Robust Sparse Fourier Transform Based on The Fourier Projection-Slice Theorem

TL;DR

This paper introduces a robust sparse Fourier transform method based on the Fourier projection-slice theorem, tailored for noisy radar signals with off-grid frequencies, improving accuracy and efficiency in multidimensional signal processing.

Contribution

The paper extends FPS-SFT into a robust version (RFPS-SFT) that effectively handles noisy, off-grid frequencies using windowing and voting-based decoding techniques.

Findings

Achieves lower frequency localization error in noisy conditions.

Reduces frequency leakage below noise level.

Demonstrates improved performance both theoretically and numerically.

Abstract

The state-of-the-art automotive radars employ multidimensional discrete Fourier transforms (DFT) in order to estimate various target parameters. The DFT is implemented using the fast Fourier transform (FFT), at sample and computational complexity of $O(N)$ and $O(N \log N)$, respectively, where $N$ is the number of samples in the signal space. We have recently proposed a sparse Fourier transform based on the Fourier projection-slice theorem (FPS-SFT), which applies to multidimensional signals that are sparse in the frequency domain. FPS-SFT achieves sample complexity of $O(K)$ and computational complexity of $O(K \log K)$ for a multidimensional, $K$-sparse signal. While FPS-SFT considers the ideal scenario, i.e., exactly sparse data that contains on-grid frequencies, in this paper, by extending FPS-SFT into a robust version (RFPS-SFT), we emphasize on addressing noisy signals that contain off-grid frequencies; such signals arise from radar applications. This is achieved by employing a windowing technique and a voting-based frequency decoding procedure; the former reduces the frequency leakage of the off-grid frequencies below the noise level to preserve the sparsity of the signal, while the latter significantly lowers the frequency localization error stemming from the noise. The performance of the proposed method is demonstrated both theoretically and numerically.