field targets, representing received signals as a low-rank third-order CP tensor.The paper also proposes structured CP decomposition, subspace-based AoA estimation, and a low-complexity iterative method for fractional delay-Doppler estimation.Source: §I · p. 2
- Simulations report lower NMSE than benchmark schemes and performance relatively close to the corresponding CRLB for estimated AoA, AoD, delay, and Doppler shift.The paper also derives a CRLB and analyzes computational complexity.Source: §I · p. 3
II. MIXED NF AND FF AFDM-ISAC SYSTEM MODEL
- The system assumes a far-field transmitter and a receiver with mixed near-field and far-field sensing, with the receiver knowing the transmitted signals through an optical-fiber backhaul.A micro-cell base station transmits AFDM signals; a nearby macro-cell base station reuses target echoes for sensing.Source: §II · p. 3
A. Transmitted AFDM ISAC Signal
- An AFDM symbol starts as a DAF-domain vector and is transformed to time domain by an inverse DAFT; its bandwidth is NcΔf and duration is Ta.The chirp-subcarrier spacing is Δf = 1/Ta, and the transmitted signal uses a chirp-periodic prefix.Source: A. Transmitted AFDM ISAC Signal · p. 3
B. Sensing Channel Model
The sensing model distinguishes near-field and far-field propagation at the receiver and represents target echoes with delay and Doppler effects. Received signals are sampled and transformed into the DAF domain for sensing.
B. Sensing Channel Model
- Each sensing target is modeled by its line-of-sight path, with delay and Doppler shift determined by target range and relative velocity.The received array response includes the target's AoA and distance from the reference antenna.Source: B. Sensing Channel Model · pp. 3–4
- The receiver is modeled as a symmetric ULA with G = 2Gx + 1 antennas, and near-field targets within the Rayleigh distance require a spherical-wave phase model that retains distance-dependent phase.The paper uses a unified array-response representation for near-field and far-field targets.Source: B. Sensing Channel Model · p. 4
C. Received AFDM ISAC Signals
- The receiver samples the time-domain echoes at interval Ts and represents them with normalized delay and Doppler parameters that include integer and fractional components.The maximum normalized delay and Doppler values are constrained relative to the AFDM subcarrier count Nc.Source: C. Received AFDM ISAC Signals · pp. 4–5
- After serial-to-parallel conversion and removal of the chirp-periodic prefix, the received sensing signal is transformed back to the DAF domain.The paper expresses the received signal in matrix form and specifies the corresponding DAF-domain noise.Source: C. Received AFDM ISAC Signals · p. 5
III. TENSOR REPRESENTATION AND ANALYSIS
The received DAF-domain signal is represented as a third-order tensor with a CP form whose factor matrices encode target parameters.
A. Low-rank Tensor Model
- The received DAF-domain signal is modeled as a third-order tensor, with the received signal and corresponding noise represented as tensors.The tensor dimensions are G × Nc × K.Source: A. Low-rank Tensor Model · p. 5
- The signal tensor admits a CP decomposition with factor matrices associated with receiver array response, delay-Doppler parameters, and transmitter-side parameters.The factor matrices are denoted AR, BC, and AT.Source: A. Low-rank Tensor Model · p. 5
B. Uniqueness Property Analysis
The paper analyzes when the structured CP decomposition is unique and develops a decomposition method that exploits the factor matrices’ structure. Spatial smoothing and the Vandermonde structure relax the uniqueness condition, including cases with rank-deficient factors.
B. Uniqueness Property Analysis
- Kruskal’s condition provides a criterion for CP uniqueness; its generic form is expressed using the Kruskal ranks of the factor matrices.The generic-case condition is stated in the section, while the structural argument relaxes it.Source: B. Uniqueness Property Analysis · p. 5
- With structural constraints, the CP uniqueness condition can hold for one target and when A_R or B_C is not full Kruskal-rank.The latter includes cases where targets share an AoA or delay and Doppler shift.Source: B. Uniqueness Property Analysis · p. 6
IV. PROPOSED ANGLE-DELAY-DOPPLER ESTIMATION SCHEME
- The paper’s two-phase scheme first decomposes a tensor model, then extracts sensing parameters from the estimated factor matrices.The decomposition exploits structural characteristics of the factor matrices.Source: §IV · p. 6
A. Structured CP Decomposition for Factor Matrices
- Spatial smoothing exploits the Vandermonde structure of A_T, expanding the dimension and addressing rank-deficient cases.The method applies smoothing to the second-mode unfolding of Y.Source: A. Structured CP Decomposition for… · p. 6
B. Sensing Parameters Estimation
The sensing stage extracts AoD and AoA from the estimated factor matrices. It estimates AoA without first distinguishing near-field from far-field targets.
B. Sensing Parameters Estimation
- The estimated factor matrices from the structured decomposition provide the inputs for sensing-parameter extraction.The method describes this as the second phase of the estimation scheme.Source: B. Sensing Parameters Estimation · pp. 6–7
1) AoD:
- AoD is estimated directly from the generators of the corresponding columns of A_T.Each column of A_T is associated with one AoD.Source: 1) AoD · p. 7
2) AoA:
- For AoA, the method forms a Toeplitz correlation matrix to decouple angle from distance under the near-field spherical wavefront.The paper states that the resulting AoA scheme does not require differentiating near-field and far-field targets.Source: 2) AoA · p. 7
- The paper motivates its AoA procedure against MUSIC’s high eigendecomposition complexity and derives the Toeplitz matrix into two parts.It then uses those parts to obtain the target AoA.Source: 2) AoA · p. 7
3) Delay and Doppler Shift:
The delay-Doppler method uses AFDM’s delay-Doppler representation in a two-stage scheme. It first obtains integer components by pulse compression, then estimates fractional components through iterative one-dimensional searches.
