Robust sparse fourier transform based on the fourier projection-slice theorem

We have recently proposed a sparse Fourier transform based on the Fourier projection-slice theorem (FPS-SFT), which is an efficient implementation of the discrete Fourier transform for multidimensional signals that are sparse in the frequency domain. For a K-sparse signal, FPS-SFT achieves sample complexity of O(K) and computational complexity of O(K log K). While FPS-SFT considers the ideal scenario, i.e., exactly sparse data that contain on-grid frequencies, in this paper, we propose a robust FPS-SFT (RFPS-SFT), which applies to noisy signals that contain off-grid frequencies; such signals arise in radar applications. RFPS-SFT employs a windowing step and a voting-based frequency decoding step; the former reduces the frequency leakage of off-grid frequencies below the noise level, thus preserving 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.