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On A Unified Cramér-Rao Bound Framework for Joint Delay-Doppler Estimation with Multi-Carrier Waveforms

Zi-Jie Wang, Xudong Wang, Giuseppe Caire

arXiv:2608.26441v1eess.SP

TL;DR

Existing sensing analyses lack a unified treatment of joint delay-Doppler CRBs across diverse multi-carrier waveforms. This paper establishes a common FIM framework, identifies the Pareto boundary through TF-symbol power allocation, and shows that CRB performance depends on the resulting power profile and waveform modulation.

  • Problem

    Existing CRB analyses are mostly waveform-specific, use simplified settings, and provide limited systematic comparison across waveforms and system configurations.

  • Method

    The paper establishes a unified FIM framework using unitary data-to-symbol mappings and optimizes TF-symbol power profiles to identify the joint delay-Doppler CRB Pareto boundary.

  • Results

    Different waveforms can yield the same joint delay-Doppler CRBs under the same TF symbol-level power profile, while waveform-specific modulation determines data-dependent sensing advantages.

  • Takeaways & Limitations

    The Pareto front represents the system’s ultimate joint delay-Doppler estimation-centric capability beyond commonly adopted isotropic-transmission CRBs.

Abstract

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In this work, we derive the Cramér-Rao bound for joint Delay-doppler estimation with multi-carrier waveforms

I. INTRODUCTION

The paper addresses missing unified estimation-theoretic analysis for joint delay-Doppler sensing across multi-carrier waveforms. It establishes a common FIM/CRB framework, characterizes performance limits through power-profile optimization, and derives tractable single-target results.

  • Research gaps: Existing CRB analyses are waveform-specific, often simplified to single-target or isotropic settings, and frequently rely on semi-analytical or numerical FIM calculations.These limitations hinder systematic comparison and obscure structure-revealing insights into joint delay-Doppler estimation.
  • Unified framework: The framework represents different waveform modulations as unitary transformations from information-bearing data to time-domain symbols, enabling a common FIM analysis.This representation supports joint delay-Doppler CRB quantification under general sensing settings.
  • Waveform effects: Under unitary modulation, different waveforms can yield identical joint delay-Doppler CRBs when their TF symbol-level power profiles are identical.Observed waveform advantages instead depend on how modulation reallocates a given data sequence’s power across the TF profile.
  • Performance limits: The Pareto boundary of the joint delay-Doppler CRB region is obtained by optimizing transmitted TF-symbol power profiles.A low-complexity numerical algorithm traces the boundary, which represents the system’s ultimate estimation-theoretic capability.
  • Special case: Closed-form FIM/CRB expressions and optimal waveform structures are derived for the single-target case.These results support analytical tractability and asymptotic analysis of joint delay-Doppler estimation.
  • Validation and scope: Theoretical findings are validated numerically, and the framework is presented as a building block for future waveform-specific processing.The authors describe it as the first framework to quantify joint delay-Doppler CRB performance for diverse multi-carrier waveforms.

III. CRAM´ER-RAO BOUND ON JOINT DELAY-DOPPLER ESTIMATION

The measurement model converts prefix-protected, pulse-shaped multi-carrier transmissions into a tractable TF-domain observation model. Under AWGN and matched-filter sampling, this model supports unified FIM and CRB derivation for joint delay-Doppler estimation.

  • Measurement Model: The pulse shape is modeled through its frequency response and autocorrelation, with rectangular, RRC, sinc, and Gaussian pulses considered.The received signal is obtained after matched filtering and sampling at the chip rate.
  • Measurement Model: A cyclic or chirp-periodic prefix at least as long as the maximum delay spread prevents inter-symbol interference and induces a circulant effective delay channel.The circulant structure permits diagonalization by the fast-time DFT matrix.
  • Measurement Model: Neglecting high-order spectral aliasing terms retains the dominant pulse-energy components because practical pulse spectra concentrate energy near the main lobe.The retained summation terms are selected according to the normalized frequency index.
  • TF-domain representation: Applying a fast-time DFT and stacking transformed columns produces TF-domain observation vectors y, z, and s.The resulting representation is used as an equivalent and more tractable characterization for subsequent FIM analysis.
  • Noise and processing: The noise is modeled as complex AWGN, and Nyquist matched-filter sampling makes the sampled noise components mutually uncorrelated.Further waveform-specific receiver processing is omitted because it adds no Fisher information when the transmitted symbol is known.
  • FIM and CRB: The Slepian-Bangs formula defines the FIM for all unknown parameters, from which delay and Doppler CRB matrices and elementwise bounds are obtained.The framework provides a principled complete FIM calculation before extracting the delay and Doppler components.

