Detection-Aware Rate--CRB Characterization for Resource Allocation in ISAC Systems

A. Related Work and Research Gap

• Prior rate–CRB formulations do not explicitly account for detection reliability or model how a frame’s detection outcome affects estimation availability and tracking updates.The paper identifies this as a gap in formulations that often treat detection and estimation as separate tasks.Source: A. Related Work and Research Gap · pp. 1–2

B. Contributions

• The paper compares DB rate–CRB, DA rate–CRB with an explicit detection requirement, and DTE rate–CRB with a detection-gated CRB.The DB–DA comparison isolates the effect of enforcing detection; DA–DTE isolates detector gating.Source: B. Contributions · p. 2
• The paper identifies that equal nominal range CRBs can yield different detection probabilities and derives a threshold for detection feasibility at the DB rate-optimal allocation.It also characterizes how the detection requirement changes optimal allocation and communication rate.Source: B. Contributions · p. 2
• For DTE, the paper derives a detection-gated range CRB and relates accepted-measurement accuracy and availability to tracking PCRB.It also formulates multi-user, multi-target max–min rate problems under DB, DA, and DTE constraints and establishes their convexity.Source: B. Contributions · p. 2

II. SYSTEM MODEL AND PERFORMANCE METRICS

• The system model serves multiple communication users and sensing targets with precoded streams that may use separate subbands or pair a user and target on one stream.Stream powers and bandwidths share total resource budgets, and communication SNR depends on the assigned power-to-bandwidth ratio.Source: §II · p. 2, A. Communication Channel and… · p. 3

B. Monostatic Sensing Model and Performance Metrics

• The analysis assumes front-end clutter suppression and models residual clutter and thermal noise as effective AWGN; signal-dependent clutter requires model-specific re-derivation or numerical evaluation.The closed-form detection, range-CRB, and selection-factor results may not apply directly to signal-dependent clutter.Source: B. Monostatic Sensing Model and… · p. 3
• The received sensing samples are modeled as a target waveform scaled by power and reflectivity, plus noise and residual clutter.The waveform depends on target delay and allocated bandwidth; residual noise and clutter are modeled jointly as complex Gaussian noise.Source: B. Monostatic Sensing Model and… · p. 3
• Under the Swerling-I model, target reflectivity is constant within a frame and independent across frames, while target parameters are approximately constant within each frame.The analysis assumes spatial or delay–Doppler separability and studies range CRB, with Doppler and angle available or estimated separately.Source: B. Monostatic Sensing Model and… · p. 3
• The energy detector applies a likelihood-ratio-equivalent threshold test to the squared magnitude of the normalized matched-filter output.A prescribed false-alarm probability determines the threshold, and detection probability follows from the target-present distribution.Source: 1) Framewise Detection · p. 3
• For the assumed rectangular spectrum, the nominal range CRB follows from the delay Fisher information and depends on the waveform’s range sensitivity.Bandwidth changes delay sensitivity without changing waveform energy, while the nominal CRB is not conditioned on the energy-detector output.Source: 2) Nominal Range CRB · p. 3, B. Monostatic Sensing Model and… · p. 3

III. DETECTION-BLIND AND DETECTION-AWARE RATE–CRB ANALYSIS

• The section frames DA as adding a detection-probability requirement alongside the nominal CRB, without specifying how the receiver couples detection and estimation.It uses the constant-CRB contour to expose resource-composition effects; conventional inequality constraints are revisited later in the paper.Source: §III · p. 4, B. DB Rate Optimization on a… · p. 4

