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Joint Waveform and Filter Designs for STAP-SLP-based MIMO-DFRC Systems

R. Liu, M. Li, Q. Liu, A. L. Swindlehurst

arXiv:2112.08849v2eess.SP

TL;DR

Existing MIMO-DFRC waveform designs often neglect Doppler and simplify away clutter, limiting their treatment of target detection in cluttered settings. This paper combines STAP and symbol-level precoding in a joint waveform and receive-filter design. Simulations report clutter suppression and satisfactory target detection, alongside communication benefits from CI-based precoding.

  • Problem

    Many MIMO-DFRC designs focus on spatial second-order waveform statistics, neglect Doppler, and assume clutter-free sensing; prior SLP work also leaves temporal waveform design and clutter suppression unaddressed.

    Source: §I · p. 2

  • Method

    The paper jointly optimizes transmit waveforms and a receive filter using STAP and CI-based SLP, with MM-neADMM algorithms for waveform constraints.

    Source: §I · p. 2

  • Results

    Simulations show effective signal-dependent clutter suppression and satisfactory target detection; the CI-based approach also improves performance over the non-CI approach and exploits interference in dense-user cases.

    Source: §V · pp. 11–12

  • Takeaways & Limitations

    The simulations demonstrate the advantages of combining STAP and CI-based SLP for MIMO-DFRC and the effectiveness of the joint-design algorithms.

    Source: §VI · p. 12

Abstract

from arXiv · show

Dual-function radar-communication (DFRC), which can simultaneously perform both radar and communication functionalities using the same hardware platform, spectral resource and transmit waveform, is a promising technique for realizing integrated sensing and communication (ISAC). Spacetime adaptive processing (STAP) in multi-antenna radar systems is the primary tool for detecting moving targets in the presence of strong clutter. The idea of joint spatial-temporal optimization in STAP-based radar systems is consistent with the concept of symbol-level precoding (SLP) for multi-input multi-output (MIMO) communications, which optimizes the transmit waveform for each of the transmitted symbols. In this paper, we combine STAP and SLP and propose a novel STAP-SLP-based DFRC system that enjoys the advantages of both techniques. The radar output signal-to-interference-plus-noise ratio (SINR) is maximized by jointly optimizing the transmit waveform and receive filter, while satisfying the communication quality-of-service (QoS) constraint and various waveform constraints including constant-modulus, similarity and peak-to-average power ratio (PAPR). An efficient algorithm framework based on majorization-minimization (MM) and nonlinear equality constrained alternative direction method of multipliers (neADMM) methods is proposed to solve these complicated non-convex optimization problems. Simulation results verify the effectiveness of the proposed STAP-SLP-based MIMO-DRFC scheme and the associate algorithms.

I. INTRODUCTION

The introduction identifies gaps in MIMO-DFRC waveform design under clutter and motivates combining STAP with symbol-level precoding. The system jointly designs transmitted waveforms and a receive filter for radar detection while carrying multiuser information.

I. INTRODUCTION

  • Existing MIMO-DFRC designs often ignore Doppler and assume clutter-free sensing; prior SLP work also omits temporal waveform properties and spatial-temporal filtering.Source: §I · p. 2
  • STAP exploits spatial and temporal degrees of freedom to suppress clutter, while symbol-level precoding uses transmitted symbols to turn multiuser interference into constructive signal power.Source: §I · p. 2
  • The proposed system jointly optimizes the transmit waveform and receive filter to maximize radar output SINR under communication QoS and radar waveform constraints.Source: §I · p. 2

II. SYSTEM MODEL AND PROBLEM FORMULATION

  • A colocated base station uses STAP for target detection amid strong signal-dependent clutter and CI-based SLP to carry user symbols on the same waveforms.Source: §II · p. 3

A. Radar Received Signal and Radar Performance Metric

  • The radar design maximizes output SINR, with waveform choices also shaped by hardware-related constant-modulus constraints, PAPR limits, and similarity to a reference waveform.Source: A. Radar Received Signal and Radar… · p. 4

B. User Received Signal and Communication Performance Metric

The communication model carries each user's symbols in the radar transmit waveform using CI-based SLP. Its QoS constraints place received signals in constructive regions of the constellation.

B. User Received Signal and Communication Performance Metric

C. Problem Formulation

The problem jointly selects the transmit waveform and receive filter to maximize radar output SINR under communication QoS, power, and waveform constraints. A closed-form filter solution concentrates the problem on waveform design, which remains non-convex.

