  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.
