The Sparse Fourier Transform - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook The Sparse Fourier Transform write by Haitham Hassanieh. This book was released on 2018-02-27. The Sparse Fourier Transform available in PDF, EPUB and Kindle. 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 Sparse Fourier Transform
The Sparse Fourier Transform - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook The Sparse Fourier Transform write by Haitham Hassanieh. This book was released on 2018-02-27. The Sparse Fourier Transform available in PDF, EPUB and Kindle. 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 Sparse Fourier Transform
The Sparse Fourier Transform - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook The Sparse Fourier Transform write by Joel Laity. This book was released on 2016. The Sparse Fourier Transform available in PDF, EPUB and Kindle. 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
The Sparse Fourier Transform
The Sparse Fourier Transform - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook The Sparse Fourier Transform write by Haitham Zuhair Al-Hassanieh. This book was released on 2016. The Sparse Fourier Transform available in PDF, EPUB and Kindle. 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.
Sparsity in the Spectrum
Sparsity in the Spectrum - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Sparsity in the Spectrum write by Craig Gross. This book was released on 2023. Sparsity in the Spectrum available in PDF, EPUB and Kindle. 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.
