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Statistical Ambiguity-Function-Aware Joint Beamforming and Subcarrier Allocation for MIMO-OFDM ISAC under Random Symbols

Wonill Kim, Yumin Kim, Hyeonsu Lyu, Hyun Jong Yang

arXiv:2610.21888v1eess.SP

TL;DR

Random communication symbols make OFDM-ISAC ambiguity sidelobes data-dependent, and existing objectives do not directly capture their statistical behavior. The paper characterizes the DP-AF and develops a CV-based joint beamforming and subcarrier-allocation design. At flat spectrum it reports higher PSLR with limited sum-rate loss, while the CV formulation approaches the direct constraint’s average rate with lower runtime.

  • Problem

    Random communication symbols make OFDM-ISAC ambiguity sidelobes data-dependent, while common sensing objectives do not explicitly capture their statistical sidelobe behavior.

    Source: §I · p. 1

  • Method

    The paper derives the MUMT MIMO–OFDM DP-AF and ESL, characterizes PSLR optimality through flat target-wise directional power, and uses CV-dependent bounds for a convex sufficient constraint.

    Source: §I · p. 2

  • Results

    At flat spectrum, PSLR gains were 9.4–11.0 dB over communication-only beamforming with 2.4–5.9% sum-rate loss; the CV design achieved 99.7% of baseline average sum-rate and runtime reductions of 10.35–12.15×.

    Source: C. Computational Burden and… · p. 9, §VI · p. 10

  • Takeaways & Limitations

    In this MUMT MIMO–OFDM setting, zero-Doppler PSLR is the AF metric that remains controllable by beamforming, and flat per-target directional power characterizes its maximum for κ > 1.

    Source: §I · p. 2, §VI · p. 10

Abstract

from arXiv · show

Ambiguity function (AF) characterizes the delay--Doppler sidelobe structure that governs target resolution and masking in radar sensing. In ISAC, however, random communication symbols make the AF data-dependent, complicating its characterization and incorporation into transmit design. This raises a fundamental question: which AF characteristic can be effectively controlled through beamforming in a practical multiuser multitarget (MUMT) MIMO--OFDM system? To answer this, we statistically characterize the discrete-periodic AF (DP-AF) and derive an exact closed-form expected sidelobe level (ESL). We observe that nonzero-Doppler ESL cuts are delay-flat and the full two-dimensional integrated sidelobe level ratio (ISLR) is invariant to beamforming, whereas the zero-Doppler peak-to-sidelobe ratio (PSLR) remains controllable. We accordingly formulate an AF-aware joint beamforming and subcarrier-allocation problem with target-wise PSLR and illumination constraints. Since the direct PSLR constraint is intractable, we characterize its global optimum and prove that, for symbol kurtosis greater than one, the PSLR is maximized if and only if each target-wise directional-power profile is flat across subcarriers. We then introduce the coefficient of variation (CV) to measure deviation from this flatness and derive tight CV-dependent PSLR bounds. The lower bound yields a convex sufficient constraint for a desired PSLR, enabling an efficient reformulation. Simulations show $9.4$--$11.0$ dB worst-case PSLR gains over communication-only beamforming at the flat-spectrum point with a $2.4$--$5.9\%$ sum-rate loss. The CV-relaxed formulation achieves $99.7\%$ of the average sum-rate of the original PSLR-constrained problem, while reducing runtime by factors of $10.35$--$12.15$ under the considered scenarios.

I. INTRODUCTION

Random communication symbols make OFDM-ISAC ambiguity sidelobes data-dependent, while existing sensing objectives do not directly capture their statistical behavior. The paper characterizes the MUMT MIMO–OFDM DP-AF and develops a joint design using beamforming and subcarrier allocation to control the beamformer-sensitive PSLR.

I. INTRODUCTION

  • Random communication symbols make the AF vary across transmissions, and existing SINR, beampattern-error, MI, and CRB objectives do not explicitly capture this statistical sidelobe behavior.Unfavorable symbol realizations can produce sidelobes that obscure weak targets or create spurious peaks.Source: §I · p. 1
  • The paper identifies zero-Doppler PSLR as the beamformer-sensitive AF metric and proposes controlling it through joint beamforming and subcarrier allocation.Its analysis finds delay-flat nonzero-Doppler ESL cuts and beamformer-invariant two-dimensional ISLR.Source: §I · p. 2
  • For κ > 1 and nonzero target illumination, PSLR is maximized if and only if each target-wise directional-power profile is flat across subcarriers.The design incorporates target-wise PSLR and minimum-illumination constraints.Source: §I · p. 2
  • CV-dependent PSLR bounds yield a convex sufficient constraint that supports an efficient reformulation of the direct PSLR constraint.The proposed alternating-optimization method achieves 99.7% of the direct PSLR-constrained baseline’s average sum-rate while reducing runtime by factors of 10.35–12.15 in the stated scenarios.Source: §I · p. 2

II. SYSTEM MODEL

  • The system uses a shared data-bearing OFDM waveform to serve K single-antenna users and illuminate L far-field targets over N subcarriers, without a superimposed dedicated radar waveform.The model combines MIMO beamforming with exclusive OFDMA subcarrier assignment.Source: §II · p. 3

1) Downlink MIMO–OFDM ISAC Transmit Signal:

B. MIMO DP-AF

The section defines a sampled periodic MIMO ambiguity function and derives its expected sidelobe level for random-symbol MIMO–OFDM signals. It sets up the ESL as the statistical quantity used to analyze sidelobe behavior across delay and Doppler.

