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MIMO OFDM Dual-Function Radar-Communication Under Error Rate and Beampattern Constraints

Jeremy Johnston, Luca Venturino, Emanuele Grossi, Marco Lops, Xiaodong Wang

arXiv:2108.10555v2eess.SPcs.ITmath.IT

TL;DR

Joint waveform and receiver design remains a challenging problem for OFDM DFRC systems that sense and communicate simultaneously. The paper develops a communication-constrained radar design with a unified iterative solver, and numerical examples illustrate tradeoffs between radar objective, leakage constraints, and subcarrier balance.

  • Problem

    Joint design of DFRC waveforms and radar and communication receivers remains a challenging and debated problem, while prior OFDM-DFRC work mainly focused on single-antenna systems or specific massive-MIMO configurations.

    Source: A. Contribution of the Work · p. 2

  • Method

    The paper jointly designs OFDM-DFRC waveforms and radar and user filters, optimizing radar merit under power, beampattern-leakage, and user error-rate constraints with an alternating, convex-restricted minorization-maximization procedure.

    Source: A. Contribution of the Work · p. 2, §II · p. 3, §VI · p. 11

  • Results

    Across radar merits, lower δ or SCR lowers the objective, while smaller p better safeguards the least-favorable subcarrier SINR at the cost of the most-favorable one.

    Source: A. Examples · pp. 9–10

  • Takeaways & Limitations

    Numerical analysis illustrates achievable radar-communication tradeoffs and the effect of prior uncertainty about target strength.

    Source: §VI · p. 11

    Designing for users that simultaneously exploit direct and indirect signals remains an open problem requiring further investigation.

    Source: §VI · p. 11

Abstract

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In this work we consider a multiple-input multiple-output (MIMO) dual-function radar-communication (DFRC) system, which senses multiple spatial directions and serves multiple users. Upon resorting to an orthogonal frequency division multiplexing (OFDM) transmission format and a differential phase shift keying (DPSK) modulation, we study the design of the radiated waveforms and of the receive filters employed by the radar and the users. The approach is communication-centric, in the sense that a radar-oriented objective is optimized under constraints on the average transmit power, the power leakage towards specific directions, and the error rate of each user, thus safeguarding the communication quality of service (QoS). We adopt a unified design approach allowing a broad family of radar objectives, including both estimation- and detection-oriented merit functions. We devise a suboptimal solution based on alternating optimization of the involved variables, a convex restriction of the feasible search set, and minorization-maximization, offering a single algorithm for all of the radar merit functions in the considered family. Finally, the performance is inspected through numerical examples.

I. INTRODUCTION

The paper addresses joint design of DFRC waveforms and radar and communication receivers in an OFDM MIMO system, where radar performance must be balanced with communication quality. It formulates a communication-constrained radar resource-allocation problem and proposes a unified iterative approach for a broad family of radar objectives.

I. INTRODUCTION

  • Prior OFDM-DFRC studies mainly considered single-antenna systems or massive-MIMO configurations with separated antenna groups or uplink reception.The paper frames joint waveform and receiver design as a challenging, still-debated problem.Source: A. Contribution of the Work · p. 2

A. Contribution of the Work

  • The design jointly optimizes the transmitter and radar and user receivers for radar performance under average-power, beampattern-leakage, and per-user error-rate constraints.The error-rate constraints safeguard link quality while beampattern constraints limit power leakage in specified directions.Source: A. Contribution of the Work · p. 2
  • The unified approach covers radar objectives including SINR means, mutual information, Fisher information, detection probability, and relative entropy.Different directions can be inspected on different subcarriers, enabling multiple-target handling and performance balancing across subcarriers.Source: A. Contribution of the Work · p. 2
  • DPSK allows incoherent user receivers without full channel-state information and makes the transmit beampattern independent of the data symbol.The paper also models direct and indirect user paths with different statistical characterizations.Source: A. Contribution of the Work · p. 2, §II · p. 3
  • The nonconvex design is addressed with a suboptimal iterative algorithm combining alternating optimization, convex restriction, and minorization-maximization.The algorithm structure remains the same across the considered radar merit functions and monotonically increases the objective at each iteration.Source: A. Contribution of the Work · p. 2

