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Precision Coexistence: Angular-Spectral Region Constrained Waveform Design for MIMO-ISAC

Syed Ali Haider Shirazi, Taejoon Kim, David J. Love, Sergiy A. Vorobyov

arXiv:2610.29364v1cs.ITmath.IT

TL;DR

In congested electromagnetic environments, MIMO-ISAC waveforms must coexist with legacy systems occupying localized angle-frequency regions while maintaining sensing and communication performance. This paper develops a constrained dual-function waveform design and solves its nonconvex optimization with MM-ADMM. Simulations show localized protection with sensing and communication performance close to the unconstrained benchmark.

  • Problem

    MIMO-ISAC must protect legacy systems occupying specific frequency bands and angular sectors while maintaining radar SCNR and communication QoS.

    Source: §I · p. 1

  • Method

    The framework imposes localized angular-spectral leakage constraints on a unified MIMO-OFDM waveform and solves the nonconvex design using MM-ADMM.

    Source: §I · pp. 1–2

  • Results

    The proposed method creates localized attenuation in protected regions and achieves about 28.09 dB and 28.02 dB SCNR at Γ = 0 and 8 dB, respectively, versus 28.31 dB and 28.22 dB for the unconstrained benchmark.

    Source: §VI · pp. 5–6

  • Takeaways & Limitations

    Simulations indicate the framework provides effective coexistence with negligible sensing and communication performance loss, and increasing gains over spectral-only baselines under stringent communication QoS.

    Source: §VII · p. 6

    The study assumes perfect channel knowledge; the effect of channel estimation error on achievable suppression remains to be examined.

    Source: §VII · p. 6

Abstract

from arXiv · show

In this paper, we develop a dual-function waveform design framework for multiple-input multiple-output integrated sensing and communication (MIMO-ISAC) systems operating in congested electromagnetic environments. In such environments, the dual-function waveform must coexist with legacy systems, such as localized positioning sensors, maritime radar networks, and satellite communication links, that occupy specific frequency bands and angular sectors. To enable precision coexistence, we incorporate localized angular-spectral region constraints that suppress beamformed energy only within the angle-frequency regions occupied by each legacy system, while preserving the spatial degrees of freedom (DoFs) available outside those regions. The resulting waveform design problem is formulated as a nonconvex optimization problem and solved using a low-complexity majorization--minimization alternating direction method of multipliers (MM--ADMM) algorithm. Simulation results demonstrate that the proposed framework provides precise legacy-system protection while maintaining robust radar sensing and multi-user communication performance.

I. INTRODUCTION

MIMO-ISAC must jointly manage sensing, multiuser communication, and coexistence with legacy systems occupying localized frequency bands and angular sectors. The paper proposes a unified MIMO-OFDM waveform with localized angular-spectral constraints and an MM-ADMM solution.

I. INTRODUCTION

  • Legacy systems occupy specific frequency bands and angular sectors, requiring joint angular-spectral control while preserving radar SCNR and communication QoS.The motivating setting includes localized positioning sensors and radar altimeter receivers.Source: §I · p. 1
  • Conventional spectral notching protects entire bands regardless of direction, while spatial nulling methods are blind to environmental spectral variation.These approaches underuse MIMO spatial degrees of freedom or do not account for spectral variation.Source: §I · p. 1
  • The proposed unified MIMO-OFDM waveform selectively suppresses energy in legacy systems’ angle-frequency regions, preserving spatial degrees of freedom elsewhere.A common base station waveform supports sensing, communication, and coexistence, avoiding separate transmitters and waveforms.Source: §I · p. 1
  • A low-complexity MM-ADMM algorithm addresses the nonconvex design, with simulations showing localized protection and negligible radar SCNR and communication QoS degradation versus the unconstrained benchmark.SCNR is the primary sensing objective, and the waveform-dependent clutter covariance contributes to the optimization’s nonconvexity.Source: §I · p. 2

II. SYSTEM MODEL

The system model describes a monostatic multiuser OFDM MIMO-ISAC link and its radar returns, user channels, and constructive-interference communication requirements. The model stacks observations across OFDM symbols to represent space-time responses.

