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. This book 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: wireless networks; mobile systems; computer graphics; medical imaging; biochemistry; and digital circuits. This is a revised version of the thesis that won the 2016 ACM Doctoral Dissertation Award.

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.

The Fourier basis has been a cornerstone of numerical approximations due in part to its amenable algebraic properties resulting in efficient algorithmic approaches. Primary among these is the Fast Fourier Transform (FFT) which transforms a collection samples of a univariate function into that function's Fourier coefficients with computational complexity linear in the number of samples (with an extra logarithmic term). Extensions based on the FFT include algorithms that take advantage of sparsity in a function's Fourier coefficients (sparse Fourier transforms or SFTs) to lower this complexity even further as well as efficient approaches for approximating certain Fourier coefficients of multivariate functions, most often those indexed over computationally friendly hyperbolic cross structures. The ability to quickly compute a function's Fourier coefficients has additionally allowed for a variety of applications including fast algorithms for numerically solving partial differential equations (PDEs) via spectral methods. This dissertation considers improvements on these three applications of the FFT to produce (1) a high-dimensional Fourier transform over arbitrary index sets with reduced sampling complexity from current state of the art methods, (2) an accurate high-dimensional, sparse Fourier transform that can dramatically drive down the sampling and computational complexity so long as a sparsity assumption is satisfied, and (3) a high-dimensional, sparse spectral method which makes use of our sparse Fourier transform to solve PDEs with multiscale structure in extremely high dimensions.All three of these applications rely on the method of rank-1 lattices for their flexibility. By using this quasi-Monte Carlo approach for sampling in high-dimensions, high-dimensional functions are converted into one-dimensional ones on which well-studied techniques can be used. We extend these approaches by first developing a fully deterministic construction of multiple, smaller, rank-1 lattices to sample over simultaneously which drive down the sampling complexity from traditional rank-1 lattice methods. Our improved technique depends only linearly on the size of the underlying set of frequencies that Fourier coefficients are computed over rather than the previously standard quadratic dependence (with additional logarithmic terms).We can push further beyond this linear dependence on the frequency set of interest by making use of univariate SFTs after the high-dimensional to one-dimensional conversion. However, to effectively integrate univariate SFT algorithms into the rank-1 lattice approach without ruining the derived computational speedups, we provide an alternative approach. Rather than employing multiple rank-1 lattice sampling sets, we need to employ multiple rank-1 lattice SFTs. The slightly inflated sampling cost allows for significant gains in coefficient reconstruction: we produce two methods whose dependence on the frequency set of interest is cast entirely into logarithmic terms. The complexity is then quadratically or linearly (depending on the chosen variation) dependent on an imposed sparsity parameter and linear in the dimension of the underlying function domain. The dependence on this sparsity is then fully characterized in near-optimal approximation guarantees for the function of interest.And just as the FFT provided the foundation for fast spectral methods for numerically approximating solutions to PDE, so too does our high-dimensional, sparse Fourier transform provide the foundation for a high-dimensional, sparse spectral method. However, to be most effective, the underlying frequency set of interest should be primarily driven by the PDE itself rather than the user. As such, we provide a technique for efficiently converting sparse Fourier approximations of the PDE data into a Fourier basis in which the solution to the PDE will be guaranteed to have a good approximation. These ingredients combined with the rich literature on spectral methods allow for us to provide error estimates in the Sobolev norm for the solution which are fully characterized by properties of the PDE, namely the Fourier sparsity of its data and conditions related to its well-posedness.Throughout the text, these proposed algorithms are accompanied with practical considerations and implementations. These implementations are then judged against a variety of numerical tests which demonstrate performance on par with the theoretical guarantees provided.

We present a sublinear randomized algorithm to compute a sparse Fourier transform for nonequispaced data. We address the situation where a signal S is known to consist of N equispaced samples, of which only L

This book presents an introduction to the principles of the fast Fourier transform. This book covers FFTs, frequency domain filtering, and applications to video and audio signal processing. As fields like communications, speech and image processing, and related areas are rapidly developing, the FFT as one of essential parts in digital signal processing has been widely used. Thus there is a pressing need from instructors and students for a book dealing with the latest FFT topics. This book provides thorough and detailed explanation of important or up-to-date FFTs. It also has adopted modern approaches like MATLAB examples and projects for better understanding of diverse FFTs.