B. Organization and Notation

  • The paper is organized from system description and resource-allocation formulation through solution, examples, conclusions, and an appendix; notation conventions are also introduced.The notation includes real and complex vector and matrix spaces and standard matrix and calculus operators.Source: B. Organization and Notation · pp. 2–3

II. SYSTEM DESCRIPTION

  • The system uses shared OFDM subcarriers to illuminate a prospective target while transmitting messages to multiple users.It comprises a DFRC transmitter, a co-located radar receiver, and multiple users, each equipped with a linear array.Source: §II · p. 3

A. Communication Side

  • The communication model permits a direct path and multiple indirect paths, with DPSK error probability decreasing as the direct-path ratio and SNR increase.Indirect-path responses are modeled as circularly symmetric Gaussian variables, while the channel response combines the paths reaching each user.Source: A. Communication Side · pp. 3–4

B. Radar Side

The radar model accounts for target returns, clutter, and disturbance on each OFDM subcarrier, then evaluates radar performance through an increasing function of per-subcarrier SINRs. The framework can also extend to simultaneous inspection of multiple directions on one subcarrier, though that generalization is omitted from the exposition.

B. Radar Side

  • The received radar signal includes the target response, clutter from J independent scatterers, and disturbance on each subcarrier.The target and clutter responses and disturbance are modeled as independent circularly symmetric Gaussian variables under a Swerling I assumption.Source: B. Radar Side · p. 4
  • Radar design evaluates an increasing merit function of the SINRs across the K subcarriers.The function may be concave or admit a concave minorizer at each point.Source: B. Radar Side · p. 4
  • The radar may inspect different directions on different subcarriers, and the method can extend to multiple directions on one subcarrier using separate receive filters.The same-subcarrier multi-direction extension is omitted from the exposition.Source: B. Radar Side · p. 4

C. Transmit Beampattern

The formulation shapes transmitted power to illuminate target and user directions while limiting leakage toward protected directions. It jointly optimizes transmit codes and radar and user filters for radar merit, subject to power, beampattern, and user error-rate constraints.

C. Transmit Beampattern

  • Transmit beampattern constraints limit power leakage toward protected directions, reducing possible interference to the radar receiver and nearby co-channel systems.The intended illumination includes prospective targets and connected users.Source: C. Transmit Beampattern · p. 4

III. PROBLEM FORMULATION

  • For a given transmit-power budget, the design maximizes a radar merit function subject to user error-rate and directional beampattern constraints.Transmit codes allocate power across subcarriers and shape the beampattern; receiver filters add degrees of freedom for interference management.Source: §III · p. 4, C. Transmit Beampattern · p. 4
  • The formulation can assign different users to different subcarriers, with OFDMA arising when a user's error constraints are removed from other subcarriers.The error-rate and leakage parameters are bounded by δ_k,ℓ ∈ [0, 1] and ϵ_k,m ∈ (0, 1/2).Source: §III · p. 4
  • The resulting optimization problem is non-convex, and the paper proposes a procedure for computing a suboptimal solution.The procedure is developed in Section IV.Source: §III · p. 5

A. Examples of Radar Merit Functions

The radar merit-function family covers objectives based on subcarrier SINRs, target information, delay estimation, detection probability, and sequential testing. Its functions include quasi-arithmetic means and weighted sums, with concavity or concave minorization supporting the design framework.

A. Examples of Radar Merit Functions

  • The p-th power mean of subcarrier SINRs is one example of a radar merit function, with smaller p placing more emphasis on the smallest SINR.The family also includes the geometric mean as a quasi-arithmetic mean.Source: A. Examples of Radar Merit Functions · p. 5
  • A weighted sum of mutual information between received signals and target responses is relevant to target classification.This example is described as increasing and concave.Source: A. Examples of Radar Merit Functions · p. 5
  • A weighted sum of Fisher information for delay estimation relates to target-ranging accuracy and admits a concave linear minorizer.The passage states that each summation term is convex and lower-bounded by its tangent line.Source: A. Examples of Radar Merit Functions · p. 5
  • A weighted sum of likelihood-ratio-test detection probabilities is another merit function, with each term admitting a quadratic lower bound.The detection probability is specified per subcarrier.Source: A. Examples of Radar Merit Functions · p. 5
  • Weighted relative entropies between the two hypotheses can control the average sample count in a sequential probability ratio test.The test is considered with given detection and false-alarm probabilities.Source: A. Examples of Radar Merit Functions · pp. 5–6

B. Handling the Error Probability Constraint

The error-rate constraint simplifies when the direct-path parameter κ is forced to either zero or infinity, corresponding to indirect-only or direct-only propagation. Enforcing those cases restricts the feasible design and can yield a suboptimal solution.

