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Super-Resolution Delay-Doppler Estimation for OFDM Passive Radar

Le Zheng, Xiaodong Wang

arXiv:1610.04218v2cs.IT

TL;DR

The paper addresses joint delay-Doppler estimation for OFDM passive radar when demodulation errors undermine the perfect-demodulation assumption. It combines continuous-domain atomic-norm sparsity with ℓ1-norm modeling of demodulation errors, formulates a convex SDP, and solves it efficiently with ADMM. Simulations report better performance than existing high-resolution compressed-sensing and 2D-MUSIC methods, especially in the presence of [the supplied passages do not complete this condition].

  • Problem

    Perfect demodulation does not hold in many wireless applications, while grid-based compressed sensing mismatches continuous delays and Doppler frequencies.

  • Method

    The method combines atomic-norm sparsity for the continuous delay-Doppler signal with ℓ1-norm sparsity for demodulation errors, then uses a convex SDP and ADMM iterations.

  • Results

    The proposed algorithm provides better performance than existing high-resolution compressed-sensing and 2D-MUSIC methods, especially in the presence of [condition incomplete in the supplied passage].

  • Takeaways & Limitations

    The paper demonstrates a high-resolution OFDM passive-radar approach that explicitly accommodates sparse demodulation errors.

Abstract

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In this paper, we consider the problem of joint delay-Doppler estimation of moving targets in a passive radar that makes use of orthogonal frequency-division multiplexing (OFDM) communication signals. A compressed sensing algorithm is proposed to achieve supper-resolution and better accuracy, using both the atomic norm and the $\ell_1$-norm. The atomic norm is used to manifest the signal sparsity in the continuous domain. Unlike previous works which assume the demodulation to be error free, we explicitly introduce the demodulation error signal whose sparsity is imposed by the $\ell_1$-norm. On this basis, the delays and Doppler frequencies are estimated by solving a semidefinite program (SDP) which is convex. We also develop an iterative method for solving this SDP via the alternating direction method of multipliers (ADMM) where each iteration involves closed-form computation. Simulation results are presented to illustrate the high performance of the proposed algorithm.

I. INTRODUCTION

The paper targets joint delay-Doppler estimation in OFDM passive radar when demodulation errors and continuous-domain parameters challenge existing methods. It proposes a super-resolution compressed-sensing receiver combining atomic- and ℓ1-norm sparsity models, with convex SDP and ADMM computation.

  • Passive radar uses communication illuminators but faces interference from direct signals, clutters, and noise, complicating target detection and parameter estimation.
  • Demodulation can improve detection and estimation over directly using the reference channel, while removing the need for a separate reference channel.
  • OFDM passive radar has attracted interest because OFDM communication signals are widely used and support matched-filter, MUSIC, and compressed-sensing receivers.
  • Perfect demodulation is unrealistic in many wireless applications; demodulation errors act as impulsive noise, increasing clutter and strong-target sidelobes and interfering with estimation.
  • Grid-based compressed sensing suffers model mismatch because delays and Doppler frequencies are continuous, motivating atomic-norm super-resolution.
  • The proposed receiver imposes atomic-norm sparsity on the delay-Doppler signal and ℓ1-norm sparsity on demodulation errors, solves a convex SDP, and develops a closed-form-per-iteration ADMM method.

II. SYSTEM DESCRIPTIONS

The system models OFDM passive-radar observations as multipath components indexed by delay and Doppler, multiplied by transmitted symbols and corrupted by demodulation errors and noise. The estimation task is to recover path coefficients and delay-Doppler parameters from these observations.

  • A. Signal Model: The OFDM transmitter divides data into blocks and sends symbols over N orthogonal subcarriers with spacing Δf = 1/T.
  • A. Signal Model: Targets, clutters, and the direct path are modeled as K multipaths with complex coefficients, delays τ_k, and Doppler frequencies f_k.
  • A. Signal Model: After Fourier processing, the received subcarrier signal is represented as a sum of path responses weighted by transmitted symbols, plus noise.
  • A. Signal Model: Unknown transmitted symbols are replaced by demodulated estimates, producing an error e_m(n) that is zero for correct decisions and nonzero otherwise.
  • A. Signal Model: Because demodulation errors are typically infrequent under normal conditions, the error signal is modeled as sparse.
  • A. Signal Model: The joint estimation problem recovers α, φ, and ψ from noisy vectorized observations, yielding path reflections, delays, and Doppler frequencies.