B. Structure and Evaluation of the FIM and CRB

The framework derives the complete FIM and corresponding CRB matrices for joint delay-Doppler estimation, including general and single-target evaluations. It shows that the single-target FIM accumulates contributions from all TF samples and is parameter-independent, unlike the multi-target case.

  • Theorem 1 provides a principled calculation of the complete FIM for all unknown parameters, from which the CRB is defined.Element-wise expressions support numerical computation.
  • The CRB matrices for delay and Doppler are obtained from the corresponding FIM-based expressions.
  • The delay and Doppler CRBs for individual targets are given by the diagonal elements of the respective CRB matrices.
  • The FIM structure depends only on the diagonal terms of the TF symbol correlation matrix, so cross-sample correlations do not affect the joint delay-Doppler CRB.
  • Single-target evaluation: For a single target, Theorem 2 gives closed-form evaluations of the FIM and CRB matrix.
  • Single-target evaluation: Under the single-target scenario, the FIM adds contributions from all Q TF samples and does not depend on the specific delay or Doppler values.This differs from the general multi-target case, where the FIM and CRB typically depend on delay and Doppler.

C. Impacts of Waveform Structure

Waveform structure affects delay-Doppler estimation through the TF-domain power profile induced by each waveform’s data-to-time mapping. Waveforms with identical TF power profiles have identical CRBs, while data-dependent projections can produce different or reversed advantages.

  • Waveform mappings: Different waveforms are represented through mappings from domain-specific sensing information X to the time-domain symbol S.The paper considers OFDM, OTFS, ODDM, AFDM, and OCDM mappings.
  • Waveform mappings: The TF symbol vector and data vector satisfy s_w = U_w x, where U_w is the waveform-dependent transformation matrix.This relationship enables waveform-specific FIM and CRB evaluation for a given data correlation matrix.
  • Power-profile effects: Waveforms sharing the same TF symbol power profile have identical joint delay-Doppler CRB performance.
  • Power-profile effects: Under isotropic TF power, different waveforms exhibit the same CRB performance.The isotropic condition is diag{Φ_s,w} = pt/Q 1_Q or equal envelope magnitude for every TF symbol.
  • Power-profile effects: For a fixed data sequence, different waveform transformations generally produce different TF power profiles and therefore different delay-Doppler estimation capabilities.Changing the data sequence can make a waveform’s advantage disappear or reverse.
  • Single-target isotropic case: In the single-target isotropic-transmission case, delay and Doppler estimation decouple, with CRBs given by the stated expressions for ε(τ) and ε(ν).The formulas are waveform-independent under isotropic transmission and are not the optimal CRBs because TF power can be further designed.

IV. FUNDAMENTAL JOINT DELAY-DOPPLER ESTIMATION-THEORETIC PERFORMANCE LIMITS

Varying the TF-symbol power profile changes the delay and Doppler CRBs under the power constraint, creating a trade-off captured by the achievable joint CRB region’s Pareto front.

  • Reducing one CRB under the power constraint necessarily increases the other, so the optimal trade-off is represented by the Pareto front.

A. Problem Formulation

The paper formulates joint delay-Doppler CRB optimization through transformed, comparable parameters and a weighted Pareto objective. It then converts the challenging rank-constrained problem into a convex formulation with reduced dimensionality, while noting practical communication constraints remain outside the sensing-only design.

  • The optimization targets the TF symbol power profile or correlation matrix to optimize delay and Doppler CRBs.
  • The parameter transformation makes FIM/CRB entries comparable by giving transformed parameters the same units and similar magnitudes.
  • A weighted sum of delay and Doppler CRBs, swept over α ∈ [0, 1], traces the complete Pareto-optimal boundary.
  • The sensing-only formulation does not impose communication constraints such as peak-to-average power ratio, modulation order, or bit error rate.
  • Proposition 3 equivalently transforms the high-dimensional rank-constrained problem into a more tractable convex optimization with greatly reduced variable dimensionality.
  • The CRB-optimal sequence retains arbitrary phase degrees of freedom, allowing (Q)PSK symbols with optimal power allocation to remain delay-Doppler estimation-optimal.

B. Identifying the Delay-Doppler CRB Pareto Boundary

The Pareto-boundary computation uses convexity and KKT conditions to iteratively optimize power allocation across FIM components. The resulting gradient-descent algorithm converges globally under a proper step size, with complexity determined by targets, components, and iterations.

  • Convexity ensures that any stationary point satisfying the KKT condition is globally optimal.
  • The algorithm initializes every component with equal power pt/Q before computing marginal utilities and updating the allocation.
  • Algorithm 1 uses gradient descent to update the power allocation and converges to the global optimum for a proper step size η.
  • The per-iteration computational complexity is O(QP^3), and total complexity is O(KiterQP^3 + Q^2P^2).