B. Handling the Error Probability Constraint

  • κ_k,m = ∞ represents a direct-only path without fading, whereas κ_k,m = 0 represents indirect-only paths with Rayleigh fading.These are the two propagation cases used to simplify the error-probability analysis.Source: B. Handling the Error Probability… · p. 6
  • Receive-filter nulling can force κ_k,m to zero or infinity when sufficient degrees of freedom are available.Zero-forcing the direct signal requires at least two user antennas; zero-forcing indirect signals requires more receive antennas than indirect paths.Source: B. Handling the Error Probability… · p. 6
  • Constraining κ_k,m to {0, ∞} can lead to a suboptimal solution because it adds a restriction to the optimization problem.The effects of changing the transmit and receive filters on the error probability are generally difficult to assess.Source: B. Handling the Error Probability… · p. 6
  • Forcing κ_k,m to zero or infinity simplifies the error-probability constraint into a lower bound on the required SNR.The simplified problem's solution is a lower bound to that of the original problem.Source: B. Handling the Error Probability… · p. 6

IV. PROPOSED ALGORITHM

The proposed suboptimal solution uses cyclic alternating maximization over three blocks of variables. Its objective does not decrease at each iteration, but convergence to a local or global optimum is not guaranteed.

IV. PROPOSED ALGORITHM

  • The algorithm cyclically maximizes the objective over {u_k}, {w_k}, and {w_k,m}, holding the other blocks fixed at their previous values.It starts from a feasible point.Source: §IV · p. 6
  • The objective value is not decreased at each iteration.Source: §IV · p. 6
  • Convergence to a local or global optimum is not guaranteed because not all subproblems are solved optimally.Source: §IV · p. 6

A. Update of the Transmit Code

The transmit-code update addresses a non-concave objective and non-convex constraints by introducing auxiliary variables and convex restrictions. It uses either a convex optimization step or minorization-maximization, depending on the objective.

A. Update of the Transmit Code

  • The transmit-code problem has a non-concave objective and a non-convex constraint, motivating auxiliary variables and convex restrictions.The restriction replaces constraints C3 and C5 with tighter forms based on the previous iteration's variables.Source: A. Update of the Transmit Code · p. 7
  • For positive x, g(u,x) = x u^H Ψ u is convex when Ψ is positive semidefinite, supporting the restriction of C5.Source: A. Update of the Transmit Code · p. 7
  • When f is concave, the restricted problem is convex, and its solution is feasible for the original problem without decreasing its objective.Source: A. Update of the Transmit Code · p. 7
  • When f is not concave but admits a concave minorizer, minorization-maximization solves convex subproblems and generates feasible points with a nondecreasing objective sequence.Because this step is nested within alternating maximization, the inner iterations may stop after one or a few steps; a useful surrogate balances close fit with computational simplicity.Source: A. Update of the Transmit Code · p. 7

B. Update of the Radar Receive Filters

The receive-filter updates use closed-form or eigenvector-based choices for radar and user filters. The user-filter procedure also compares feasible candidates, while a proposition gives feasibility conditions for constraint C3.

B. Update of the Radar Receive Filters

C. Update of the User Receive Filters

  • When Problem (30) is feasible, an optimal user receive filter is proportional to a maximum-eigenvalue eigenvector of Ξ^d.The procedure compares candidates associated with Ξ^d and Ξ^i when both are feasible, selecting the one with lower error probability.Source: C. Update of the User Receive Filters · p. 8
  • Proposition 5 gives a necessary condition for C3 feasibility and a sufficient condition when Rank{U_k} = N_t.Source: C. Update of the User Receive Filters · p. 8

D. Selection of the Starting Point

The starting-point procedure alternates updates to find a feasible initialization for Algorithm 1. Numerical analysis then evaluates the method under a specified OFDM, channel, radar, and beampattern setup.

