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Adaptive Interference Removal for Un-coordinated Radar/Communication Co-existence

Le Zheng, Marco Lops, Xiaodong Wang

arXiv:1706.03151v2eess.SYcs.SYstat.AP

TL;DR

In un-coordinated radar/communication coexistence, a communication receiver must remove interference from potentially active radars whose waveforms and relevant parameters are unknown. The paper proposes iterative joint estimation and demodulation algorithms based on on-grid CS and atomic norms; simulations report improved SER over original demodulation, with CS-AN outperforming CS-L1 in reported comparisons.

  • Problem

    In un-coordinated coexistence, the communication receiver must estimate and remove interference from potentially active radars despite unknown waveforms, delays, and coupling coefficients.

    Source: §I · p. 1

  • Method

    The paper iteratively estimates radar interference and communication demodulation errors using sparse on-grid CS or atomic-norm formulations.

    Source: §I · p. 2, §III · p. 4, B. Joint Waveform Estimation and… · p. 5

  • Results

    Simulations show better SER than original demodulation, and CS-AN outperforms CS-L1 in the reported comparisons.

    Source: C. Example · p. 7, B. Performance · p. 10, §VI · p. 11

  • Takeaways & Limitations

    The reported examples show CS-AN outperforming CS-L1, including cases where on-grid basis mismatch produces more false alarms.

    Source: C. Example · p. 7

    The fast-algorithm result has a technical N ≥ 257 requirement, although simulations suggest it still holds without that condition.

    Source: §IV · p. 8

Abstract

from arXiv · show

Most existing approaches to co-existing communication/radar systems assume that the radar and communication systems are coordinated, i.e., they share information, such as relative position, transmitted waveforms and channel state. In this paper, we consider an un-coordinated scenario where a communication receiver is to operate in the presence of a number of radars, of which only a sub-set may be active, which poses the problem of estimating the active waveforms and the relevant parameters thereof, so as to cancel them prior to demodulation. Two algorithms are proposed for such a joint waveform estimation/data demodulation problem, both exploiting sparsity of a proper representation of the interference and of the vector containing the errors of the data block, so as to implement an iterative joint interference removal/data demodulation process. The former algorithm is based on classical on-grid compressed sensing (CS), while the latter forces an atomic norm (AN) constraint: in both cases the radar parameters and the communication demodulation errors can be estimated by solving a convex problem. We also propose a way to improve the efficiency of the AN-based algorithm. The performance of these algorithms are demonstrated through extensive simulations, taking into account a variety of conditions concerning both the interferers and the respective channel states.

I. INTRODUCTION

Spectrum sharing creates radar-to-communication interference, while many existing coexistence methods depend on coordination and shared information. This paper addresses an un-coordinated setting with sparse, iterative joint interference estimation and data demodulation, using on-grid CS or an atomic-norm approach.

I. INTRODUCTION

  • Spectrum sharing can produce inter-system interference that degrades both systems, with high radar transmit power particularly able to degrade communication performance.Existing mitigation options include interference mitigation, precoding or spatial separation, and waveform design.Source: §I · p. 1
  • Many prior approaches rely on coordinated systems sharing information such as waveform or channel information, with some also requiring centralized design.A blind null-space estimation extension is described, though adaptively adjusted radar waveforms can pose real-time processing difficulties and some performance loss for radar.Source: §I · p. 1
  • Here, a communication receiver faces potentially active, fully overlapping radars with unknown, possibly time-varying waveforms, delays, and channels, and must estimate and remove interference while demodulating data.The receiver is a single TX/RX pair and cannot rely on collaborative broadcasting.Source: §I · p. 1
  • The proposed joint process exploits sparse interference representations and sparse demodulation errors, using on-grid CS or atomic-norm constraints in convex estimation algorithms.Atomic-norm methods address continuous parameters such as delays, for which simple discretization can cause model mismatch; the paper also proposes an efficiency improvement and evaluates the algorithms in simulations.Source: §I · p. 2

II. PROBLEM FORMULATION

The formulation models the received communication data as a communication component mixed with radar interference and noise. The receiver must estimate unknown radar waveforms and delays jointly with communication demodulation errors under stated waveform and channel assumptions.

