PaperlayerView exact revision on arXiv

Source-linked AI summary

Joint Design of Overlaid Communication Systems and Pulsed Radars

Le Zheng, Marco Lops, Xiaodong Wang, Emanuele Grossi

arXiv:1703.10184v2cs.IT

TL;DR

The paper addresses how a communication system and pulsed radar can share bandwidth despite asymmetric, intermittent interference. It jointly optimizes the radar waveform and communication encoding through a compound-rate objective subject to radar SINR and power constraints, deriving closed-form solutions for two scattering models and assessing their performance. The conclusions identify both the potential and the practical boundaries of the co-existing architecture.

  • Problem

    The paper studies how overlapping-bandwidth radar and communication systems can jointly preserve radar SINR while safeguarding communication rate under intermittent radar interference.

  • Method

    The paper jointly optimizes the radar waveform and communication encoding matrix by maximizing a compound rate under radar SINR, interference, noise, and power constraints.

  • Results

    The paper derives closed-form transmit policies for coherent and incoherent scattering and characterizes communication rates with and without interference.

  • Takeaways & Limitations

    Co-design of the radar waveform and communication encoding is key to guaranteeing performance for both systems in the shared-band architecture.

  • Takeaways & Limitations

    The setup assumes slow-time radar coding and frame synchronism between the radar and communication receiver.

Abstract

from arXiv · show

The focus of this paper is on co-existence between a communication system and a pulsed radar sharing the same bandwidth. Based on the fact that the interference generated by the radar onto the communication receiver is intermittent and depends on the density of scattering objects (such as, e.g., targets), we first show that the communication system is equivalent to a set of independent parallel channels, whereby pre-coding on each channel can be introduced as a new degree of freedom. We introduce a new figure of merit, named the {\em compound rate}, which is a convex combination of rates with and without interference, to be optimized under constraints concerning the signal-to-interference-plus-noise ratio (including {\em signal-dependent} interference due to clutter) experienced by the radar and obviously the powers emitted by the two systems: the degrees of freedom are the radar waveform and the afore-mentioned encoding matrix for the communication symbols. We provide closed-form solutions for the optimum transmit policies for both systems under two basic models for the scattering produced by the radar onto the communication receiver, and account for possible correlation of the signal-independent fraction of the interference impinging on the radar. We also discuss the region of the achievable communication rates with and without interference. A thorough performance assessment shows the potentials and the limitations of the proposed co-existing architecture.

I. INTRODUCTION

The paper develops a joint design framework for radar and communication systems sharing bandwidth, exploiting intermittent radar interference at the communication receiver. It models the communication link and formulates co-design of the radar waveform and communication encoding under radar SINR and power constraints.

  • Motivation: Prior work generally protected radar detection and estimation performance by designing waveforms that limit interference to communications.The paper instead combines radar detection objectives with explicit protection of the communication rate.
  • Motivation: Radar interference at the communication receiver is intermittent, depending on waveform duty cycle and the number of reflecting objects.The communication system is exposed to spurious radar reflections, whereas the radar can use range-gating and clutter-reduction devices.
  • System model: The system model treats the radar as an amplitude-modulated pulse train whose pulse duration matches the communication symbol duration.The radar and communication systems share bandwidth, with K range cells defined over each pulse repetition interval.
  • System model: Radar detection is constrained by communication interference, signal-dependent clutter, and additive signal-independent receiver noise.The radar observation model includes a scattering coefficient for unwanted reflectors and a covariance model for signal-dependent interference at the communication system.
  • System model: Radar echoes from targets and other reflectors produce delayed interference at the communication receiver, with unknown interfering objects and random reflection coefficients.The model includes direct-path interference as a possible special case, while unknown target or reverberation reflections are identified as especially damaging.
  • Problem formulation: Because communication transmission can use a covariance matrix as an additional design degree of freedom, the paper jointly designs its codebook with the radar waveform.This extends independent Gaussian coding by introducing a precoding-related covariance choice in the co-existing architecture.

A. Performance measures

The paper evaluates coexistence using radar SINR and a compound communication rate that combines rates with and without radar interference. Intermittent interference is modeled through parallel channels, making communication precoding part of the design.

  • Radar measure: Radar performance is measured by SINR, which is also linked to detection-oriented Kullback–Leibler divergences under the stated observation model.Maximizing SINR is equivalent to maximizing those divergences when g is deterministic and a is Gaussian.
  • Communication measure: The communication rate interpolates between an additive white Gaussian noise channel and an interference channel with covariance σ_v^2 IN + S R_f,k S^H.The interference is modeled as Gaussian for design purposes, using a worst-case reflector covariance R_f.
  • Communication measure: The compound rate is a convex combination of the communication rates with and without interference, weighted by β.The weighting parameter satisfies β ∈ [0, 1].
  • Communication measure: Radar interference is represented by independent parallel communication channels whose interference indicators occur with probability α.When β = α and the interference-free rates are identical across channels, N CR equals the conditional mutual information per channel.
  • Communication measure: The compound rate differs from the mutual information per channel use by less than 1/N, although it is not directly achievable with the proposed encoding scheme.For sufficiently large N, the mutual information per channel use represents the maximum achievable transmission rate.