II. SYSTEM MODEL

  • The monostatic system uses collocated transmit and receive antennas, N orthogonal subcarriers, and L OFDM symbols per coherent processing interval.The radar receiver removes the cyclic prefix and applies an N-point DFT to obtain its complex baseband observations.Source: §II · p. 2
  • The target and clutter are parameterized by angle, Doppler, delay, and complex return coefficient, with half-wavelength ULA steering vectors for transmit and receive arrays.Target delay is τ0 = 2R0/ζ, where R0 is range and ζ is the speed of light.Source: §II · p. 2
  • Stacking received vectors across L symbols captures Doppler phase progression in a space-time steering vector for each subcarrier.The stacked observation has dimension C^(NRL×1), and its steering vector combines Doppler, receive-array, and transmit-array responses.Source: §II · p. 2
  • The radar observation includes a target return, clutter-scatterer returns, and noise; target and clutter share a channel structure.The target return includes a range-dependent phase rotation across subcarriers and a space-time response at the target angle and Doppler.Source: §II · p. 2
  • Each user’s known channel carries independently drawn Ω-PSK symbols, and symbol-level precoding uses constructive interference to meet per-user decision margins.The QoS requirement δn,ℓ,k ≥ γk applies across subcarriers, OFDM symbols, and users; γk depends on the QoS threshold and receiver noise variance.Source: §II · pp. 2–3

III. ANGULAR–SPECTRAL REGION CONSTRAINTS

The coexistence constraint limits aggregate radiated energy within each legacy system’s continuous protected angle-frequency region. Spatial and spectral exposure are represented by matrices combined into a joint quadratic energy measure.

III. ANGULAR–SPECTRAL REGION CONSTRAINTS

  • Each legacy system’s protected region is a continuous angular sector paired with a frequency interval, and aggregate radiated energy there is limited.The allowable interference-energy threshold is ηp > 0.Source: §III · p. 3
  • The angular exposure matrix Bp integrates transmit steering-vector correlation across the protected sector.Bp is positive semidefinite and captures angular exposure for legacy system p.Source: §III · p. 3
  • The spectral matrix integrates transmitted energy over the protected continuous band, accounting for the unitary IFFT mapping from subcarrier coefficients to time-domain samples.RI,p = F_N SI,p F_N^H is positive semidefinite and captures in-band energy and spectral sidelobe leakage from finite OFDM symbol duration, unlike subcarrier-selective masking.Source: §III · p. 3
  • The Kronecker product combines spatial and spectral matrices so the waveform’s quadratic form measures energy within the protected angle-frequency region.The resulting joint matrix is Qp = RI,p ⊗ Bp.Source: §III · p. 3

IV. PROBLEM FORMULATION

The problem maximizes radar SCNR subject to communication QoS, angular-spectral leakage, PAPR, and transmit-power constraints. Its nonconvexity motivates a low-complexity MM-ADMM algorithm.

IV. PROBLEM FORMULATION

  • The design uses a linear radar filter to produce per-subcarrier SCNR, with clutter-plus-noise covariance entering the SCNR expression.For a fixed transmit waveform, the MVDR filter maximizes the SCNR.Source: §IV · p. 3
  • The feasible set enforces constructive-interference QoS for all users and bounds leakage into protected regions using Qp and threshold ηp.The formulation also constrains time-domain peak amplitude per antenna and normalizes total transmit power to PT.Source: §IV · pp. 3–4
  • A global PAPR constraint limits instantaneous time-domain amplitude to mitigate nonlinear distortions in radar RF chains.The stated bound is Apapr = (1 + ε)PT /(LNNT ), with tolerance ε ≥ 0.Source: §IV · p. 4
  • The resulting optimization is highly nonconvex, and the paper develops a low-complexity MM-ADMM algorithm to solve it.The nonconvexity includes waveform dependence in the clutter covariance.Source: §IV · pp. 3–4

V. PROPOSED ALGORITHM

The waveform-dependent clutter covariance is the primary challenge in solving the nonconvex problem; the proposed response is a double-loop MM-ADMM algorithm.

V. PROPOSED ALGORITHM

  • The primary solution challenge is the waveform-dependent clutter covariance R_n.It is identified as the main difficulty in solving the nonconvex problem.Source: §V · p. 4
  • The proposed algorithm uses a double-loop iterative framework based on MM and ADMM.Source: §V · p. 4
  • The framework is developed for computationally efficient implementation.The section introduces preliminaries to support that implementation.Source: §V · p. 4

A. Preliminaries

The preliminaries isolate waveform dependence in the clutter covariance and form a tractable SCNR surrogate; low-rank updates reduce the cost of repeated covariance inversion.