1) General MIMO DP-AF:

  • The DP-AF analysis considers integer delay–Doppler offsets after cyclic-prefix removal, assuming the maximum round-trip delay does not exceed the CP duration.The matched-filter output is formed for hypothesized target parameters with the azimuth assumed known.Source: 1) General MIMO DP-AF · p. 3
  • The general MIMO DP-AF is defined from the transmitted antenna signals and target response, with the receive-array factor canceling in normalized AF ratios.The formulation uses periodic time- and frequency-shift operations.Source: 1) General MIMO DP-AF · pp. 3–4

2) Sidelobe Level of AF:

  • Because random communication symbols cause AF fluctuations, the expected sidelobe level is defined as E[|χ[τ, ν; θ]|²] and derived in closed form.The expression is stated in terms of per-subcarrier directional powers; the section also gives a separate expression for the zero-Doppler cut.Source: 2) Sidelobe Level of AF · p. 4

C. AF-based Sensing Metrics

The expected sidelobe level analysis shows that delay–Doppler coupling is absent and ISLR is beamformer-invariant, while zero-Doppler PSLR remains controllable. This motivates a joint design with target-wise PSLR and illumination constraints.

C. AF-based Sensing Metrics

1) DD-coupling Analysis:

  • Every nonzero-Doppler ESL cut is delay-flat, so directional MIMO beamforming introduces no delay–Doppler coupling.The delay-dependent ESL term contributes only on the zero-Doppler cut.Source: 1) DD-coupling Analysis · p. 4

2) Closed-form Derivation of ISLR:

  • The full two-dimensional ISLR is fixed at N −1, independent of the beamformer, and is omitted from optimization.This invariance extends the AF volume property to direction-dependent MIMO transmit beamforming.Source: 2) Closed-form Derivation of ISLR · p. 4

3) Closed-form Derivation of the Zero-Doppler PSLR:

  • Range sidelobes are evaluated on the zero-Doppler ESL cut, where the section defines the target-angle PSLR.Nonzero-Doppler cuts contain no range-sidelobe structure because they are delay-independent.Source: 3) Closed-form Derivation of the… · p. 4

D. Problem Formulation

  • The proposed joint beamforming and subcarrier-allocation design imposes target-wise PSLR and minimum-illumination constraints.The illumination floor prevents target illumination from collapsing while shaping the scale-invariant PSLR constraint.Source: D. Problem Formulation · p. 5

III. PSLR OPTIMALITY AND CV-BASED REFORMULATION

The direct PSLR constraint is computationally difficult, but its optimum has a flat-spectrum characterization for κ > 1. The paper uses CV-dependent bounds to replace the direct constraint with a convex sufficient condition.

A. Intractability of the PSLR Constraint

  • The direct PSLR constraint remains computationally costly after alternating optimization separates allocation and beamforming.Expanding it into L(N −1) SOC constraints substantially increases constraint count; retaining its compact DC form requires repeated SCA solves.Source: A. Intractability of the PSLR Constraint · p. 5

B. Flat Directional Power for Optimal AF Shaping

  • For κ > 1 and positive target illumination, PSLR is globally maximized if and only if directional power is flat across subcarriers.The corresponding maximum is 1 + N/(κ −1); for κ = 1, PSLR has no finite upper bound.Source: B. Flat Directional Power for Optimal… · p. 5
  • The coefficient of variation measures directional-power variation, and the PSLR admits tight CV-dependent bounds.For even N, the theorem gives a two-sided bound; both analytic bounds decrease strictly as CV increases in the reported comparison.Source: B. Flat Directional Power for Optimal… · p. 6
  • Requiring the lower PSLR bound to meet the desired threshold yields a sufficient CV constraint expressible as one convex SOC constraint.The CV constraint is convex in the beamforming covariance matrices because directional power is affine in them.Source: B. Flat Directional Power for Optimal… · p. 6

C. Proposed CV-Based Reformulation

The proposed reformulation replaces the direct PSLR threshold with a sufficient CV-based constraint. Its design still handles discrete allocation and rank constraints separately.