D. Selection of the Starting Point

  • If the starting-point problem yields t ≥ 1, its variables provide a feasible initialization for Algorithm 1; the NP-hard problem is handled by alternating maximization.The updates alternate between {t, u_k} and {w_k,m}.Source: D. Selection of the Starting Point · p. 8

V. NUMERICAL ANALYSIS

  • The numerical setup uses 20 dBW shared-subcarrier power, a 2 GHz center frequency, and 100 kHz subcarrier spacing, with two direct and two indirect users unless stated otherwise.It also specifies radar directions, clutter, protected directions, and algorithm initialization and stopping parameters.Source: §V · pp. 8–9

A. Examples

The examples show radar–communication tradeoffs across radar objectives and operating conditions, alongside transmit and receive patterns shaped to serve radar and users. The conclusion summarizes a unified iterative design and identifies simultaneous use of direct and indirect signals by users as an open problem.

A. Examples

  • Radar performance drops sharply when the user error-rate constraint falls below 10^-5, but is only marginally restrained when it exceeds 10^-3.5.This tradeoff is reported across the arithmetic-mean SINR, detection-probability, and relative-entropy objectives; lower δ or SCR also lowers the objective by consuming degrees of freedom to reduce leakage.Source: Figure 1 · p. 9, A. Examples · p. 9
  • The transmit pattern peaks toward the target and nulls protected directions, while indirect users receive significant power and direct users are served by sidelobes.For ǫ = 10^-5, the stated required SNR is approximately 10 dB for direct users and 40 dB for indirect users; radar and receive patterns also cooperate to mitigate clutter.Source: A. Examples · p. 9, Figure 2 · p. 10, Figure 3 · p. 10
  • Reducing ǫ from 10^-2 to 10^-5 changes the transmit pattern from a target-centered mainlobe to three similarly high lobes aimed at the target and indirect users.Some power directed toward the radar in the ǫ = 10^-2 case is reallocated to indirect users at ǫ = 10^-5.Source: A. Examples · p. 9, Figure 4 · p. 10
  • Designing for a nominal SCR of −20 dB yields a robust radar-SINR design, whereas a 20 dB design degrades rapidly as true SCR falls.The −20 dB design prioritizes clutter nulling, making true clutter strength less consequential.Source: Figure 5 · p. 10, A. Examples · p. 10

VI. CONCLUSIONS

  • The proposed non-convex design is solved suboptimally by alternating optimization, convex restriction, and minorization-maximization, for a broad family of radar merit functions.The design selects transmit waveforms and radar and user receive filters under radiated-power, user-error-rate, and directional beampattern constraints.Source: §VI · pp. 10–11

A. Examples

  • Adding indirect users causes a severe radar-SINR loss; indirect users require more physical resources than direct users to counteract channel fading.The comparison varies user count and considers L = 0, 2, and 4 indirect paths.Source: A. Examples · p. 10, Figure 6 · p. 11
  • Smaller p in the p-power mean better safeguards radar operation on the least favorable subcarrier, at the cost of performance on the most favorable one.The comparison uses p = −20, −10, −5, −1, and 1.Source: A. Examples · p. 10, Figure 7 · p. 11

VI. CONCLUSIONS

  • Simultaneous use of direct and indirect signals by users remains an open problem for future investigation.Future work may also consider reconfigurable intelligent surfaces and differential space-time-frequency coding.Source: §VI · p. 11

APPENDIX

The appendix supplies proofs for Propositions 1, 3, 4, and 5. The included fragments establish concavity conditions for Proposition 1 and convexity for Proposition 3.

APPENDIX

  • The appendix provides proofs of Propositions 1, 3, 4, and 5.Source: APPENDIX · p. 11

A. Proof of Proposition 1

  • For Proposition 1, the proof relates concavity of g to convexity of γ′/γ′′ under the stated derivative-sign conditions.The proof also uses inverse-function differentiation and the Cauchy–Schwarz inequality.Source: A. Proof of Proposition 1 · pp. 11–12

B. Proof of Proposition 3

C. Proof of Proposition 4

The proofs characterize optimal communication filters and conditions relevant to satisfying constraint C3. They identify nonzero projected steering vectors as necessary in the direct- and indirect-path cases.

C. Proof of Proposition 4

  • When κk,m = ∞, the optimal wk,m maximizes SNRk,m over the stated constraint set.The associated bound is tight when wk,m is proportional to the eigenvector for the largest eigenvalue of Ξd; the κk,m = 0 case follows similarly.Source: C. Proof of Proposition 4 · p. 12

D. Proof of Proposition 5

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