II. PROBLEM FORMULATION

  • The communication signal is modeled using data vector b, known matrix A, and channel matrix H; the model includes CDMA and OFDM examples.The receiver data vector is x = HAb.Source: §II · p. 2
  • The model assumes full bandwidth overlap, communication synchronization, a shared SRRC pulse, and radar waveforms in a known low-dimensional subspace with K ≪ N.For simplicity, each active radar has one path to the communication receiver; this is justified for narrow-band systems or direct-path-dominated interference.Source: §II · pp. 2–3
  • The receiver observes communication data mixed with radar interference and noise, without knowing active radar delays or waveforms, and must estimate interference alongside demodulation errors.The residual includes both radar interference and errors from mis-demodulated symbols, motivating joint estimation.Source: §II · p. 3

III. PROPOSED ALGORITHMS

The proposed algorithms jointly estimate radar interference and communication demodulation errors because the two are coupled in the received measurements. They exploit sparsity in both the radar signal representation and the error vector.

III. PROPOSED ALGORITHMS

  • Interference removal and data demodulation are coupled: estimating radar components requires estimating the communication error vector, and poor performance in either estimate harms the other.The algorithms use an iterative process based on the current demodulated symbols and residual measurements.Source: §III · pp. 3–4
  • The radar matrix X is modeled as a combination of J complex exponentials, with J ≪ N.The exponentials have unknown modulation.Source: §III · p. 4
  • The demodulation-error vector should ideally be zero, so each iteration seeks a sparse error vector with far fewer than N nonzero entries.The error vector is defined relative to the current demodulated block.Source: §III · p. 4

A. Joint Waveform Estimation and Demodulation Based on On-grid CS Algorithm

The on-grid CS algorithm approximates unknown radar delays on a discrete grid and jointly estimates radar components and demodulation errors. It uses sparse recovery, re-demodulation, and iteration to refine the symbol decisions.

A. Joint Waveform Estimation and Demodulation Based on On-grid CS Algorithm

  • The algorithm places candidate radar delays on a uniformly spaced grid and represents radar components with a sparse coefficient vector whose nonzero entries correspond to active delays.The grid is densely sampled when its number of points is sufficiently large.Source: A. Joint Waveform Estimation and… · p. 4
  • It estimates radar coefficients and communication errors jointly under sparsity constraints, replacing impractical ℓ0 constraints with ℓ1 regularization.The formulation relaxes a non-convex constellation-difference constraint to obtain a linear parameter-estimation problem.Source: A. Joint Waveform Estimation and… · p. 4
  • After estimation, the algorithm corrects symbols and re-demodulates using the communication constellation, then repeats the joint estimation and correction steps.Iterations continue until successive symbol estimates agree or a maximum iteration count is reached; the error vector is expected to become sparser as iterations proceed.Source: A. Joint Waveform Estimation and… · pp. 4–5
  • A modified CS-L1 version assumes known radar delays, using periodically transmitted training pilots for channel sensing to assess the cost of radar-position uncertainty.With known delays, the coefficient vector is not sparse, but the iterative procedure remains the same.Source: A. Joint Waveform Estimation and… · p. 5

B. Joint Waveform Estimation and Demodulation Based on Off-grid CS Algorithm

The CS-AN approach represents off-grid radar interference in a continuous domain, avoiding the grid mismatch possible with discrete CS. Its convex optimization is used iteratively for joint interference estimation and demodulation, while additional procedures recover radar delays and channel information.

B. Joint Waveform Estimation and Demodulation Based on Off-grid CS Algorithm

  • The atomic-norm formulation represents off-grid complex exponentials without discretization, avoiding the model mismatch that can degrade on-grid CS.The resulting optimization is convex and has per-iteration computational load O((N + K)^6) with an interior-point method.Source: B. Joint Waveform Estimation and… · pp. 5–6
  • Each iteration estimates the interference matrix and demodulation-error vector from the processed received signal through a convex optimization problem.The estimates are updated iteratively until the demodulated block stops changing.Source: B. Joint Waveform Estimation and… · pp. 5–6
  • Radar delays can also be estimated from rows of the recovered interference matrix using MUSIC or Prony’s method, followed by association across rows.The association procedure groups delay estimates within threshold δ and combines their associated waveform parameters.Source: B. Joint Waveform Estimation and… · p. 6
  • Active-radar delays can be identified where the dual polynomial has modulus λ, while γ-level dual magnitudes flag mistaken demodulation.The dual solution thus supports both delay estimation and demodulation-error detection.Source: B. Joint Waveform Estimation and… · pp. 6–7
  • The algorithm estimates channel states generated by active radar transmitters without requiring those radars to transmit pilots.The estimates concern the active radars’ delays and channel states.Source: B. Joint Waveform Estimation and… · p. 7

C. Example

In the example, the first iteration substantially clarifies the interference picture for both algorithms. CS-AN is superior to CS-L1 at both tested initial SER values, with CS-L1 producing more false alarms because of basis mismatch.

