The sparse fourier transform : theory & 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 FFT (Fast Fourier Transform) 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. This thesis addresses 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. Specifically, on the theory front, the thesis 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 thesis introduces several new algorithms which encompass two main axes: * Runtime Complexity: The thesis presents nearly optimal Sparse Fourier Transform algorithms that are faster than FFT and have the lowest runtime complexity known to date. " Sampling Complexity: The thesis 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. On the systems front, the thesis develops software and hardware architectures for leveraging the Sparse Fourier Transform to address practical problems in applied fields. Our systems customize the theoretical algorithms to capture the structure of sparsity in each application, and hence maximize the resulting gains. We prototype all of our systems and evaluate them in accordance with the standard's of each application domain. The following list gives an overview of the systems presented in this thesis. " Wireless Networks: The thesis demonstrates how to use the Sparse Fourier Transform to build a wireless receiver that captures GHz-wide signals without sampling at the Nyquist rate. Hence, it enables wideband spectrum sensing and acquisition using cheap commodity hardware. * Mobile Systems: The thesis uses the Sparse Fourier Transform to design a GPS receiver that both reduces the delay to find the location and decreases the power consumption by 2 x. " Computer Graphics: Light fields enable new virtual reality and computational photography applications like interactive viewpoint changes, depth extraction and refocusing. The thesis shows that reconstructing light field images using the Sparse Fourier Transform reduces camera sampling requirements and improves image reconstruction quality. * Medical Imaging: The thesis enables efficient magnetic resonance spectroscopy (MRS), a new medical imaging technique that can reveal biomarkers for diseases like autism and cancer. The thesis shows how to improve the image quality while reducing the time a patient spends in an MRI machine by 3 x (e.g., from two hours to less than forty minutes). * Biochemistry: The thesis demonstrates that the Sparse Fourier Transform reduces NMR (Nuclear Magnetic Resonance) experiment time by 16 x (e.g. from weeks to days), enabling high dimensional NMR needed for discovering complex protein structures. * Digital Circuits: The thesis 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 40 x less power than prior FFT VLSI implementations.