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The Sparse Fourier Transform
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Book Synopsis The Sparse Fourier Transform by : Haitham Hassanieh
Download or read book The Sparse Fourier Transform written by Haitham Hassanieh and published by Morgan & Claypool. This book was released on 2018-02-27 with total page 282 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis The Sparse Fourier Transform by : Joel Laity
Download or read book The Sparse Fourier Transform written by Joel Laity and published by . This book was released on 2016 with total page 65 pages. Available in PDF, EPUB and Kindle. Book excerpt: Some functions can be well approximated by taking their Fourier transforms and discarding the terms that have small Fourier coefficients. The sparse Fourier transform is an algorithm that computes such an approximation more efficiently than computing the entire Fourier transform. The sparse Fourier transform has many applications to problems in mathematics and engineering. For example, in mathematics the sparse Fourier transform can be used to solve the chosen multiplier hidden number problem. In engineering, the sparse Fourier transform can be used to compress audio or video data. In Chapter 3 we present an algorithm that computes the sparse Fourier transform. This algorithm generalises and unifies the sparse fast Fourier transforms in [19] and [21]. These algorithms are of particular importance as they are the earliest algorithms for computing the sparse Fourier transform. The final chapter develops a method for reducing the problem of calculating the sparse Fourier transform over Zn to calculating it over Z2k where k is the smallest integer such that n
Book Synopsis The Sparse Fourier Transform by : Haitham Zuhair Al-Hassanieh
Download or read book The Sparse Fourier Transform written by Haitham Zuhair Al-Hassanieh and published by . This book was released on 2016 with total page 250 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Sparsity in the Spectrum by : Craig Gross
Download or read book Sparsity in the Spectrum written by Craig Gross and published by . This book was released on 2023 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Sub-Linear Sparse Fourier Transform Algorithm by : Ruochuan Zhang
Download or read book Sub-Linear Sparse Fourier Transform Algorithm written by Ruochuan Zhang and published by . This book was released on 2017 with total page 96 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis A Sublinear Algorithm of Sparse Fourier Transform for Nonequispaced Data by :
Download or read book A Sublinear Algorithm of Sparse Fourier Transform for Nonequispaced Data written by and published by . This book was released on 2005 with total page 28 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
Book Synopsis Numerical Fourier Analysis by : Gerlind Plonka
Download or read book Numerical Fourier Analysis written by Gerlind Plonka and published by Springer. This book was released on 2019-02-05 with total page 618 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis On the Butterfly Sparse Fourier Transform by : Stefan Kunis
Download or read book On the Butterfly Sparse Fourier Transform written by Stefan Kunis and published by . This book was released on 2011 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis A Wavelet Tour of Signal Processing by : Stephane Mallat
Download or read book A Wavelet Tour of Signal Processing written by Stephane Mallat and published by Elsevier. This book was released on 1999-09-14 with total page 663 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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
Book Synopsis Fast Fourier Transform - Algorithms and Applications by : K.R. Rao
Download or read book Fast Fourier Transform - Algorithms and Applications written by K.R. Rao and published by Springer Science & Business Media. This book was released on 2011-02-21 with total page 437 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Theory of Discrete and Continuous Fourier Analysis by : H. Joseph Weaver
Download or read book Theory of Discrete and Continuous Fourier Analysis written by H. Joseph Weaver and published by Wiley-Interscience. This book was released on 1989-01-17 with total page 328 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Imaging Applications of the Sparse FFT by : Lixin Shi (S.M.)