C. Example

  • The example tests an OFDM communication system with N = 65 against J = 2 active radars drawn from K = 3 potential radars.The initial symbol error rates are 0.1 and 0.3, and the comparison focuses on the first iteration.Source: C. Example · p. 7
  • The initial symbol error rate is a key factor because interference estimation and data demodulation are coupled.The example uses BPSK and randomly places incorrect symbols in the initial demodulated block.Source: C. Example · p. 7
  • The first iteration substantially clears the interference picture, and CS-AN outperforms CS-L1 at both initial SER values, 0.1 and 0.3.CS-L1 produces more false alarms because of basis mismatch.Source: C. Example · p. 7

IV. FAST ALGORITHM BASED ON NON-CONVEX FACTORIZATION

The section replaces off-the-shelf SDP solving for the atomic-norm method with a low-rank non-convex factorization and conjugate-gradient solver. Simulations report a solution close to CVX while reducing runtime compared with ADMM.

IV. FAST ALGORITHM BASED ON NON-CONVEX FACTORIZATION

  • Under the lemma's conditions, the SDP solution Z has rank equal to the estimated number of active radars, enabling low-rank factorization.The stated conditions include N ≥ 257 and a minimum separation between estimated delays of at least 4/(N−1).Source: §IV · p. 8
  • With a known upper bound on active radars, the method imposes a rank constraint, factorizes Z, and uses conjugate gradient on a smoothed objective.The ℓ1 term is approximated by a differentiable function, and Armijo line search makes the objective non-increasing toward a stationary point of the surrogate problem.Source: §IV · pp. 8–9
  • The stated N ≥ 257 condition is technical; simulations report that the result still holds without it.The condition originates from Theorem 1.3 of, while the simulations cited in find the result holds without it.Source: §IV · p. 8
  • If the upper bound on active radars is below the true count, some radars may go undetected, leaving interference uncanceled and reducing demodulation performance.An upper bound above the true count preserves performance but incurs additional computational cost.Source: §IV · p. 8
  • The factorized solver's per-iteration cost is O((N + K)^2 J̄), with J̄ much smaller than N + K, avoiding the classical eigenvalue-decomposition cost for large problems.The passage states that the lower per-iteration complexity can make the nonconvex approach faster than ADMM and projected gradient descent on large-dimensional problems.Source: §IV · pp. 7, 9
  • With w = 0.01, the proposed algorithm’s nonconvex solver uses CG to solve (35); after 1,000 iterations, its result is close to CVX and much faster than ADMM.Source: §IV · p. 9

A. Simulation Setup

The simulations model an OFDM communication receiver sharing spectrum with radar interferers, using specified channel and waveform settings. They assess radar estimation and waveform error, alongside demodulation comparisons.

A. Simulation Setup

  • The communication link uses OFDM with 10 kHz subcarrier spacing, 129 subcarriers across 1.29 MHz, 100 µs symbols, and QPSK.The simulations consider one communication receiver and multiple radars.Source: A. Simulation Setup · p. 9
  • The channel uses a Ricean model, while radars are point sources with fixed-magnitude, random-phase path gains; SIR is defined as communication-signal power over radar-interference power.The fading fluctuation is varied through ρ; the default setting is ρ/σ_h = 3 unless otherwise specified.Source: A. Simulation Setup · p. 9
  • Radar pulses use Gaussian random codes of length 32 in a K = 5-dimensional waveform subspace, with delays randomly drawn from 10 µs to 70 µs.The code entries have distribution CN(0, 1/N′) for the first N′ = 32 entries and are zero thereafter.Source: A. Simulation Setup · pp. 9–10
  • CS-L1 uses a grid of cardinality 4N, while CS-AN uses the proposed nonconvex solver with J̄ = 10, μ = 0.01, and ϱ = 5.The nonconvex solver stops when the gradient norm is below 0.01.Source: A. Simulation Setup · p. 10
  • Evaluation reports delay RMSE and relative waveform MSE, conditioning the delay RMSE on correct radar identification within T/4.“Iteration 0” is the raw-data demodulator output used as the initial point for the iterative algorithms.Source: A. Simulation Setup · p. 10