B. Existing Methods

Existing OFDM passive-radar methods include matched filtering, MUSIC, and grid-based compressed sensing, but their assumptions and discretization can limit generality and resolution.

  • B. Existing Methods: Matched filtering with conventional FFT processing is described as having low resolution, motivating MUSIC and compressed-sensing super-resolution methods.
  • B. Existing Methods: Prior work assumes perfect demodulation, with delays and Doppler frequencies estimated when the demodulation error vector is zero.
  • B. Existing Methods: The cited method assumes unit-magnitude PSK symbols, whereas OFDM communications typically use QAM symbols with different magnitudes.

1) Super-resolution Receiver based on MUSIC:

The MUSIC-based receiver estimates target and clutter delays and Doppler frequencies through spatial smoothing, subspace decomposition, and spectral peak localization. Its performance is vulnerable to impulsive noise from passive-radar demodulation errors, while grid-based compressed sensing also suffers from off-grid mismatch and unmodeled demodulation error.

  • MUSIC receiver: MUSIC requires spatial smoothing to construct an observation matrix from multiple waveform snapshots.With N′ < N and M′ < M, smoothing produces Nsnap = (N − N′ + 1)(M − M′ + 1) snapshots.
  • MUSIC receiver: The smoothed observation matrix is decomposed by SVD to obtain signal and noise subspaces.The signal subspace corresponds to eigenvectors associated with the largest eigenvalues.
  • MUSIC receiver: 2D-MUSIC estimates target and clutter delays and Doppler frequencies by locating poles in the MUSIC spectrum.After the parameters are identified, amplitudes are estimated using least squares.
  • MUSIC receiver: MUSIC offers good frequency-estimation resolution and accuracy, but sparsity-based optimization can outperform it in noisy environments.MUSIC is designed for small Gaussian-like perturbations, whereas passive-radar demodulation errors introduce impulsive noise that can severely degrade performance.
  • Compressed-sensing comparison: Grid-based compressed sensing linearizes joint parameter estimation, but off-grid targets and ignored demodulation errors can significantly degrade performance.The method uses a densely sampled overcomplete dictionary and identifies delays and Doppler frequencies from nonzero sparse coefficients.

III. CS-BASED HIGH-RESOLUTION PASSIVE RADAR USING ATOMIC NORM

The proposed receiver models both continuous-domain target and clutter frequencies and sparse demodulation errors, then estimates them through a convex atomic-norm and ℓ1-norm formulation. The resulting SDP is convex and yields signal and error estimates, with CS-ANL1 defined as the receiver using both norms.

  • Problem formulation: The formulation exploits two sparsity structures: few target and clutter sinusoids and few mistakenly demodulated symbols.The assumptions are K ≪ MN for signal frequencies and ∥ē∥0 = J for demodulation errors.
  • Convex optimization: The optimization combines atomic-norm regularization for the signal with ℓ1-norm regularization for sparse demodulation errors.The weights λ > 0 and µ > 0 control the corresponding regularization terms.
  • Atomic norm formulation: Atomic norm regularization represents the complex-sinusoidal signal when frequencies lie in a continuous domain rather than on discrete grids.The atom set contains normalized 2D complex sinusoids, and the atomic norm enforces sparsity over that set.
  • Atomic norm formulation: The atomic norm is converted to an equivalent trace-based form using a block Toeplitz matrix representation.U is a (2M − 1) × (2N − 1) matrix, and T(U) is formed from Toeplitz blocks.
  • Convex optimization: The resulting problem is a convex semidefinite program whose solutions provide the reconstructed signal and demodulation-error vector.Demodulation errors are identified from nonzero entries of ê, and the receiver is named CS-ANL1 when µ ≠ 0.

B. Example

The example compares grid-based CS-L1 with continuous-domain CS-AN and demonstrates dual-based frequency estimation and demodulation-error detection. CS-AN reduces basis-mismatch effects, while the dual polynomial identifies estimated frequencies and the dual solution detects mistaken symbols.

  • The atomic norm enforces sparsity in the continuous domain, avoiding the off-grid problem.
  • CS-L1 produces many false alarms from basis mismatch, making weak targets difficult to distinguish from spurious detections.
  • Increasing grid density reduces discretization mismatch but increases sensing-matrix coherence, so targets remain split into multiple scatterers.
  • CS-AN identifies target frequencies accurately with far fewer false alarms than CS-L1, despite one noise-induced false alarm.
  • Estimated delays and Doppler frequencies are obtained by locating points where the dual polynomial has magnitude λ.
  • With M = 8, N = 8, K = 2, λ = 0.16, and µ = 0.02, the dual solution correctly detects two demodulation errors and satisfies |ν̂_j| = µ when ê_j ≠ 0.The example uses BPSK-modulated data carriers and observes ||e||_0 = 2.

