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18.327 Wavelets, Filter Banks and Applications (MIT) 18.327 Wavelets, Filter Banks and Applications (MIT)

Description

Wavelets are localized basis functions, good for representing short-time events. The coefficients at each scale are filtered and subsampled to give coefficients at the next scale. This is Mallat's pyramid algorithm for multiresolution, connecting wavelets to filter banks. Wavelets and multiscale algorithms for compression and signal/image processing are developed. Subject is project-based for engineering and scientific applications. Wavelets are localized basis functions, good for representing short-time events. The coefficients at each scale are filtered and subsampled to give coefficients at the next scale. This is Mallat's pyramid algorithm for multiresolution, connecting wavelets to filter banks. Wavelets and multiscale algorithms for compression and signal/image processing are developed. Subject is project-based for engineering and scientific applications.

Subjects

Discrete-time filters | Discrete-time filters | convolution | convolution | Fourier transform | Fourier transform | owpass and highpass filters | owpass and highpass filters | Sampling rate change operations | Sampling rate change operations | upsampling and downsampling | upsampling and downsampling | ractional sampling | ractional sampling | interpolation | interpolation | Filter Banks | Filter Banks | time domain (Haar example) and frequency domain | time domain (Haar example) and frequency domain | conditions for alias cancellation and no distortion | conditions for alias cancellation and no distortion | perfect reconstruction | perfect reconstruction | halfband filters and possible factorizations | halfband filters and possible factorizations | Modulation and polyphase representations | Modulation and polyphase representations | Noble identities | Noble identities | block Toeplitz matrices and block z-transforms | block Toeplitz matrices and block z-transforms | polyphase examples | polyphase examples | Matlab wavelet toolbox | Matlab wavelet toolbox | Orthogonal filter banks | Orthogonal filter banks | paraunitary matrices | paraunitary matrices | orthogonality condition (Condition O) in the time domain | orthogonality condition (Condition O) in the time domain | modulation domain and polyphase domain | modulation domain and polyphase domain | Maxflat filters | Maxflat filters | Daubechies and Meyer formulas | Daubechies and Meyer formulas | Spectral factorization | Spectral factorization | Multiresolution Analysis (MRA) | Multiresolution Analysis (MRA) | requirements for MRA | requirements for MRA | nested spaces and complementary spaces; scaling functions and wavelets | nested spaces and complementary spaces; scaling functions and wavelets | Refinement equation | Refinement equation | iterative and recursive solution techniques | iterative and recursive solution techniques | infinite product formula | infinite product formula | filter bank approach for computing scaling functions and wavelets | filter bank approach for computing scaling functions and wavelets | Orthogonal wavelet bases | Orthogonal wavelet bases | connection to orthogonal filters | connection to orthogonal filters | orthogonality in the frequency domain | orthogonality in the frequency domain | Biorthogonal wavelet bases | Biorthogonal wavelet bases | Mallat pyramid algorithm | Mallat pyramid algorithm | Accuracy of wavelet approximations (Condition A) | Accuracy of wavelet approximations (Condition A) | vanishing moments | vanishing moments | polynomial cancellation in filter banks | polynomial cancellation in filter banks | Smoothness of wavelet bases | Smoothness of wavelet bases | convergence of the cascade algorithm (Condition E) | convergence of the cascade algorithm (Condition E) | splines | splines | Bases vs. frames | Bases vs. frames | Signal and image processing | Signal and image processing | finite length signals | finite length signals | boundary filters and boundary wavelets | boundary filters and boundary wavelets | wavelet compression algorithms | wavelet compression algorithms | Lifting | Lifting | ladder structure for filter banks | ladder structure for filter banks | factorization of polyphase matrix into lifting steps | factorization of polyphase matrix into lifting steps | lifting form of refinement equationSec | lifting form of refinement equationSec | Wavelets and subdivision | Wavelets and subdivision | nonuniform grids | nonuniform grids | multiresolution for triangular meshes | multiresolution for triangular meshes | representation and compression of surfaces | representation and compression of surfaces | Numerical solution of PDEs | Numerical solution of PDEs | Galerkin approximation | Galerkin approximation | wavelet integrals (projection coefficients | moments and connection coefficients) | wavelet integrals (projection coefficients | moments and connection coefficients) | convergence | convergence | Subdivision wavelets for integral equations | Subdivision wavelets for integral equations | Compression and convergence estimates | Compression and convergence estimates | M-band wavelets | M-band wavelets | DFT filter banks and cosine modulated filter banks | DFT filter banks and cosine modulated filter banks | Multiwavelets | Multiwavelets

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18.327 Wavelets, Filter Banks and Applications (MIT)

Description

Wavelets are localized basis functions, good for representing short-time events. The coefficients at each scale are filtered and subsampled to give coefficients at the next scale. This is Mallat's pyramid algorithm for multiresolution, connecting wavelets to filter banks. Wavelets and multiscale algorithms for compression and signal/image processing are developed. Subject is project-based for engineering and scientific applications.