C. Proposed CV-Based Reformulation

  • The proposed problem replaces the direct PSLR threshold with a CV upper-bound condition.The sufficient condition bounds CV by the inverse lower-bound function evaluated at the desired PSLR threshold.Source: C. Proposed CV-Based Reformulation · p. 6
  • The reformulated problem retains binary subcarrier allocation and rank-one constraints, handled by alternating updates and rank-one recovery.For each fixed allocation, the rank-one constraints are relaxed before recovery.Source: C. Proposed CV-Based Reformulation · p. 6
  • The illumination floor reserves transmit power and spatial degrees of freedom toward target directions, while the CV surrogate's certification uncertainty is bounded by a PSLR bracket width.The section evaluates the illumination-floor impact and the surrogate's performance degradation separately.Source: C. Proposed CV-Based Reformulation · p. 6

IV. PROPOSED JOINT DESIGN ALGORITHMS

The joint design uses alternating optimization to update allocation and beamforming. It combines dual-based subcarrier assignment with a CV-constrained beamforming update and a direct-PSLR baseline.

A. Subcarrier Allocation via Lagrangian-Dual Decomposition

  • With beamforming fixed, dual decomposition relaxes binary allocation and decouples the optimal subcarrier assignment across subcarriers.A greedy repair addresses QoS shortfalls, followed by feasibility checking after rank-one recovery.Source: A. Subcarrier Allocation via… · p. 7

B. Proposed CV-Constrained Beamforming Optimization

  • For fixed allocation, the CV-constrained beamforming update is a convex conic program because its CV constraint is SOC-representable.Rank-one beamformers are recovered by dominant-eigenvector extraction, using Gaussian randomization if needed.Source: B. Proposed CV-Constrained Beamforming… · p. 7

C. Direct PSLR-Constrained Baseline

  • The direct PSLR-constrained baseline uses an inner SCA loop, adding iterative convexified SDP solves to its beamforming update.The proposed CV-based update instead uses a single SDP solve per beamforming update.Source: C. Direct PSLR-Constrained Baseline · p. 7

V. SIMULATIONS AND NUMERICAL RESULTS

The simulations compare the proposed CV-constrained design with communication-only, direct PSLR-constrained, and conventional sensing baselines. They show a substantial worst-case PSLR gain at the flat-spectrum point, with a limited sum-rate loss, while the CV reformulation closely tracks the direct baseline.

A. Simulation Setup

  • The evaluation uses frequency-domain Rayleigh fading channels and a half-wavelength uniform linear array in a downlink serving multiple single-antenna users and target directions.The compared schemes include communication-only, flat-spectrum sensing-optimal, direct PSLR-constrained, CRB-inspired, and MI-inspired designs.Source: A. Simulation Setup · pp. 7–8

B. PSLR Pareto Frontiers

  • 9.4–11.0 dB worst-case PSLR gains over communication-only beamforming at the flat-spectrum point cost 2.4–5.9% of its sum-rate.The Pareto frontiers average 100 channel realizations; Fig. 3 plots sum-rate against worst-case PSLR.Source: B. PSLR Pareto Frontiers · p. 8, Figure 3 · p. 8
  • The proposed CV-reformulated scheme achieves 99.7% of the direct PSLR-constrained baseline’s average sum-rate across four cases.The direct baseline’s average-rate advantage is 0.18–0.69 bps/Hz, with a maximum pointwise average difference of 1.74 bps/Hz.Source: B. PSLR Pareto Frontiers · pp. 8–9
  • The MI-inspired design reaches a maximum PSLR of 7.36–9.96 dB, while the CRB-inspired design’s sum-rate drops to 60.40–92.90 bps/Hz.The results characterize both conventional baselines as yielding only marginal PSLR gains.Source: B. PSLR Pareto Frontiers · p. 9
  • At comparable sum-rate operating points, the proposed design suppresses localized delay–Doppler peaks and lowers the worst-direction range-sidelobe envelope relative to conventional baselines.Fig. 4 shows representative AF surfaces, and Fig. 5 compares zero-Doppler range sidelobes across two realizations.Source: Figure 4 · p. 8, Figure 5 · p. 8, B. PSLR Pareto Frontiers · p. 9

C. Computational Burden and Feasibility Under System Stress

The simulations assess computational feasibility, neural-network constraint enforcement, and the effect of target illumination floors. The CV-based design maintains feasibility under the tested stress cases, and the illumination floor trades communication resources for target-directed power.