A. Preliminaries

  • The MVDR receive filter is defined for a given waveform, and Lemma 1 enables a standard MM identity to construct a tractable radar-SCNR surrogate.The surrogate separates target gain at the target angle and Doppler from spatial clutter leakage through a fixed receive filter.Source: A. Preliminaries · p. 4
  • Lemma 1 rewrites waveform dependence in the clutter covariance as an explicit quadratic form of the per-subcarrier waveform vector.The waveform vector stacks the per-symbol transmit vectors across all L symbols.Source: A. Preliminaries · p. 4
  • Computing the surrogate terms requires repeated inversion of the full-rank clutter covariance, identified as a major bottleneck.The low-rank inverse methodology addresses this online inversion burden.Source: A. Preliminaries · p. 4
  • Online inversion complexity drops from O((NRL)^3) to O(C(NRL)^2), with C ≪ NRL, using a low-rank methodology.The approximate inverse is computed online at each MM-ADMM iteration via Sherman–Morrison recursion.Source: A. Preliminaries · p. 4

B. Proposed MM-ADMM Solution

The solution pairs an outer MM update with inner ADMM iterations, exploiting separability across OFDM symbols while enforcing the per-symbol constraints and total power condition.

B. Proposed MM-ADMM Solution

  • An outer MM loop and inner ADMM loop form the proposed double-loop framework.The MM surrogate and constraints separate across the L OFDM symbols, yielding independent per-symbol subproblems.Source: B. Proposed MM-ADMM Solution · p. 4
  • Each per-symbol feasible set combines CI-QoS, angular-spectral coexistence, PAPR, and power inequality constraints.The separate total-power equality constraint is nonconvex and is handled using a consensus variable.Source: B. Proposed MM-ADMM Solution · p. 4
  • The inner ADMM recursion updates the per-symbol waveform, consensus variable, and scaled dual variable.The waveform update is a convex quadratic program over the feasible set and is solved by CVX.Source: B. Proposed MM-ADMM Solution · pp. 4–5
  • The figure presents angular-spectral power maps for the proposed, spectral-only, and unconstrained methods, plus a radar-SCNR versus communication-QoS plot.Source: Figure 2 · p. 5

VI. SIMULATION RESULTS

The simulations show that localized angular–spectral protection preserves radar SCNR better than spectral-only suppression as communication QoS tightens, while Proposed closely tracks Unconstrained. Antenna-sweep results show SCNR increasing with array size, with Proposed remaining above Spectral-only.

VI. SIMULATION RESULTS

  • The comparison uses three protected angle–frequency regions with −20 dB suppression and includes Proposed, Spectral-only, Unconstrained, and Radar-only benchmarks.The Radar-only benchmark removes communication constraints.Source: §VI · p. 5
  • At Γ = 5 dB, Proposed forms deep notches in three protected regions, while Spectral-only suppresses full bands and Unconstrained leaves them unprotected.The spectral-only suppression spans the full 180° azimuth extent.Source: §VI · p. 5
  • Across 50 channel trials, Proposed Γ⋆ exceeds 8 dB; Spectral-only falls below 8 dB in some trials, requiring transmit power above 15 W.The feasibility test uses CI, PAPR, transmit-power, and −20 dB angular–spectral leakage constraints.Source: §VI · pp. 5–6
  • From 14 to 20 antennas, Proposed SCNR rises from approximately 29.33 to 30.99 dB, remaining below Unconstrained but above Spectral-only.At those endpoints, Radar-only achieves approximately 29.36 and 31.30 dB, respectively.Source: Figure 3 · p. 6, §VI · p. 6
  • At Γ = 8 dB, Proposed SCNR is approximately 28.02 dB versus 26.22 dB for Spectral-only, a 1.79 dB loss for the latter.Proposed closely tracks Unconstrained, which achieves approximately 28.22 dB at Γ = 8 dB.Source: §VI · p. 6

VII. CONCLUSION AND FUTURE WORKS

The paper concludes that its MM–ADMM waveform design supports localized legacy-system coexistence while retaining sensing and communication performance. Future work will examine larger dimensions, channel-estimation error, and cell-free massive MIMO.

VII. CONCLUSION AND FUTURE WORKS

  • The MM–ADMM framework uses spatial DoFs to confine beamformed leakage to legacy systems’ localized angle–frequency regions.Simulations show effective coexistence with negligible sensing and communication performance loss, and increasing gains over spectral-only baselines under stringent communication QoS requirements.Source: §VII · p. 6
  • Scaling to larger space–frequency dimensions may be bottlenecked by the per-symbol convex subproblem; the study also assumes perfect channel knowledge.The authors plan to study suppression under channel-estimation error and extend the formulation to cell-free massive MIMO with joint constraints across distributed access points.Source: §VII · p. 6
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