A companion volume to Weaver's Applications of Discrete and Continuous Fourier Analysis (Wiley, 1983). Addresses the theoretical and analytical aspects of Fourier analysis, including topics usually found only in more advanced treatises. Provides background information before going on to cover such topics as existence of the inner product, distribution theory, Fourier series representation of complex functions, properties and behavior of the Fourier transform, Fourier transform of a distribution, physical interpretation of convolution, the fast Fourier transform, sampling a function, and much more. Includes exercises, problems, applications, over 150 illustrations, and a Fourier transform FORTRAN subroutine.

This book offers a unified presentation of Fourier theory and corresponding algorithms emerging from new developments in function approximation using Fourier methods. It starts with a detailed discussion of classical Fourier theory to enable readers to grasp the construction and analysis of advanced fast Fourier algorithms introduced in the second part, such as nonequispaced and sparse FFTs in higher dimensions. Lastly, it contains a selection of numerical applications, including recent research results on nonlinear function approximation by exponential sums. The code of most of the presented algorithms is available in the authors’ public domain software packages. Students and researchers alike benefit from this unified presentation of Fourier theory and corresponding algorithms.

Fast Fourier Transform (FFT) is one of the most important numerical algorithms widely used in numerous scientific and engineering computations. With the emergence of big data problems, however, in which the size of the processed data can easily exceed terabytes, it is challenging to acquire, process and store a sufficient amount of data to compute the FFT in the first place. The recently developed \textit{sparse} FFT (sFFT) algorithm provides a solution to this problem. The sFFT can compute a compressed Fourier transform by using only a small subset of the input data, thus achieves significant performance improvement. Modern homogeneous and heterogeneous multicore and manycore architectures are now part of the mainstream computing scene and can offer impressive performance for many applications. The computations that arise in sFFT lend it naturally to efficient parallel implementations. In this dissertation, we present efficient parallel implementations of the sFFT algorithm on three state-of-the-art parallel architectures, namely multicore CPUs, GPUs and a heterogeneous multicore embedded system. While the increase in the number of cores and memory bandwidth on modern architectures provide an opportunity to improve the performance through sophisticated parallel algorithm design, the sFFT is inherently complex, and numerous challenges need to address to deliver the optimal performance. In this dissertation, various parallelization and performance optimization techniques are proposed and implemented. Our parallel sFFT is more than 5x and 20x faster than the sequential sFFT on multicore CPUs and GPUs, respectively. Compared to full-size FFT libraries, the parallel sFFT achieves more than 9x speedup on multicore CPUs and 12x speedup on GPUs for a broad range of signal spectra.

This book is intended to serve as an invaluable reference for anyone concerned with the application of wavelets to signal processing. It has evolved from material used to teach "wavelet signal processing" courses in electrical engineering departments at Massachusetts Institute of Technology and Tel Aviv University, as well as applied mathematics departments at the Courant Institute of New York University and ÉcolePolytechnique in Paris. - Provides a broad perspective on the principles and applications of transient signal processing with wavelets - Emphasizes intuitive understanding, while providing the mathematical foundations and description of fast algorithms - Numerous examples of real applications to noise removal, deconvolution, audio and image compression, singularity and edge detection, multifractal analysis, and time-varying frequency measurements - Algorithms and numerical examples are implemented in Wavelab, which is a Matlab toolbox freely available over the Internet - Content is accessible on several level of complexity, depending on the individual reader's needs New to the Second Edition - Optical flow calculation and video compression algorithms - Image models with bounded variation functions - Bayes and Minimax theories for signal estimation - 200 pages rewritten and most illustrations redrawn - More problems and topics for a graduate course in wavelet signal processing, in engineering and applied mathematics