V. NUMERICAL EXAMPLES

The numerical examples validate the theoretical findings using IEEE 802.11p-based parameters and MATLAB. They examine how target count affects CRBs and compare waveform-dependent joint delay-Doppler bounds under a particular data realization.

  • The numerical results are presented to validate the paper’s theoretical findings.
  • The waveform-related parameters are based on IEEE 802.11p, with default settings summarized in Table I.
  • Figure 3 plots delay-Doppler CRBs versus transmit SNR for different numbers of targets under isotropic transmission.
  • For the example with P = 4, delays, Dopplers, and channel coefficients are specified, while additional pulse and chirp parameters define the waveform settings.
  • Figure 4 compares joint delay-Doppler CRBs for different waveforms under a particular realization of the data sequence.

B. Results

The results show that DD estimation performance depends on pulse shape, SNR, target count, and the TF-domain power profile rather than waveform labels alone. Pareto-optimal power allocation substantially outperforms isotropic transmission, while the derived CRB predicts estimator performance at moderate-to-high SNR.

  • B. Results: Root CRBs for delay and Doppler decrease linearly with transmit SNR in dB, while increasing target count degrades per-target accuracy.The reported example uses an RRC pulse and isotropic transmission.
  • B. Results: For a fixed data sequence, waveform modulation determines the TF symbol-level power distribution and its alignment with the FIM, producing different DD CRBs.OTFS and ODDM yield identical CRBs in the considered setup because their modulation matrices are equal.
  • B. Results: Different pulse shapes produce different DD CRBs and threshold SNRs, while the derived CRB accurately predicts estimator performance in the moderate-to-high SNR regime.The MSE is obtained with an approximately maximum-likelihood estimator based on a low-complexity two-stage search.
  • B. Results: The joint DD CRB Pareto front is distinctly convex for all pulse shapes, and isotropic-transmission CRBs lie considerably far from the optimal boundary.This gap indicates that optimized power allocation can exceed the commonly used isotropic benchmark.
  • B. Results: The Pareto-optimal normalized TF power profile is sparse and shifts from slow-time extremes toward range-frequency indices as the DD preference α moves from 0 to 1.At α = 0, allocation minimizes Doppler CRB; at α = 1, it minimizes delay CRB.
  • B. Results: The unified framework quantifies joint DD CRBs across waveforms and identifies the Pareto front through a family of power-allocation problems.The associated appendix develops the underlying FIM calculations.

Q BTR

This material develops blockwise FIM calculations involving delay, Doppler, gain, and pulse-shape terms. The derivation uses diagonal gain structure and related FIM blocks to obtain the stated expressions.

  • Q BTR: The cross-delay-Doppler FIM block is calculated from indexed matrix products involving the pulse-shape derivative terms.The derivation relates Jτν to its corresponding block expression.
  • Q BTR: Diagonal structure of G simplifies the delay-related FIM blocks because only matching index blocks contribute.The derivation uses G{p,p} = gpIQ and zero off-diagonal blocks.
  • Q BTR: The gain-related blocks Jgigi and Jgrgi are expressed directly in terms of Jgg through real and signed-real transformations.The stated relationships are Jgigi = 2 σ2 ℜ{Jgg} and Jgrgi = −2…

APPENDIX B

Appendix B expands the FIM into element-wise blocks and derives the equivalent FIM for delay and Doppler. It then specializes the expressions to ideal sinc pulses and isotropic transmission and reformulates the power-allocation objective.

  • APPENDIX B: The FIM sub-blocks are expanded element-wise using the definitions of V, G, Tps, and the BTR operator.The expansion applies for indices 1 ≤ p1, p2 ≤ P.
  • APPENDIX B: For a single target, each FIM sub-block becomes a scalar, simplifying the proof of the corresponding CRB expressions.The appendix illustrates this using Jττ with p1 = p2 = 1.
  • APPENDIX B: The equivalent FIM for τ and ν follows from the block structure of Theorem 1 after matrix algebra.The resulting expression is used to obtain the associated CRB.
  • APPENDIX B: Under an ideal sinc pulse and isotropic transmission, the off-diagonal equivalent-FIM elements vanish, so delay and Doppler estimation decouple.The specialization uses Υsinc(m) = 1 and uniform symbol power.
  • APPENDIX B: The optimization objective is rewritten as a trace involving A(α), and the FIM dependence on Φs is reduced to its diagonal terms.This permits optimization over the TF-symbol power profile while temporarily neglecting the rank-1 constraint.
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