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

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

A High-dimensional Sparse Fourier Transform in the Continuous Setting

In this paper, we theoretically propose a new hashing scheme to establish the sparse Fourier transform in high-dimensional space. The estimation of the algorithm complexity shows that this sparse Fourier transform can overcome the curse of dimensionality. To the best of our knowledge, this is the first polynomial-time algorithm to recover the high-dimensional continuous frequencies

Multidimensional radar signal processing based on Sparse Fourier Transforms

The conventional radar signal processing typically employs the Fast Fourier Transform (FFT) to detect targets and identify their parameters. The sample and computational complexity of the N-point FFT are O(N) and O(N log N), respectively. In modern Digital Beamforming (DBF) and Multiple-Input Multiple-Output (MIMO) radars, $N$ is large due to the increased dimensions of processing (i.e., range, Doppler and angle) and the need for high radar resolution in each dimension. Hence, the FFT-based radar processing is still challenging for DBF/MIMO radars of constrained computation resources, such as the state-of-the-art automotive radars. Sparse Fourier Transform (SFT) is a family of low-complexity algorithms that implement Discrete Fourier Transform (DFT) for frequency-domain sparse signals. State-of-the-art SFT algorithms achieve sample complexity of O(K) and computational complexity of O(K log K) for a K-sparse signal. When K<<N, the sample and computational savings of SFT are significant as compared with that of the FFT. In radar applications, the number of radar targets is usually much smaller than the number of resolution cells in the multidimensional frequency domain, i.e., the radar signal is sparse in the frequency domain; thus, it is tempting to replace the FFT with SFT to reduce sample and computational complexity of signal processing. However, applying SFT in radar signal processing is not trivial for the following reasons: 1) Most existing SFT algorithms are designed for one-dimensional, ideal signals, which are noiseless and contain on-grid frequencies; those SFT algorithms are not practical for radar applications as the radar signals are multidimensional, noisy, and contain off-grid frequencies. 2) The signal processing schemes of different radar architectures need to be properly designed to support the application of SFT. When the radar signal is not naturally sparse, proper preprocessing is required to sparsify the signal. 3) The application of SFT in radar signal processing involves tradeoffs between sample/computational savings and radar detection performance. Such tradeoff needs to be characterized and the design of various parameters of SFT algorithms need to be investigated to achieve the optimal tradeoff. This dissertation aims to formulate SFT-based frameworks for radar signal processing and address the above issues by proposing two new SFT algorithms, and adapting them to DBF and MIMO radars. The proposed SFT algorithms are the Robust Sparse Fourier Transform (RSFT) and MultidimensionAl Random Slice based Sparse Fourier Transform (MARS-SFT). RSFT extends the basic SFT algorithm to multidimensional, noisy signals that contain off-grid frequencies. By incorporating Neyman-Pearson detection, frequency detection in the RSFT does not require knowledge of the exact sparsity of the signal and is robust to noise. The computational savings versus detection performance tradeoff is investigated, and the optimal threshold is found by solving a constrained optimization problem. The application of RSFT in DBF and MIMO radars is investigated. A uniform processing scheme based on RSFT is proposed for MIMO radar that employs fast-time coded and slow-time coded pulse-compression waveform. Although RSFT-based radar signal processing achieves significant computational savings as compared to FFT-based processing, it does not offer sample complexity savings. To reduce sample as well as computational complexity, we propose MARS-SFT, a sparse Fourier transform for multidimensional, frequency-domain sparse signals, inspired by the idea of the Fourier projection-slice theorem. MARS-SFT identifies frequencies by operating on one-dimensional slices of the discrete-time domain data, taken along specially designed lines; those lines are parametrized by slopes that are randomly generated from a set at runtime. The DFTs of the data slices represent DFT projections onto the lines along which the slices were taken. On designing the line lengths and slopes so that they allow for orthogonal and uniform frequency projections, the multidimensional frequencies can be recovered from their projections with low sample and computational complexity. To apply MARS-SFT to real-world radar signal processing, which involves noisy signals and off-grid frequencies, we propose the robust MARS-SFT, and demonstrate its performance in digital beamforming automotive radar signal processing. In that context, the robust MARS-SFT is used to identify range, velocity and angular parameters of targets with low sample and computational complexity. Finally, we propose a new automotive radar architecture. Such radar achieves high resolution in range, range rate, azimuth and elevation angles of extended targets by leveraging two orthogonally-placed digital beamforming linear arrays of a few channels. A deep learning based beam matching method is developed for the proposed radar to address the beam association challenges. In sparse scenarios, the proposed robust MARS-SFT can be employed in the beamforming, and range-Doppler imaging procedures to reduce computation.Ph.D.Includes bibliographical reference