Subjects

Discrete-time filters | convolution | Fourier transform | owpass and highpass filters | Sampling rate change operations | upsampling and downsampling | ractional sampling | interpolation | Filter Banks | time domain (Haar example) and frequency domain | conditions for alias cancellation and no distortion | perfect reconstruction | halfband filters and possible factorizations | Modulation and polyphase representations | Noble identities | block Toeplitz matrices and block z-transforms | polyphase examples | Matlab wavelet toolbox | Orthogonal filter banks | paraunitary matrices | orthogonality condition (Condition O) in the time domain | modulation domain and polyphase domain | Maxflat filters | Daubechies and Meyer formulas | Spectral factorization | Multiresolution Analysis (MRA) | requirements for MRA | nested spaces and complementary spaces; scaling functions and wavelets | Refinement equation | iterative and recursive solution techniques | infinite product formula | filter bank approach for computing scaling functions and wavelets | Orthogonal wavelet bases | connection to orthogonal filters | orthogonality in the frequency domain | Biorthogonal wavelet bases | Mallat pyramid algorithm | Accuracy of wavelet approximations (Condition A) | vanishing moments | polynomial cancellation in filter banks | Smoothness of wavelet bases | convergence of the cascade algorithm (Condition E) | splines | Bases vs. frames | Signal and image processing | finite length signals | boundary filters and boundary wavelets | wavelet compression algorithms | Lifting | ladder structure for filter banks | factorization of polyphase matrix into lifting steps | lifting form of refinement equationSec | Wavelets and subdivision | nonuniform grids | multiresolution for triangular meshes | representation and compression of surfaces | Numerical solution of PDEs | Galerkin approximation | wavelet integrals (projection coefficients | moments and connection coefficients) | convergence | Subdivision wavelets for integral equations | Compression and convergence estimates | M-band wavelets | DFT filter banks and cosine modulated filter banks | Multiwavelets

License

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18.327 Wavelets, Filter Banks and Applications (MIT)

Description

Wavelets are localized basis functions, good for representing short-time events. The coefficients at each scale are filtered and subsampled to give coefficients at the next scale. This is Mallat's pyramid algorithm for multiresolution, connecting wavelets to filter banks. Wavelets and multiscale algorithms for compression and signal/image processing are developed. Subject is project-based for engineering and scientific applications.

Subjects

Discrete-time filters | convolution | Fourier transform | owpass and highpass filters | Sampling rate change operations | upsampling and downsampling | ractional sampling | interpolation | Filter Banks | time domain (Haar example) and frequency domain | conditions for alias cancellation and no distortion | perfect reconstruction | halfband filters and possible factorizations | Modulation and polyphase representations | Noble identities | block Toeplitz matrices and block z-transforms | polyphase examples | Matlab wavelet toolbox | Orthogonal filter banks | paraunitary matrices | orthogonality condition (Condition O) in the time domain | modulation domain and polyphase domain | Maxflat filters | Daubechies and Meyer formulas | Spectral factorization | Multiresolution Analysis (MRA) | requirements for MRA | nested spaces and complementary spaces; scaling functions and wavelets | Refinement equation | iterative and recursive solution techniques | infinite product formula | filter bank approach for computing scaling functions and wavelets | Orthogonal wavelet bases | connection to orthogonal filters | orthogonality in the frequency domain | Biorthogonal wavelet bases | Mallat pyramid algorithm | Accuracy of wavelet approximations (Condition A) | vanishing moments | polynomial cancellation in filter banks | Smoothness of wavelet bases | convergence of the cascade algorithm (Condition E) | splines | Bases vs. frames | Signal and image processing | finite length signals | boundary filters and boundary wavelets | wavelet compression algorithms | Lifting | ladder structure for filter banks | factorization of polyphase matrix into lifting steps | lifting form of refinement equationSec | Wavelets and subdivision | nonuniform grids | multiresolution for triangular meshes | representation and compression of surfaces | Numerical solution of PDEs | Galerkin approximation | wavelet integrals (projection coefficients | moments and connection coefficients) | convergence | Subdivision wavelets for integral equations | Compression and convergence estimates | M-band wavelets | DFT filter banks and cosine modulated filter banks | Multiwavelets

License

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2.160 Identification, Estimation, and Learning (MIT) 2.160 Identification, Estimation, and Learning (MIT)

Description

This course provides a broad theoretical basis for system identification, estimation, and learning. Students will study least squares estimation and its convergence properties, Kalman filters, noise dynamics and system representation, function approximation theory, neural nets, radial basis functions, wavelets, Volterra expansions, informative data sets, persistent excitation, asymptotic variance, central limit theorems, model structure selection, system order estimate, maximum likelihood, unbiased estimates, Cramer-Rao lower bound, Kullback-Leibler information distance, Akaike's information criterion, experiment design, and model validation. This course provides a broad theoretical basis for system identification, estimation, and learning. Students will study least squares estimation and its convergence properties, Kalman filters, noise dynamics and system representation, function approximation theory, neural nets, radial basis functions, wavelets, Volterra expansions, informative data sets, persistent excitation, asymptotic variance, central limit theorems, model structure selection, system order estimate, maximum likelihood, unbiased estimates, Cramer-Rao lower bound, Kullback-Leibler information distance, Akaike's information criterion, experiment design, and model validation.