Download or read book Imaging Applications of the Sparse FFT written by Lixin Shi (S.M.) and published by . This book was released on 2013 with total page 81 pages. Available in PDF, EPUB and Kindle. Book excerpt: The sparse Fourier transform leverages the intrinsic sparsity of the frequency spectrum in many natural applications to compute the discrete Fourier Transform (DFT) in sub-linear time. Consequently, it has the potential to enable Big Data applications. In this thesis, we focus on extending the sparse Fourier transform (sparse FFT) to two imaging applications: 4D Light Field and Magnetic Resonance Spectroscopy. Directly applying sparse FFT to these applications however will not work. We need to extend the sparse FFT algorithm to address the following challenges: First, both applications are sample-intensive. It is time consuming, costly, and difficult to acquire samples. So, we need a new sparse FFT algorithm that minimizes the number of required input samples instead of purely focusing on the running time. Second, for these applications the spectra are not very sparse in the discrete Fourier domain. The sparsity is much greater in the continuous Fourier domain. Hence, we need a new sparse FFT algorithm that can leverage the sparsity in the continuous domain as opposed to the discrete domain. In this thesis, we design a sparse FFT algorithm suitable for our imaging applications. Our algorithm contains two phases: it first reconstructs a coarse discrete spectrum and then refines it using gradient descent in the continuous Fourier domain. In our experiments, we showed high-quality reconstruction of 4D light field with only 10% 20% of the samples, and a reduction of the MRS acquisition time by a factor of 3x 4x.
Book Synopsis High Performance Sparse Fourier Transform on Parallel Architectures by : Cheng Wang
Download or read book High Performance Sparse Fourier Transform on Parallel Architectures written by Cheng Wang and published by . This book was released on 2016 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Sparse Recovery and Fourier Sampling by : Eric C. Price
Download or read book Sparse Recovery and Fourier Sampling written by Eric C. Price and published by . This book was released on 2013 with total page 160 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the last decade a broad literature has arisen studying sparse recovery, the estimation of sparse vectors from low dimensional linear projections. Sparse recovery has a wide variety of applications such as streaming algorithms, image acquisition, and disease testing. A particularly important subclass of sparse recovery is the sparse Fourier transform, which considers the computation of a discrete Fourier transform when the output is sparse. Applications of the sparse Fourier transform include medical imaging, spectrum sensing, and purely computation tasks involving convolution. This thesis describes a coherent set of techniques that achieve optimal or near-optimal upper and lower bounds for a variety of sparse recovery problems. We give the following state-of-the-art algorithms for recovery of an approximately k-sparse vector in n dimensions: -- Two sparse Fourier transform algorithms, respectively taking ... time and ... samples. The latter is within log^e log n of the optimal sample complexity when ... -- An algorithm for adaptive sparse recovery using ... measurements, showing that adaptivity can give substantial improvements when k is small. -- An algorithm for C-approximate sparse recovery with ... measurements, which matches our lower bound up to the log* k factor and gives the first improvement for ... In the second part of this thesis, we give lower bounds for the above problems and more.
Book Synopsis Emerging Trends in Computing and Expert Technology by : D. Jude Hemanth
Download or read book Emerging Trends in Computing and Expert Technology written by D. Jude Hemanth and published by Springer Nature. This book was released on 2019-11-07 with total page 1642 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents high-quality research papers that demonstrate how emerging technologies in the field of intelligent systems can be used to effectively meet global needs. The respective papers highlight a wealth of innovations and experimental results, while also addressing proven IT governance, standards and practices, and new designs and tools that facilitate rapid information flows to the user. The book is divided into five major sections, namely: “Advances in High Performance Computing”, “Advances in Machine and Deep Learning”, “Advances in Networking and Communication”, “Advances in Circuits and Systems in Computing” and “Advances in Control and Soft Computing”.
Book Synopsis High Performance Sparse Fast Fourier Transform by : Jörn Schumacher
Download or read book High Performance Sparse Fast Fourier Transform written by Jörn Schumacher and published by . This book was released on 2013 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Computational Frameworks for the Fast Fourier Transform by : Charles Van Loan
Download or read book Computational Frameworks for the Fast Fourier Transform written by Charles Van Loan and published by SIAM. This book was released on 1992-01-01 with total page 285 pages. Available in PDF, EPUB and Kindle. Book excerpt: The author captures the interplay between mathematics and the design of effective numerical algorithms.
