CP-OFDM Achieves Lower Ranging CRB Than Frequency-Spread Waveforms in the Large-Sample Regime

2026-05-14

AI Summary

This work addresses the fluctuation in ranging accuracy caused by the randomness of communication symbols in integrated sensing and communication systems, necessitating a characterization of how modulation waveforms affect the Cramér–Rao bound (CRB) for ranging based on random data symbols. By decomposing the Fisher information matrix (FIM) for joint time-delay and amplitude estimation, the authors separate the deterministic Jacobian induced by target geometry from the random frequency-domain power introduced by data symbols, thereby establishing a universal lower bound on the CRB. Leveraging Jensen’s inequality, asymptotic perturbation analysis, random matrix theory, and Riemannian optimization, the study theoretically demonstrates that CP-OFDM with PSK constellations exactly achieves this bound and, under large-sample conditions, outperforms orthogonal waveforms such as SC, OTFS, and AFDM for QAM and other constellations, exhibiting asymptotic local optimality within the unitary group. Numerical experiments confirm its CRB advantage in delay, amplitude, and joint estimation.

Abstract

The inherent randomness of communication symbols creates a fundamental tension in Integrated Sensing and Communications (ISAC). On the one hand, they enable data transmission while allowing sensing to fully reuse communication resources. On the other hand, their randomness induces waveform-dependent fluctuations that directly affect sensing accuracy. This paper investigates a foundational question arising from this tradeoff: \textit{How does the modulation waveform affect the ranging Cramér--Rao Bound (CRB) when sensing reuses random data symbols?} We address this question by revealing a structural factorization of the Fisher information matrix (FIM) for joint delay-amplitude estimation, which separates the deterministic Jacobian of the target geometry from the random frequency-domain signal power induced by the data symbols. This structure yields a Jensen-type universal lower bound on the CRB, which is exactly attained by CP-OFDM under PSK constellations. For QAM and broader sub-Gaussian constellations, we develop an asymptotic perturbation analysis of the inverse FIM and prove that, when the number of transmitted symbols $N$ grows large, CP-OFDM achieves a lower ranging CRB than any frequency-spread orthogonal waveform over the almost-sure event where the random FIM is invertible. This superiority is further extended to amplitude estimation and full joint delay-amplitude estimation. We also characterize the local geometry of the stochastic CRB minimization problem over the unitary group. The analysis reveals that CP-OFDM is a stationary point for finite $N$, and its Riemannian Hessian is positive semidefinite for sufficiently large $N$, establishing its asymptotic local optimality. Numerical results confirm that OFDM outperforms representative waveforms including SC, OTFS, and AFDM.

Problem

• Integrated Sensing and Communications
• Cramér-Rao Bound
• CP-OFDM
• waveform design
• ranging accuracy

Innovation

• CP-OFDM
• Cramér–Rao Bound
• Integrated Sensing and Communications
• Fisher Information Matrix
• Asymptotic Optimality