C. Problem Formulation

  • The design maximizes radar output SINR while meeting communication QoS, total-power, and one of the specified waveform constraints.The feasible waveform set may be non-convex.Source: C. Problem Formulation · p. 5
  • For a fixed transmit waveform, the filter optimization becomes an MVDR problem with a closed-form optimal solution.The solution is substituted into the original problem to form a concentrated transmit-waveform problem, rather than using cyclic joint optimization.Source: C. Problem Formulation · p. 5
  • The concentrated waveform problem is non-convex, so the paper uses MM and neADMM to obtain tractable subproblems and solve them iteratively.The stated sources of non-convexity are the objective and waveform constraints.Source: C. Problem Formulation · p. 6

III. CONSTANT-MODULUS WAVEFORM DESIGN

This section develops an MM-neADMM algorithm for constant-modulus waveform design. The constant-modulus constraint makes the separate total-power constraint unnecessary, and the approach is stated to generalize to other waveform constraints.

III. CONSTANT-MODULUS WAVEFORM DESIGN

  • The section develops an MM-neADMM algorithm for constant-modulus waveform design in the STAP-SLP-based MIMO-DFRC system.The approach is stated to generalize to the other waveform constraints discussed earlier.Source: §III · p. 6
  • The constant-modulus constraint is substituted into the concentrated problem to formulate the waveform-design problem.Source: §III · p. 6
  • The total-power constraint is dropped because the constant-modulus constraints naturally satisfy it.Source: §III · p. 6

A. Reformulation

The reformulation makes the waveform dependence of the objective more explicit and amenable to optimization. It uses reshaping and permutation relationships, alongside a low-rank eigenvalue representation of the inner CCM matrix.

A. Reformulation

  • Because the matrix objective obscures its dependence on the waveform vector and is not amenable to optimization, it is rewritten in terms of that vector.Source: A. Reformulation · p. 6
  • Reshaping and permutation-matrix relationships are used to rewrite terms involving X and the waveform vector.The derivation uses vectorized and reshaped matrix forms to express Xu0.Source: A. Reformulation · p. 6
  • The inner CCM matrix is positive semidefinite and is expressed through its nonzero eigenvalues and corresponding eigenvectors.Its rank is generally small in practice.Source: A. Reformulation · p. 6

B. MM Transformation

The MM transformation upper-bounds the non-convex objective with a tractable surrogate around the current iterate. The resulting waveform problem still has a non-convex constant-modulus equality constraint, which the method addresses directly with neADMM.

B. MM Transformation

  • At each iterate, MM constructs a tractable upper-bounding surrogate for the non-convex objective around the current solution.The surrogate is minimized in the next iteration, using a first-order linear expansion for the stated concave function.Source: B. MM Transformation · p. 7
  • The surrogate-based transmit-waveform problem is formulated at the current iterate before the equality-constrained solution step.Source: B. MM Transformation · p. 7
  • The resulting problem remains non-convex because of the constant-modulus equality constraint, so the method uses neADMM to handle it directly.The cited rationale is that relaxing the equality and projecting the solution back causes significant performance loss.Source: B. MM Transformation · p. 7

C. neADMM Transformation

The formulation introduces an auxiliary variable to separate constraints, then uses a feasibility indicator and augmented Lagrangian to make alternating updates possible.

C. neADMM Transformation

  • An auxiliary variable y decouples the convex constraint (44b) from the non-convex constraint (44c) with respect to x.Source: C. neADMM Transformation · p. 7
  • The feasible region of inequality constraints (45b) and (45c) is represented by set C and its indicator function IC(x).Source: C. neADMM Transformation · p. 7
  • The augmented Lagrangian uses penalty parameter ρ and dual variables λ and µ, enabling alternating updates of x, y, λ, and µ.Source: C. neADMM Transformation · p. 8

D. Block Update

The block-update procedure alternates between solving a convex SOCP for x and separable elementwise updates for y, then updates the dual variables. The resulting algorithm initializes the variables and repeats these updates until convergence.

D. Block Update

  • The x update is transformed into a convex SOCP whose optimum can be obtained with off-the-shelf optimization methods.The Hooke-Jeeves Pattern Search method with a Lagrangian dual is also identified as an efficient solution approach.Source: D. Block Update · p. 8
  • With x fixed, the non-convex y update separates into MNNt elementwise subproblems because its objective contains an absolute-value operation.Source: D. Block Update · p. 8
  • Algorithm 1 initializes x from (57), sets y to x and both dual variables to zero, then iterates the x, y, λ, and µ updates until convergence.Source: D. Block Update · p. 8

E. Summary, Initialization, and Complexity Analysis

The constant-modulus design alternates waveform and dual-variable updates until the radar SINR increase falls below a threshold, then computes the optimal receive filter. The section also specifies a QoS-based initialization and identifies the x update as the computational bottleneck.

