WaveMax: Radar Waveform Design via Convex Maximization of FrFT Phase Retrieval

2025-01-24

AI Summary

This paper addresses the non-convex inverse problem of recovering time-frequency signal phase from the ambiguity function (AF) magnitude in radar waveform design. We propose a novel convexification framework based on the fractional Fourier transform (FrFT). Its core innovation lies in exploiting the rotational equivalence of the AF in the FrFT domain, thereby transforming the original bi-variate non-convex optimization into a tractable convex problem. Theoretically, we prove that unique waveform reconstruction is guaranteed with AF samples of size at most three times the signal’s degrees of freedom. The method inherently accommodates bandlimited and timelimited constraints and supports both sparse and random AF sampling. Experiments demonstrate that our approach significantly outperforms conventional non-convex algorithms under both noiseless and noisy conditions, maintaining strong robustness even at low sampling rates—empirically validating the theoretical reconstruction bound.

Technology Category

• Search and Optimization: Non-convex Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

The ambiguity function (AF) is a critical tool in radar waveform design, representing the two-dimensional correlation between a transmitted signal and its time-delayed, frequency-shifted version. Obtaining a radar signal to match a specified AF magnitude is a bi-variate variant of the well-known phase retrieval problem. Prior approaches to this problem were either limited to a few classes of waveforms or lacked a computable procedure to estimate the signal. Our recent work provided a framework for solving this problem for both band- and time-limited signals using non-convex optimization. In this paper, we introduce a novel approach WaveMax that formulates waveform recovery as a convex optimization problem by relying on the fractional Fourier transform (FrFT)-based AF. We exploit the fact that AF of the FrFT of the original signal is equivalent to a rotation of the original AF. In particular, we reconstruct the radar signal by solving a low-rank minimization problem, which approximates the waveform using the leading eigenvector of a matrix derived from the AF. Our theoretical analysis shows that unique waveform reconstruction is achievable with a sample size no more than three times the signal frequencies or time samples. Numerical experiments validate the efficacy of WaveMax in recovering signals from noiseless and noisy AF, including scenarios with randomly and uniformly sampled sparse data.

Problem

• Radar Signal Optimization
• Target Identification
• Computational Limitations

Innovation

• Fractional Fourier Transform
• Waveform Recovery
• Optimization Problem