B. Performance

The simulations compare SER and interference-estimation accuracy across iteration, SNR, SIR, radar count, dictionary size, and channel conditions. CS-AN generally outperforms CS-L1, with performance depending on SNR, SIR, and the number of active radars.

B. Performance

  • At 15 dB SNR, both algorithms converge monotonically, but CS-AN reaches a smaller SER; convergence usually occurs within 10 iterations.Fig. 5 plots SER against iteration number.Source: B. Performance · p. 10, Figure 5 · p. 11
  • With known radar delays, SER improves visibly, at the cost of pilot-signal overhead and greater complexity from system coordination.The known-delay case is a modified CS-L1 algorithm.Source: B. Performance · p. 10
  • At 8 dB SNR, fading limits both algorithms’ performance gains; even interference-free OFDM/QPSK performs poorly, especially at low SIR.The authors attribute the low-SIR result to coupling between data demodulation and interference removal.Source: B. Performance · p. 10
  • At SNRs above 14 dB, CS-AN significantly outperforms CS-L1, especially at SIR = -5 dB, where basis mismatch limits CS-L1 error correction.Both methods restore the original OFDM/QPSK performance faster than Iteration 0 as SIR increases.Source: B. Performance · p. 10
  • CS-AN yields better relative waveform MSE and delay RMSE than CS-L1, with higher SNR aiding estimation through improved demodulation.The estimation accuracy also depends on radar-signal power and demodulator performance, so it need not increase monotonically with SIR.Source: B. Performance · pp. 10–11
  • More active radars degrade both methods, with CS-AN ineffective at J ≥ 5; CS-AN is more robust to K and restores error probability near 10^-2 when ρ/σh > 2.Fig. 8 varies J and K; Fig. 9 examines the Ricean factor with J = 2, SIR = 0 dB, and SNR = 18 dB.Source: B. Performance · p. 11

VI. CONCLUSIONS

The paper proposes two joint waveform-estimation and demodulation algorithms for overlaid radar and communication systems, plus a faster solver for the atomic-norm version. Simulations show better SER than original demodulation; extending the algorithms to fast time-varying environments remains future work.

VI. CONCLUSIONS

  • The paper proposes an on-grid CS algorithm using ℓ1-norm and a CS algorithm using atomic norm for radar components and ℓ1-norm for demodulation errors.Both address joint waveform estimation and demodulation in overlaid communication and radar systems.Source: §VI · p. 11
  • A fast algorithm is derived to compute the solution to the formulated CS-AN problem.Source: §VI · p. 11
  • Simulations show that the proposed algorithms provide better SER than original demodulation; future work will consider fast time variation, channel-estimation mismatch, and radar Doppler.Source: §VI · p. 11

A. Accuracy Analysis of (10)

This block analyzes the approximation underlying (10), gives a condition for unique atomic decomposition, and derives gradients used in the computation.

A. Accuracy Analysis of (10)

  • The approximation error is small for large T′; the approximation is rigorous for 0 ≤ β ≤ 1/N and holds approximately for small to moderate excess bandwidths.For small β, the error is attributed to extending a finite-range integration to an infinite range.Source: A. Accuracy Analysis of (10) · pp. 11, 13

B. Proof of Lemma 2

  • The atomic decomposition is unique when N ≥ 257 and the separation between estimated delays satisfies Δτ′ ≥ 1/(N−1)/4.The proof relates the atomic norm to the trace expression for the recovered signal.Source: B. Proof of Lemma 2 · p. 13

C. The calculation of ∇v ˜ζ(V V H, v) and ∇V ˜ζ(V V H, v)

  • The implementation section rewrites the objective in quadratic form and derives its gradient with respect to Z, then uses the chain rule to obtain the gradient with respect to V.It also gives the direct gradient with respect to v.Source: C. The calculation of ∇v ˜ζ(V V H, v)… · pp. 13–14
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