E. Summary, Initialization, and Complexity Analysis

  • The waveform updates stop when the relative increase in radar output SINR falls below a convergence threshold, after which the optimal receive filter is calculated.Source: E. Summary, Initialization, and… · p. 8
  • Initialization maximizes the minimum communication-user QoS using available transmit power, with a relaxed convex power constraint enabling efficient solution.The initialization problem is convex and can be solved by an interior point method or CVX.Source: E. Summary, Initialization, and… · pp. 8–9
  • The x update dominates waveform-design computational complexity; the y and dual-variable updates each require O{MNNt} operations.Source: E. Summary, Initialization, and… · p. 9

IV. GENERALIZATIONS TO OTHER WAVEFORM CONSTRAINTS

The MM-neADMM framework is generalized to other waveform constraints, with the supplied derivation detailing the PAPR-constrained design. That design introduces an auxiliary variable and alternates SOCP and closed-form updates.

IV. GENERALIZATIONS TO OTHER WAVEFORM CONSTRAINTS

  • The proposed MM-neADMM framework is generalized to waveform designs with PAPR or similarity constraints.Source: §IV · p. 9

A. PAPR-Constrained Waveform Design

  • For the PAPR-constrained design, the total-power equality constraint couples all elements of x, requiring modifications to the constant-modulus algorithm.Source: A. PAPR-Constrained Waveform Design · p. 9
  • The PAPR algorithm decouples constraints with auxiliary y and iterates an SOCP x update, a y update, and a λ update.Its initialization is convex; the closed-form y and λ updates have complexity O{MNNt}, lower than the x update.Source: A. PAPR-Constrained Waveform Design · p. 9

B. Constant-Modulus and Similarity-Constrained Waveform Design

The design incorporates constant-modulus and similarity constraints into joint waveform optimization, using MM surrogates and neADMM updates. Simulations report target-centered mainlobes and clutter-ridge nulls, alongside trade-offs in radar SINR as communication and waveform constraints vary.

B. Constant-Modulus and Similarity-Constrained Waveform Design

  • The constrained design follows the constant-modulus procedure, adding convex similarity constraints and using neADMM to update the waveform, auxiliary variable, and dual variables.The waveform update becomes an MNNt-dimensional SOCP.Source: B. Constant-Modulus and… · pp. 9–10

V. SIMULATION RESULTS

  • Radar SINR rises monotonically over iterations; PAPR converges fastest and reaches the highest SINR, while the constant-modulus-and-similarity constraint converges slowest and yields the lowest.The stricter the waveform constraint, the slower convergence.Source: §V · p. 10
  • Radar SINR decreases as communication QoS increases; the PAPR-constrained design performs best, the combined constant-modulus-and-similarity design worst, and constant-modulus alone lies between them.The combined design also matches a reference waveform to retain other desired radar-sensing properties.Source: §V · pp. 10–11
  • At Γ = 10dB, three users receive the required QoS at about 2dB radar-performance loss, while CI-based SLP improves performance by about 2dB over non-CI.As the number of users increases, the gap between CI-based and non-CI approaches grows.Source: §V · p. 11
  • Higher transmit power and more antennas improve radar SINR, while a significant SINR notch near zero normalized Doppler reflects difficulty distinguishing slowly moving targets from strong clutter.The antenna gain is attributed to increased waveform diversity and beamforming gains.Source: §V · pp. 11–12
  • Radar SINR increases with PAPR threshold and similarity level; as the threshold rises, performance gains and gaps between schemes become smaller.Increasing similarity level trades reference-waveform properties against radar output SINR.Source: §V · p. 12, Figure 10 · p. 12
  • Target-centered mainlobes and deep nulls spanning the clutter ridge indicate effective clutter suppression and satisfactory target detection.The space-time cross-ambiguity plots use 100 clutter patches per cell.Source: §V · p. 12

VI. CONCLUSIONS

The paper investigates joint transmit-waveform and receive-filter design for STAP-SLP-based MIMO-DFRC, maximizing radar output SINR under waveform and communication-QoS constraints. Simulations demonstrate the system’s advantages and the effectiveness of the proposed algorithms; practical implementation issues remain for future work.

VI. CONCLUSIONS

  • The design maximizes radar output SINR under waveform and constructive-interference constraints that guarantee satisfactory communication QoS.The paper jointly designs the transmit waveform and receive filter for MIMO-DFRC systems.Source: §VI · p. 12
  • The proposed MM- and neADMM-based algorithms solve the resulting non-convex optimization problems.Source: §VI · p. 12
  • Simulations demonstrate the advantages of STAP and CI-based SLP for MIMO-DFRC and the effectiveness of the joint-design algorithms.Further work will address low-complexity algorithms, hardware imperfections, and robust designs.Source: §VI · p. 12

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