B. DB Rate Optimization on a Constant-CRB Contour

• On a constant-CRB contour, the DB rate–CRB optimum can have detection probability far below the prescribed detection level.The figure presents rate and detection probability along the contour and at the DB optimum across sensing-channel gains and CRB requirements.Source: B. DB Rate Optimization on a… · p. 4, Figure 1 · p. 5
• Along the constant-CRB contour PB^2 = κ, power trades against bandwidth, but detection depends on sensing SCNR and has no corresponding bandwidth lever.This permits the rate maximizer to move toward lower power and higher bandwidth without changing the nominal CRB, while detection can deteriorate.Source: B. DB Rate Optimization on a… · p. 4
• Under the interior-DB regime, the optimum is detection-feasible if and only if the prescribed CRB meets the derived threshold, equivalently when κ ≥ κ_DF.The condition applies to the single-link constant-CRB interior optimum.Source: 1) Detection-Feasible CRB Constraint · p. 5
• Tightening the CRB can indirectly raise sensing SCNR and detection probability, but CRB tightening is an inefficient way to obtain detection reliability.Reducing the prescribed CRB by a factor of eight only doubles sensing SCNR; stricter CRB requirements also draw more bandwidth.Source: 1) Detection-Feasible CRB Constraint · p. 5

C. Detection-Aware Rate–CRB Characterization

• The DA problem imposes ΠD ≥ Πth alongside the nominal-CRB requirement, assuming the prescribed CRB is achievable and Pth ≤ Pmax.The rate characterization combines the DB optimum with the detection-imposed power floor.Source: C. Detection-Aware Rate–CRB… · p. 5
• A prescribed-CRB contour can contain no detection-feasible point even when the resource bounds permit jointly satisfying detection and CRB by over-delivering the CRB.If no allocation satisfies both requirements within the available resources, the DA requirements are jointly infeasible.Source: C. Detection-Aware Rate–CRB… · p. 5
• DA retains the DB allocation when it meets the detection requirement; otherwise it moves to the detection power floor on the CRB contour or selects (Pth, Bmin) off-contour.The off-contour choice over-delivers the prescribed CRB when the contour cannot reach Pth while respecting B ≥ Bmin.Source: C. Detection-Aware Rate–CRB… · pp. 5–6
• The achieved detection probability parameterizes rate along a fixed-CRB contour; when its corresponding point lies outside the feasible contour, the off-contour extension applies.Fig. 2 compares rate-optimal DB and DA allocations on constant-CRB contours and their relative signed rate gap.Source: 1) Rate–Detection Relationship · p. 6, 2) Signed Rate Gap and Operating Regimes · p. 6
• The signed DA–DB rate gap is zero when the DB optimum is detection-feasible, positive when DA slides along the constant-CRB contour, and can be negative under over-delivery at unequal sensing accuracy.In the negative-gap case, DA uses higher power and attains a lower CRB, so the gap is not a rate gain at the same CRB.Source: Figure 2 · p. 6, 2) Signed Rate Gap and Operating Regimes · p. 6

IV. ISAC Performance Under Detection-Gated Estimation and Tracking in a DTE Receiver

• In a detect-then-estimate (DTE) receiver, only frames passing an energy-detector threshold are sent to the estimator, and missed-detection frames provide no estimate.Source: §IV · p. 6
• The analysis derives a closed-form detection-gated range CRB under the adopted ISAC signal and resource model.Source: §IV · p. 6
• The section studies changes to feasible resource boundaries and optimal rate, then examines how detection probability and estimation error jointly affect tracking performance.Source: §IV · p. 6

A. Derivation of Detection-Gated Range CRB

• The DTE range CRB bounds the MSE of an estimator locally unbiased under the accepted-observation distribution.The receiver conditions the likelihood and Fisher information on the event that the energy detector accepts a frame.Source: A. Derivation of Detection-Gated Range… · p. 7
• At the cell center, the matched-filter statistic under the target-present model is complex Gaussian with variance 1 + γs, yielding ΠD = exp[−δ/(1 + γs)].Under the noise-only model, the threshold is δ = −ln ΠFA.Source: A. Derivation of Detection-Gated Range… · p. 7
• At the predicted range-cell center, the local detection probability is stationary with respect to range, so the detection-derivative term in the conditional information vanishes.The local evaluation point does not imply that the receiver knows the current range; small within-cell offsets affect detection and conditional information only to second order.Source: A. Derivation of Detection-Gated Range… · p. 7
• Detector gating scales accepted-frame range information by h(γs) while PB2 remains the underlying range-information factor; h peaks at γs = 1 and tends to 1 at both extremes.Missed frames still provide no measurement, so the conditional information increase does not imply better per-transmitted-frame accuracy; detection probability separately represents availability.Source: A. Derivation of Detection-Gated Range… · p. 7
• The Monte Carlo estimate closely agrees with h(γs) for ΠFA = 10−4, supporting the predicted target-energy enrichment in accepted frames.The simulation forms noisy target returns and accepts frames whose detector statistic exceeds δ.Source: 1) Monte Carlo Validation of the… · p. 7, Figure 3 · p. 7