IV. AN ADMM-BASED ALGORITHM

The proposed SDP is solved with an ADMM procedure designed to replace slow general-purpose solvers for real-time processing. The updates use closed-form operations, including proximal shrinkage and projection onto the positive semidefinite cone.

  • The high-resolution receiver formulation is equivalent to a semidefinite program that can be solved by off-the-shelf SDP solvers.
  • Because general-purpose solvers can be slow for large problems, the paper derives a fast ADMM method for real-time signal processing.
  • The ADMM formulation rewrites the SDP and dualizes its equality constraint through an augmented Lagrangian.
  • The demodulation-error variable is updated using the proximal operator associated with the ℓ1 penalty.
  • The matrix variable is projected onto the positive semidefinite cone by eigenvalue decomposition and setting negative eigenvalues to zero.
  • ADMM also provides the dual solution through a submatrix of its dual variables, enabling subsequent dual-based frequency recovery.

A. Simulation Setup

Simulations compare CS-ANL1 with CS-L1, 2D-MUSIC, and CS-AN across low- and high-clutter scenarios, with and without demodulation errors. Results assess target identification, range and velocity accuracy, and robustness as BER increases.

  • Simulation scenarios: Two scenarios contain three targets each, with 5 clutters in Scenario 1 and 80 clutters in Scenario 2.These represent low and high clutter density, respectively.
  • Evaluation: The simulations compare CS-ANL1 against CS-L1, 2D-MUSIC, and CS-AN using target detection and range-velocity accuracy.Delays and Doppler frequencies are estimated for comparison, while range and velocity RMSE are evaluated for correctly identified targets.
  • Limitations: All algorithms fail to distinguish clutters when their separation is too small, especially in the high-density Scenario 2.This limitation does not affect target delay-Doppler estimation when higher-Doppler targets are separated from near-zero-velocity clutter.
  • Demodulation errors: At BER=0.02, demodulation errors substantially degrade CS-L1, 2D-MUSIC, and CS-AN, producing false alarms, defocused mainlobes, ghost peaks, and low detection probability.The degradation is reported in both low- and high-clutter scenarios.
  • Error-free communication: When BER=0, CS-AN and CS-ANL1 have similar accuracy and outperform CS-L1 and 2D-MUSIC.CS-L1 also produces more false alarms because of model mismatch, while CS-AN contains fewer ghost peaks than CS-L1.

VI. CONCLUSIONS

The paper concludes with a high-resolution OFDM passive-radar algorithm that combines atomic-norm and ℓ1-norm sparsity modeling and an ADMM solver. Simulations report better performance than existing compressed-sensing and 2D-MUSIC high-resolution methods, especially with demodulation errors.

  • Contributions: The proposed CS-based algorithm uses the atomic norm for multipath signal sparsity and the ℓ1-norm for demodulation-error sparsity.The two norms address sparsity in different components of the signal model.
  • Contributions: An ADMM-based fast algorithm computes the solution to the formulated compressed-sensing problem.The conclusion identifies this as the computational method for the proposed formulation.
  • Results: Simulation results show better performance than existing compressed-sensing and 2D-MUSIC high-resolution methods, especially in the presence of demodulation errors.The reported comparison concerns the proposed algorithm's overall simulation performance.

APPENDIX

The appendix establishes optimality conditions for the joint signal and demodulation-error estimate and connects the primal and dual formulations. It uses convex optimality arguments involving the objective, Lagrangian, dual atomic norm, and KKT conditions.

  • Optimality analysis: Lemma 3 states that the estimated pair (ẑ, ê) solves the optimization problem if and only if the stated optimality conditions hold.The proof is deferred to Appendix B.
  • Primal-dual connection: The derivation uses vanishing duality gap and KKT conditions to relate the primal estimate to the dual solution.The appendix explicitly connects ν̂ and the dual certificate through these conditions.
  • Optimality analysis: The proof rewrites the necessary-and-sufficient optimality inequality using atomic-norm and ℓ1-norm differences for arbitrary candidate estimates.The argument then invokes the definition of the dual atomic norm to obtain the required conditions.
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