The Sparse Fourier Transform: Theory and Practice

Abstract

The Fourier transform is one of the most fundamental tools for computing the frequency representation of signals. It plays a central role in signal processing, communications, audio and video compression, medical imaging, genomics, astronomy, as well as many other areas. Because of its widespread use, fast algorithms for computing the Fourier transform can benefit a large number of applications. The fastest algorithm for computing the Fourier transform is the Fast Fourier Transform (FFT) which runs in near-linear time making it an indispensable tool for many applications. However, today, the runtime of the FFT algorithm is no longer fast enough especially for big data problems where each dataset can be few terabytes. Hence, faster algorithms that run in sublinear time, i.e., do not even sample all the data points, have become necessary. The Sparse Fourier Transform: Theory and Practice book shows how to address the above problem by developing the Sparse Fourier Transform algorithms and building practical systems that use these algorithms to solve key problems in six different applications. Part I of the book focuses on the theory front. It introduces the Sparse Fourier Transform algorithms: a family of sublinear time algorithms for computing the Fourier transform faster than FFT. The Sparse Fourier Transform is based on the insight that many real-world signals are sparse, i.e., most of the frequencies have negligible contribution to the overall signal. Exploiting this sparsity, the book introduces several new algorithms which encompass two main axes: runtime complexity and sampling complexity. The book presents nearly optimal Sparse Fourier Transform algorithms that are faster than FFT and have the lowest runtime complexity known to date. It also presents Sparse Fourier Transform algorithms with optimal sampling complexity in the average case and the same nearly optimal runtime complexity. These algorithms use the minimum number of input data samples and hence, reduce acquisition cost and I/O overhead. Part II of the book focuses on the systems front. It describes software and hardware architectures for leveraging the Sparse Fourier Transform to address practical problems in six applied fields. In the area of wireless networks, it demonstrates how to use the Sparse Fourier Transform to build a wireless receiver that captures GHz-wide signals without sampling at the Nyquist rate enabling wideband spectrum sensing and acquisition using cheap commodity hardware. In mobile systems, the book introduces a new GPS receiver design that both reduces the delay to find the location and decreases the power consumption. In computer graphics, light fields enable new virtual reality and computational photography applications like interactive viewpoint changes, depth extraction and refocusing. The book shows that reconstructing light field images using the Sparse Fourier Transform reduces camera sampling requirements and improves image reconstruction quality. In the area of medical imaging, the book described how to enable efficient magnetic resonance spectroscopy (MRS), a new medical imaging technique that can reveal biomarkers for diseases like autism and cancer. In Biochemistry, it shows how the Sparse Fourier Transform can reduce Nuclear Magnetic Resonance (NMR) experiment time from weeks to days, enabling high dimensional NMR needed for discovering complex protein structures. Finally, in digital circuits, the book develops a chip with the largest Fourier Transform to date for sparse data. It delivers a 0.75 million point Sparse Fourier Transform chip that consumes 40x less power than prior FFT VLSI implementations. All these systems customize the theoretical algorithms to capture the structure of sparsity in each application, and hence maximize the resulting gains. The book also presents prototypes and evaluations of these systems in accordance with the standards of each application domain. This book is a revised version of the author's 2016 ACM Dissertation Award winning Ph.D. thesis. The revisions include editing, formatting, and minor corrections.