B. DTE Rate–CRB Performance Implications and Resource-

• The DTE constant-CRB contour is determined by Ph(GsP)B2, while detection feasibility separately requires P ≥ Pth.The power dependence of h prevents a closed-form solution; the bounded one-dimensional optimization over feasible power values is solved numerically.Source: B. DTE Rate–CRB Performance… · p. 8
• When the DTE equality contour is feasible, it meets the prescribed conditional-CRB target; otherwise the least-over-delivery choice is (Pth, Bmin).The same over-delivery convention is applied to the DTE feasible set.Source: B. DTE Rate–CRB Performance… · p. 8
• For κ1 < κ0 < κ2, DTE requires less bandwidth than DA to attain the prescribed conditional accuracy; the bandwidth difference peaks and then decreases as required SCNR rises and h(GsP) approaches 1.For κ0 ≤ κ1, the rate difference is zero and DTE reduces bandwidth.Source: B. DTE Rate–CRB Performance… · p. 8
• DTE’s lower bandwidth requirement concerns accuracy on accepted estimates, not all-frame accuracy, because missed frames provide no estimate and availability remains governed by ΠD.Source: B. DTE Rate–CRB Performance… · p. 8

C. Tracking with Intermittent Measurements

• The tracker models accepted range estimates as locally unbiased Gaussian measurements with variance equal to the prescribed CRB, available independently each frame with probability ΠD.Its random Riccati recursion yields a sequence-averaged steady-state PCRB; each realized availability sequence has a conditional PCRB.Source: C. Tracking with Intermittent… · p. 8
• Lower prescribed CRB reduces steady-state range PCRB at every detection probability, while lower ΠD increases PCRB and worsens tracking relative to assuming a measurement every frame.The DB markers quantify degradation at detection probabilities induced by CRB-only allocations.Source: Figure 5 · p. 9, C. Tracking with Intermittent… · p. 9
• Missed frames form Bernoulli gaps with Pr{N = n} = (1 − ΠD)^nΠD, so lower ΠD both reduces average information arrivals and increases the chance of long prediction-only gaps.The tracker updates on successful measurements rather than assuming one arrives in every frame.Source: C. Tracking with Intermittent… · pp. 8–9
• For gated receivers, DTE can require less bandwidth than DA for the same accepted-measurement accuracy, with their optima converging as the required detection probability approaches one.The DA–DTE gap becomes more pronounced when the detection-probability requirement is relaxed.Source: C. Tracking with Intermittent… · p. 9
• DA is the most conservative formulation in resource feasibility, with its feasible resource set contained in both the DB and DTE feasible sets.The stated set relations are FDA ⊆ FDB and FDA ⊆ FDTE.Source: C. Tracking with Intermittent… · p. 9

V. MULTI-TARGET AND MULTI-USER RESOURCE ALLOCATION

• The extension maximizes minimum per-user achievable rate under sensing constraints for DB, DA, and DTE in shared and orthogonal resource deployments.The setting has multiple communication users and sensing targets sharing power and bandwidth budgets.Source: §V · p. 9

A. Problem Formulation

• The problem formulation uses a max–min achievable-rate objective to promote user fairness, subject to sensing-service constraints for DB, DA, or DTE under shared power and bandwidth budgets.DB is the reference CRB-only formulation commonly used in ISAC resource allocation.Source: A. Problem Formulation · p. 9