3) Delay and Doppler Shift:
- A two-stage super-resolution scheme estimates joint delay and Doppler, using a one-dimensional iterative process to handle fractional components at low complexity.The estimated factor matrix B_C contains the delay and Doppler information.Source: 3) Delay and Doppler Shift · p. 7
- Stage one uses one-dimensional pulse compression to obtain integer normalized delay and Doppler components.The peak index in the compressed vector provides the location used to obtain these components.Source: 3) Delay and Doppler Shift · pp. 7–8
- Fig. 4 compares the DAF-domain vector with the vector after one-dimensional pulse compression in an example with one target.The example uses β = 8.13, ν = 1.67, SNR = 15 dB, and 256 subcarriers.Source: 3) Delay and Doppler Shift · p. 7, Figure 4 · p. 8
- Stage two applies iterative one-dimensional golden-section searches to estimate the fractional parts of normalized delay and Doppler.The off-grid search is initialized from the integer components obtained in stage one.Source: 3) Delay and Doppler Shift · p. 8
A. CRLB
The paper derives a CRLB for parameter estimation and analyzes the proposed scheme’s computational complexity. The bound includes parameters that the estimator does not itself estimate.
A. CRLB
- The CRLB is used as a benchmark and is calculated by inverting the Fisher information matrix.It serves as a minimum variance limit for unbiased estimators.Source: A. CRLB · pp. 8–9
- The CRLB parameter vector includes AoA, distance, delay, Doppler shift, AoD, and target amplitude.Distance and amplitude remain in the bound even though the proposed scheme does not estimate them.Source: A. CRLB · p. 9
B. Computational Complexity
- The truncated SVD of Υ has computational complexity O(K3L3GNcR).The section also analyzes the complexity of factor-matrix and sensing-parameter estimation.Source: B. Computational Complexity · p. 9
VI. NUMERICAL RESULTS
In mixed near- and far-field AFDM-ISAC simulations, the proposed scheme outperforms benchmark methods and approaches the CRLB. Results cover angle, delay, and Doppler estimation, convergence, and complexity.
VI. NUMERICAL RESULTS
- Simulations use a 60 GHz carrier, a micro-cell ULA with K = 8 and macro-cell ULA with Gx = 50, and three mixed NF/FF targets—one near-field and two far-field.The maximum considered speed is 540 km/h, with 256 subcarriers and maximum delay ℓmax = 12.Source: §VI · p. 9
- For AoD estimation, the proposed scheme has the lowest NMSE among the compared MUSIC-based and OMP methods and remains relatively close to the CRLB.The authors attribute its gain to multidimensional structure and gridless processing, which avoids grid discretization error; MUSIC is constrained by transmitted antenna count and OMP by grid resolution.Source: §VI · pp. 9–10
- For AoA estimation with mixed near- and far-field targets, the proposed scheme outperforms the FF-based and subspace-based methods and closely approaches the CRLB.The FF-based method neglects spherical-wave propagation, while the proposed method also avoids inter-target interference.Source: §VI · p. 10
- Delay and Doppler estimation performance improves with iteration and converges after around 3 iterations.The simulations use T = 3 unless otherwise noted.Source: §VI · p. 10
- For delay and Doppler estimation versus SNR, the proposed scheme approaches the CRLB, while CS is limited to integer normalized delay and Doppler components.AML performance depends on its search resolution; finer resolution improves estimation but increases computational complexity.Source: §VI · p. 10
- The proposed method matches AML with ετ = εf = 0.001 at much lower complexity in the reported simulation setup.The passage reports AML complexity of O(10^12) and proposed-method complexity of O(10^4).Source: §VI · p. 11
VII. CONCLUSION
- The proposed scheme outperforms the benchmark schemes and is close to the corresponding CRLB in the reported simulations.The conclusion covers estimation of AoD, AoA, delay, and Doppler shift in mixed near- and far-field scenarios.Source: §VII · p. 11
APPENDIX DERIVATION OF CRLB
The appendix derives a CRLB for the estimated parameters by constructing the Fisher information matrix from likelihood derivatives and noise statistics.
APPENDIX DERIVATION OF CRLB
- The derivation forms the log-likelihood for Ψ and specifies the Fisher information matrix using the noise variance.The passage also identifies Y as the received signal and W as the noise.Source: APPENDIX DERIVATION OF CRLB · p. 11
- The appendix computes likelihood derivatives with respect to the parameters, including θr, and uses them to obtain entries of the Fisher information matrix.It specifies Re(·) as real-part extraction and er as the canonical vector with nonzero entry at index r.Source: APPENDIX DERIVATION OF CRLB · p. 11
- The CRLB for the estimated parameters is calculated from the resulting matrix using equation (45).The derivation also describes computing components in principal and off-principal minors.Source: APPENDIX DERIVATION OF CRLB · pp. 11–12