B. Problem Formulation

The design jointly chooses the communication covariance matrix and radar waveform to maximize compound rate while meeting radar SINR and transmit-power constraints. The formulation assumes a narrow-band, flat-fading coexistence model but can be extended to resolvable multipath.

  • Joint design: The optimization variables are the communication covariance matrix R_x and radar waveform s, with compound rate as the objective.The constraints include minimum radar SINR, radar and communication power limits, and R_x ⪰ 0.
  • Constraints: The radar SINR threshold ρ_min and power limits P_r and P_c constrain the joint design.The power constraints are expressed as average waveform energy and average trace of R_x.
  • Model assumptions: The narrow-band assumption makes the channel flat-fading and models each reflector as producing a single resolvable path.With resolvable multipath, the communication model can be updated by treating the environment as denser while leaving single-range-cell SINR unchanged.
  • Model assumptions: The radar vector s represents a slow-time pulse code, although an analogous discrete-time model can represent fast-time sub-pulse coding.In the fast-time interpretation, a pulse contains N sub-pulses and s contains their amplitudes.

III. WAVEFORM OPTIMIZATION

Waveform optimization is analyzed through two limiting interference models: coherent and incoherent scattering. The paper focuses on white radar noise in these cases and then extends the discussion to colored noise and achievable rate regions.

  • Interference models: A general closed-form solution is difficult for arbitrary reflector covariance R_f, so the analysis considers coherent and incoherent interference as limiting cases.Coherent interference includes coherent targets, whereas incoherent interference includes scintillating scattering.
  • Interference models: The principal analysis assumes white noise at the radar receiver for both interference cases.Colored radar noise is treated separately, including its relationship to the two limiting cases.
  • Optimization formulation: The communication rate can be reformulated using the determinant identity det(I_N + p q^H) = 1 + p^H q.This reformulation is inserted into the compound-rate objective and constrained optimization problem.
  • Optimization formulation: The joint design maximizes compound rate subject to radar SINR, radar-power, communication-power, and positive-semidefinite covariance constraints.Separate optimizations with fixed s or fixed R_x are discussed for completeness, while joint optimization is the main result.

1) Fixed communication codebook:

With the communication covariance fixed, the radar waveform is optimized under the radar SINR and power constraints. Under joint optimization, the communication covariance and radar waveform exhibit a structure similar to the fixed-system solutions.

  • The radar waveform optimization admits a solution only under a condition involving the smallest eigenvalue of the fixed communication covariance.
  • The optimal radar waveform transmits with minimum compatible power in the least-interfered direction of the communication signal space.
  • When the radar waveform is fixed, the communication covariance allocates power between N−1 interference-free eigenvectors and the radar-signal direction.
  • The fraction of communication power assigned to the interfered direction is determined by γ∗N and depends on system parameters and constraints.
  • Under joint optimization, the communication covariance and radar waveform have a structure similar to the corresponding fixed-system solutions.

B. Incoherent interference

For incoherent interference, the communication rate is reformulated within a constrained optimization that includes radar SINR and power limits. Closed-form solutions are available in key cases, but practical waveform constraints and incoherent colored-noise interference limit analytic tractability.

  • Incoherent interference: The optimization constrains radar SINR, radar power, and communication covariance while maximizing the communication objective under incoherent interference.
  • Incoherent interference: The incoherent-interference model represents communication links affected by scintillating interferers and can also model uniformly distributed normalized Doppler shifts.
  • Incoherent interference: A solution exists only under a stated feasibility condition, after which optimal communication covariance and radar waveform expressions are obtained.
  • Incoherent interference: The resulting optimized compound rate matches the coherent-scattering case, but the eigenvector freedom is lost.
  • Incoherent interference: A single-pulse waveform may violate practical PAPR limits; adding a PAPR constraint removes the available closed-form solution and requires approximation or numerical optimization.
  • Colored noise: With colored noise, coherent-interference optimization depends on the smallest eigenvalue and eigenvector of the noise covariance, whereas incoherent interference requires numerical methods.

D. Achievable communication rates

The achievable-rate analysis characterizes communication performance through the rate region and compound rate under radar SINR, interference, power, and waveform constraints. Results show how scattering density, clutter, interference strength, PAPR, and pulse number shape optimized communication performance.

  • Achievable-rate region: The achievable rate region determines transmission policies for merit functions increasing in the rates with and without interference.The selected policy lies on the boundary point touched by the largest level set of the merit function.
  • Parameter effects: Larger scattering density β reduces optimized compound rate, while larger SCR can sharply restrict feasibility because the radar SINR constraint may become unsatisfiable.This behavior is reported for N = 8, INR = 10 dB, α = β, and white radar noise.
  • Design comparisons: The disjoint design is nearly optimal at small minimum SINR but becomes catastrophic at larger minimum SINR, motivating joint optimization over orthogonal and noncooperative alternatives.The comparison uses N = 8, INR = 10 dB, SCR = 20 dB, and β = α = 0.1.

