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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.
We propose a multi-dimensional (M-D) sparse Fourier transform inspired by the idea of the Fourier projection-slice theorem, called FPS-SFT. FPS-SFT extracts samples along lines (1-dimensional slices from an M-D data cube), which are parameterized by
The Discrete Fourier Transform (DFT) is a fundamental computational primitive, and the fastest known algorithm for computing the DFT is the FFT (Fast Fourier Transform) algorithm. One remarkable feature of FFT is the fact that its runtime depends onl
By a novel reciprocal space analysis of the measurement, we report a calibrated in situ observation of the bunching effect in a 3D ultracold gas. The calibrated measurement with no free parameters confirms the role of the exchange symmetry and the Ha
In this paper, we redefine the Graph Fourier Transform (GFT) under the DSP$_mathrm{G}$ framework. We consider the Jordan eigenvectors of the directed Laplacian as graph harmonics and the corresponding eigenvalues as the graph frequencies. For this pu
In this paper we revisit the deterministic version of the Sparse Fourier Transform problem, which asks to read only a few entries of $x in mathbb{C}^n$ and design a recovery algorithm such that the output of the algorithm approximates $hat x$, the Di