B. Convex Reformulation and Solution

• All three variants are convex; DB and DA are solved with CVX, while DTE uses SQP because CVX does not recognize its detection-factor expression as DCP-compliant.The reported DTE solver converged with first-order optimality and maximum constraint violation below 10^-9 in every simulated instance.Source: 1) Convexity and Solver · p. 10

C. Simulation Setup

• The five-target setup uses heterogeneous sensing gains and detection floors, comparing shared budgets of (10 W, 70 MHz) with orthogonal budgets of (20 W, 200 MHz).The detection settings are ΠFA,q = 10^-5 and, unless otherwise stated, ΠD,q = 0.5.Source: C. Simulation Setup · p. 10

D. Numerical Results and Discussion

• Shared deployments place paired targets T1–T3 inside the sensing-feasible region, while all orthogonal targets use contour-sliding allocations that meet both sensing constraints.The shared streams for T1–T3 also carry communication; orthogonal allocations have no over-delivery.Source: Figure 6 · p. 11, 1) Allocations and Operating Regimes · p. 11
• Higher detection-probability requirements increase the rate cost of DA versus DB in the orthogonal deployment.The detection requirement draws more power to sensing streams; Fig. 7(a) compares the DA and DB optima.Source: Figure 7 · p. 11, 2) Gap between Rate Optima: DA–DB and… · p. 11
• Approximately 3% is the rate margin approached by all three DTE-versus-DA curves as detection requirements rise.Weak-target streams remain near the minimum detection SCNR, so DTE’s bandwidth reduction for accepted frames persists.Source: Figure 7 · p. 11, 2) Gap between Rate Optima: DA–DB and… · p. 11
• The DB rate–CRB optimum is detection-feasible for stronger targets T1–T3 but not weaker targets T4–T5, so CRB constraints alone do not guarantee detection.The reported pattern holds across both deployment settings.Source: 1) Allocations and Operating Regimes · p. 11
• The shared-resource simulations have a negligible shared-stream rate gap because communication is paired with stronger, stricter-CRB targets.Weaker targets instead use dedicated sensing streams and exhibit contour sliding.Source: 2) Gap between Rate Optima: DA–DB and… · p. 11

• Identical nominal range CRBs can arise from power–bandwidth allocations with substantially different detection probabilities.Source: §VI · p. 12
• The single-link analysis characterizes when DB rate–CRB optima satisfy detection and how enforcing detection changes optimal allocations and communication rate.Source: §VI · p. 12
• For DTE receivers, the detection-gated CRB of accepted frames links conditional estimate accuracy and detection availability to tracking performance.Multi-user, multi-target results also show that detection awareness can materially change resource distribution.Source: §VI · p. 12

APPENDIX A COMPLETE BOX-CONSTRAINED DB SOLUTION

• On a constant-CRB contour, substituting P = κ/B^2 reduces the DB problem to a bandwidth search over BL(κ) ≤ B ≤ Bmax.The stated reference bandwidth is B0(κ) := (Gcκ/u⋆)^(1/3).Source: Appendix A · p. 12

APPENDIX B COMPLETE DETECTION-AWARE FEASIBILITY

• The DA feasible set is defined by intersecting the constant-CRB set C(κ) with the detection-power constraint P ≥ Pth.The appendix gives a complete operating point under the interior-DB assumption.Source: Appendix B · p. 12

APPENDIX C GAP BETWEEN THE DA AND DB RATE OPTIMA

• The DA–DB rate gap changes sign between the stated low-κ and contour-sliding cases, so its behavior depends on the operating regime.Source: Appendix C · p. 12
• For detector-accepted frames, conditioning on the energy threshold selects higher-energy echo-bearing observations, reflected in the factor h(γs).The acceptance event is D = {|r|^2 > δ}, with δ = −ln ΠFA.Source: Appendix C · p. 12