V. CONCLUSIONS

The paper frames radar–communication coexistence through a compound rate that captures intermittent radar interference and uses joint waveform and encoding design to protect both systems. It also identifies frame synchronism as a scope assumption and points toward fast-time coding and timing-free extensions.

  • V. CONCLUSIONS: The compound rate is a conditional mutual information for parallel channels that may or may not experience radar interference.It reflects the intermittent nature of interference from a pulsed radar.
  • V. CONCLUSIONS: The framework allows arbitrary correlation of radar interference and considers perfectly coherent and totally incoherent radar interference.
  • V. CONCLUSIONS: Jointly designing the radar waveform and communication encoding matrix under a radar SINR constraint is key to protecting both coexisting systems.
  • V. CONCLUSIONS: The setup assumes slow-time radar coding and frame synchronism, while fast-time coding and timing-free schemes are identified as extensions.

APPENDIX

The appendix establishes supporting lemmas and connects radar SINR optimization with divergence maximization. It then derives waveform directions and feasibility conditions for the optimization problems.

  • APPENDIX: The appendix contains proofs connecting the radar SINR to Kullback–Leibler divergences and deriving solutions for Problems (20), (29), and (35).
  • APPENDIX: For a positive-definite matrix, the relevant eigenvalue bound reaches equality when the vector aligns with the corresponding minimum-eigenvalue eigenvector.
  • APPENDIX: The two Kullback–Leibler divergences increase with SINR, so maximizing SINR is equivalent to maximizing those divergences.
  • APPENDIX: For fixed waveform energy, choosing the waveform along the minimum-eigenvalue eigenvector yields the largest SINR and compound rate, subject to feasibility.
  • APPENDIX: The appendix uses eigenvalue decompositions and convexity arguments to reduce covariance optimization to lower-dimensional scalar problems.

2) Fixed radar waveform:

With the radar waveform fixed, the communication covariance is optimized through its eigenvalues and orientation. Jensen’s inequality equalizes the remaining eigenvalues, while convex scalar optimization determines the feasible optimum.

  • 2) Fixed radar waveform:: For fixed smallest eigenvalue γN, Jensen’s inequality maximizes the objective by setting γi = (NPc−γN)/(N−1) for i = 1,…,N−1.
  • 2) Fixed radar waveform:: The reduced objective is continuously differentiable and convex, so its solution is either a critical point or a boundary point determined by the associated polynomial roots.
  • 2) Fixed radar waveform:: The covariance orientation can be chosen so the radar waveform aligns with the smallest-eigenvalue direction while preserving feasibility and not decreasing the compound rate.
  • 2) Fixed radar waveform:: The optimization is feasible only when the stated condition holds, with γN constrained to the interval [0, γ̄].
  • 2) Fixed radar waveform:: The same optimization structure yields the optimal covariance matrix and radar waveform in (26) and (27) under the stated feasibility condition.

C. Solution to Problem (29)

For the correlated-noise case, the solution rotates the radar waveform and communication covariance toward the minimum-eigenvalue direction of the radar-noise covariance. The remaining eigenvalues and waveform energy follow the same reduced optimization with the minimum eigenvalue substituted for the white-noise variance.

  • C. Solution to Problem (29): For the fixed-eigenvalue reduction, the remaining eigenvalues are equalized as γi = (NPc−γN)/(N−1) for i = 1,…,N−1.
  • C. Solution to Problem (29): Rotating the radar waveform into the smallest-eigenvalue direction of the interference covariance preserves feasibility and can improve the compound rate.
  • C. Solution to Problem (29): The covariance orientation is selected with a unitary matrix whose last column is the minimum-eigenvalue eigenvector vN.
  • C. Solution to Problem (29): The remaining covariance eigenvalues and waveform norm are obtained by solving Problem (55) after replacing σ2w with the smallest eigenvalue φN.
  • C. Solution to Problem (29): The resulting optimal covariance matrix and radar waveform are those in (37) and (38), respectively, when the feasibility condition is satisfied.

E. Proof of Lemma 1

The proof constructs candidate transmit vectors and covariance matrices from minimum-eigenvalue eigenvectors, then verifies the constraints in (39), including the SINR constraint. It concludes that the second construction achieves a rate at least as large as the first.

  • Candidate constructions: The first candidate sets ŝ = √ϵu_N, where u_N is the eigenvector associated with the smallest eigenvalue of R′.This construction is used to satisfy the constraints in (39).
  • Constraint verification: Both candidate constructions satisfy the constraints in (39), including the SINR constraint, using the stated eigenvalue inequalities and Lemma 2.The proof invokes γ_N ≤ σ²_w and inequality (67) in the second verification.
  • Candidate constructions: A second candidate uses the smallest-eigenvalue eigenvector v_N of M, with s′′ = √ϵv_N and x = U*Γ(U*)^H.U* is unitary and has v_N as its last column.
  • Rate comparison: R′′_1 = R̂_1 ≥ R′_1, so the second construction provides a rate no smaller than the first.The comparison is stated for the corresponding rate quantities.
Loading 1703.10184v2…