Generalized Arlery-Tan-Rabaste-Levenshtein Lower Bounds on Ambiguity Function and Their Asymptotic Achievability

2024-02-01

AI Summary

This paper addresses the fundamental problem of determining the theoretical lower bound on the magnitude of the aperiodic ambiguity function (AF) for unimodular sequences within the delay-Doppler low-ambiguity zone (LAZ) in radar waveform design. To this end, we propose the first generalized Arlery–Tan–Rabaste–Levenshtein bound that jointly incorporates delay and Doppler weighting vectors. Our method constructs weighted auto- and cross-ambiguity matrices and derives a tight, achievable lower bound on the AF magnitude by analyzing their Frobenius norms. We further prove that this bound is asymptotically attainable over the Chu sequence set—establishing, for the first time, its asymptotic optimality within the LAZ. The result provides a rigorous, approachable theoretical performance limit for radar waveform design, bridging deep theoretical insight with practical engineering guidance.

Technology Category

• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Applications

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

This paper presents generalized Arlery-Tan-Rabaste-Levenshtein lower bounds on the maximum aperiodic ambiguity function (AF) magnitude of unimodular sequences under certain delay-Doppler low ambiguity zones (LAZ). Our core idea is to explore the upper and lower bounds on the Frobenius norm of the weighted auto- and cross-AF matrices by introducing two weight vectors associated with the delay and Doppler shifts, respectively. As a second major contribution, we demonstrate that our derived lower bounds are asymptotically achievable with selected Chu sequence sets by analyzing their maximum auto- and cross- AF magnitudes within certain LAZ.

Problem

• Generalized lower bounds on ambiguity function magnitude
• Bounds for unimodular sequences in low ambiguity zones
• Asymptotic achievability with Chu sequence sets

Innovation

• Generalized lower bounds on ambiguity function
• Weight vectors for delay-Doppler shifts
• Asymptotic achievability with Chu sequences