C. Computational Burden and Feasibility Under System Stress

  • The CV-based formulation meets the 3-s runtime budget for every tested instance, while the direct baseline’s average feasibility is 40.9%, 0.0%, and 64.8% in S1–S3.Across the displayed sweep, CV-based runtime reductions are factors of 10.68, 12.15, and 10.35 in S1–S3; the 3-s deadline is a computational benchmark, not a real-time protocol requirement.Source: C. Computational Burden and… · p. 9, Figure 7 · p. 9

D. Neural Network-Based Evaluation of Constraint Representations

  • The CV model satisfies its constraint on all 2,200 validation channels, whereas direct-PSLR feasibility rises from near zero at CVmax ≤ 0.3 to 100% at CVmax ≥ 0.8.The CV model’s feasible sum-rates scale from 58.19 to 84.68 bps/Hz; its deterministic safety layer preserves predicted illumination levels.Source: D. Neural Network-Based Evaluation of… · pp. 9–10

E. Impact of the Illumination Constraint

  • For illumination floors n = 0, 1, 2, and 3, the sum-rate ranges are 163.61–165.93, 163.50–165.89, 159.51–162.60, and 148.64–152.32 bps/Hz, respectively.Higher floors increase target-directed power while consuming power and spatial degrees of freedom otherwise available to communication; CV controls subcarrier-wise variation while Pdes controls mean directional energy.Source: E. Impact of the Illumination Constraint · p. 10, Figure 8 · p. 10, Figure 9 · p. 10

VI. CONCLUSION

  • At the flat-spectrum point, PSLR improves by 9.4–11.0 dB over communication-only beamforming with a 2.4–5.9% sum-rate loss.The design also attains 99.7% of the direct PSLR-constrained baseline’s average sum-rate while reducing runtime by factors of 10.35–12.15.Source: §VI · pp. 10–11

APPENDIX A PROOF OF LEMMA 1

The proof evaluates the expected squared magnitude of the discrete MIMO ambiguity function by handling expectations over independent random symbols. It reduces the resulting sums to a closed-form ESL expression.

APPENDIX A PROOF OF LEMMA 1

  • The derivation rewrites the AF using the unitary DFT representation and expresses ESL as the expectation of its magnitude squared.The Kronecker-product representation is used to handle the expectation over the random data symbols.Source: Appendix A · p. 11
  • Independence, zero mean, unit variance, and kurtosis κ restrict the symbol-moment expectation to specified index conditions.For a ≠ b, the nonzero case requires a = i and b = j and contributes Pa(θ)Pb(θ); for a = b, it requires i = j.Source: Appendix A · p. 11
  • The proof evaluates the remaining double sums using DFT orthogonality and modular index cases, yielding a circular-correlation term and the final closed-form ESL.The third-term sum is nonzero only when a − i − ν ≡ 0 (mod N), producing the circular-correlation form Σ_a Pa(θ)Pa−ν(θ).Source: Appendix A · p. 11

APPENDIX B DERIVATION OF THE TOTAL AF ENERGY

The appendix uses DFT orthogonality and circular indexing to derive the total expected AF energy, then establishes that the full two-dimensional ISLR is invariant to the directional power profile. It also identifies the equality condition for the PSLR maximum.

APPENDIX B DERIVATION OF THE TOTAL AF ENERGY

  • DFT orthogonality and circular indexing simplify the total expected AF energy, proving that the full two-dimensional ISLR equals N−1 for every directional power profile.Source: Appendix B · pp. 11–12

APPENDIX C PROOF OF THEOREM 1

  • The Cauchy–Schwarz equality condition requires equal directional powers across all subcarriers, which also makes every nonzero-delay correlation term zero.The global maximum is achieved exclusively by a flat target-wise directional-power profile, or CVl = 0.Source: §2 · p. 12, Appendix C · p. 12

APPENDIX D PROOF OF THEOREM 2

The proof expresses nonzero-delay sidelobes through the DFT of the zero-mean deviation sequence, then derives PSLR bounds by minimizing or maximizing the largest sidelobe peak. The equality profiles and lower-bound peak concentration depend on whether the subcarrier count is even or odd.

APPENDIX D PROOF OF THEOREM 2

  • The zero-mean deviation sequence determines nonzero-delay sidelobes; Parseval’s theorem supplies their total energy for the PSLR expression.The constant mean has zero DFT at nonzero delays.Source: Appendix D · p. 12
  • The upper bound follows by distributing total sidelobe energy uniformly over the N −1 nonzero delay bins, with equality for a frequency-domain impulse deviation.The equality profile is nonnegative provided CV_l ≤ N −1.Source: Appendix D · p. 12
  • The lower bound maximizes the largest sidelobe peak, constrained by conjugate symmetry of the real deviation sequence; peak concentration depends on N’s parity.The proof sets c = 1 for even N and c = 1/2 for odd N; it proceeds with c = 1 because practical OFDM subcarrier counts are typically even.Source: Appendix D · p. 12
  • The lower bound is attained when variance energy is maximally concentrated in delay, giving an alternating profile for even N and a sinusoidal profile for odd N.The even-N alternating profile is valid for CV_l ≤ 1; the odd-N profile is valid for CV_l ≤ 1/2.Source: Appendix D · p. 12

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