[Paper Review] Phase Retrieval for Radar Waveform Design

Samuel Pinilla, Kumar Vijay Mishra+2 more|arXiv (Cornell University)|Jan 27, 2022

Advanced X-ray Imaging Techniques4 citations

TL;DR

This paper proposes a trust-region algorithm for radar waveform design by solving the phase retrieval problem to reconstruct a transmit signal from its ambiguity function (AF) magnitude. Using a smoothed non-convex least-squares optimization with iterative spectral initialization and gradient refinement, the method achieves unique signal recovery with as few as three times the number of signal samples, demonstrating near-perfect reconstruction (MSE ≈ 1×10⁻⁶) even from sparse, noisy AF samples.

ABSTRACT

The ability of a radar to discriminate in both range and Doppler velocity is completely characterized by the ambiguity function (AF) of its transmit waveform. Mathematically, it is obtained by correlating the waveform with its Doppler-shifted and delayed replicas. We consider the inverse problem of designing a radar transmit waveform that satisfies the specified AF magnitude. This process may be viewed as a signal reconstruction with some variation of phase retrieval methods. We provide a trust-region algorithm that minimizes a smoothed non-convex least-squares objective function to iteratively recover the underlying signal-of-interest for either time- or band-limited support. The method first approximates the signal using an iterative spectral algorithm and then refines the attained initialization based on a sequence of gradient iterations. Our theoretical analysis shows that unique signal reconstruction is possible using signal samples no more than thrice the number of signal frequencies or time samples. Numerical experiments demonstrate that our method recovers both time- and band-limited signals from sparsely and randomly sampled, noisy, and noiseless AFs.

Motivation & Objective

• To address the inverse problem of designing radar waveforms with a specified ambiguity function (AF) magnitude.
• To overcome the non-convex and ill-posed nature of phase retrieval in radar signal reconstruction.
• To enable unique signal recovery from AF magnitude samples with minimal sampling requirements.
• To develop a robust, iterative optimization framework that combines spectral initialization and gradient refinement for time- and band-limited signals.

Proposed method

• The method formulates the waveform design as a non-convex phase retrieval problem minimizing a smoothed least-squares objective function.
• It employs an iterative spectral algorithm to generate a high-quality initialization for the signal of interest.
• A trust-region approach is used to refine the initialization via successive gradient iterations, ensuring convergence and stability.
• The algorithm leverages the structure of the ambiguity function and exploits the relationship between the signal and its time-delayed/Doppler-shifted replicas.
• Key components include the use of a smoothed objective function and a convergence analysis based on singular values of the Gram matrix.
• The method is validated for both time-limited and band-limited signals, with theoretical guarantees on uniqueness and convergence.

Experimental results

Research questions

• RQ1 Can a radar transmit waveform be uniquely reconstructed from its ambiguity function magnitude using a minimal number of samples?
• RQ2 What optimization strategy ensures stable and accurate recovery of the signal phase in the presence of noise and sparsity?
• RQ3 How can the non-convexity of the phase retrieval problem in radar be effectively mitigated for practical waveform design?
• RQ4 What is the theoretical minimum number of AF samples required for unique signal reconstruction?
• RQ5 How does the proposed trust-region algorithm compare in performance to existing phase retrieval methods in radar applications?

Key findings

• The method achieves a mean-square error (MSE) of 1×10⁻⁶ in reconstructing time- and band-limited signals from full, noiseless ambiguity function samples.
• For sparse, noisy AF samples, the reconstruction MSE is 9×10⁻², demonstrating robustness to data scarcity and noise.
• Theoretical analysis confirms that unique signal reconstruction is possible with no more than thrice the number of signal samples or frequency bins.
• Convergence is guaranteed under the condition that the product of the step size and the smallest non-zero singular value of the Gram matrix exceeds 1/2.
• The trust-region algorithm with spectral initialization significantly outperforms naive phase retrieval approaches in terms of accuracy and stability.
• The algorithm successfully recovers both time-limited and band-limited signals, validating its broad applicability in radar waveform design.