Subjects

system identification; estimation; least squares estimation; Kalman filter; noise dynamics; system representation; function approximation theory; neural nets; radial basis functions; wavelets; volterra expansions; informative data sets; persistent excitation; asymptotic variance; central limit theorem; model structure selection; system order estimate; maximum likelihood; unbiased estimates; Cramer-Rao lower bound; Kullback-Leibler information distance; Akaike?s information criterion; experiment design; model validation. | system identification; estimation; least squares estimation; Kalman filter; noise dynamics; system representation; function approximation theory; neural nets; radial basis functions; wavelets; volterra expansions; informative data sets; persistent excitation; asymptotic variance; central limit theorem; model structure selection; system order estimate; maximum likelihood; unbiased estimates; Cramer-Rao lower bound; Kullback-Leibler information distance; Akaike?s information criterion; experiment design; model validation. | system identification | system identification | estimation | estimation | least squares estimation | least squares estimation | Kalman filter | Kalman filter | noise dynamics | noise dynamics | system representation | system representation | function approximation theory | function approximation theory | neural nets | neural nets | radial basis functions | radial basis functions | wavelets | wavelets | volterra expansions | volterra expansions | informative data sets | informative data sets | persistent excitation | persistent excitation | asymptotic variance | asymptotic variance | central limit theorem | central limit theorem | model structure selection | model structure selection | system order estimate | system order estimate | maximum likelihood | maximum likelihood | unbiased estimates | unbiased estimates | Cramer-Rao lower bound | Cramer-Rao lower bound | Kullback-Leibler information distance | Kullback-Leibler information distance | Akaike?s information criterion | Akaike?s information criterion | experiment design | experiment design | model validation | model validation

License

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18.337J Parallel Computing (MIT) 18.337J Parallel Computing (MIT)

Description

This is an advanced interdisciplinary introduction to applied parallel computing on modern supercomputers. It has a hands-on emphasis on understanding the realities and myths of what is possible on the world's fastest machines. We will make prominent use of the Julia Language, a free, open-source, high-performance dynamic programming language for technical computing. This is an advanced interdisciplinary introduction to applied parallel computing on modern supercomputers. It has a hands-on emphasis on understanding the realities and myths of what is possible on the world's fastest machines. We will make prominent use of the Julia Language, a free, open-source, high-performance dynamic programming language for technical computing.

Subjects

cloud computing | cloud computing | dense linear algebra | dense linear algebra | sparse linear algebra | sparse linear algebra | N-body problems | N-body problems | multigrid | multigrid | fast-multipole | fast-multipole | wavelets | wavelets | Fourier transforms | Fourier transforms | partitioning | partitioning | mesh generation | mesh generation | applications oriented architecture | applications oriented architecture | parallel programming paradigms | parallel programming paradigms | MPI | MPI | data parallel systems | data parallel systems | Star-P | Star-P | parallel Python | parallel Python | parallel Matlab | parallel Matlab | graphics processors | graphics processors | virtualization | virtualization | caches | caches | vector processors | vector processors | VHLLs | VHLLs | Very High Level Languages | Very High Level Languages | Julia programming language | Julia programming language | distributed parallel execution | distributed parallel execution

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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2.160 Identification, Estimation, and Learning (MIT)

Description

This course provides a broad theoretical basis for system identification, estimation, and learning. Students will study least squares estimation and its convergence properties, Kalman filters, noise dynamics and system representation, function approximation theory, neural nets, radial basis functions, wavelets, Volterra expansions, informative data sets, persistent excitation, asymptotic variance, central limit theorems, model structure selection, system order estimate, maximum likelihood, unbiased estimates, Cramer-Rao lower bound, Kullback-Leibler information distance, Akaike's information criterion, experiment design, and model validation.

Subjects

system identification; estimation; least squares estimation; Kalman filter; noise dynamics; system representation; function approximation theory; neural nets; radial basis functions; wavelets; volterra expansions; informative data sets; persistent excitation; asymptotic variance; central limit theorem; model structure selection; system order estimate; maximum likelihood; unbiased estimates; Cramer-Rao lower bound; Kullback-Leibler information distance; Akaike?s information criterion; experiment design; model validation. | system identification | estimation | least squares estimation | Kalman filter | noise dynamics | system representation | function approximation theory | neural nets | radial basis functions | wavelets | volterra expansions | informative data sets | persistent excitation | asymptotic variance | central limit theorem | model structure selection | system order estimate | maximum likelihood | unbiased estimates | Cramer-Rao lower bound | Kullback-Leibler information distance | Akaike?s information criterion | experiment design | model validation

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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18.335J Numerical Methods of Applied Mathematics I (MIT) 18.335J Numerical Methods of Applied Mathematics I (MIT)

Description

IEEE-standard, iterative and direct linear system solution methods, eigendecomposition and model-order reduction, fast Fourier transforms, multigrid, wavelets and other multiresolution methods, matrix sparsification. Nonlinear root finding (Newton's method). Numerical interpolation and extrapolation. Quadrature.Technical RequirementsFile decompression software, such as Winzip or StuffIt, is required to open the .tar files found on this course site. The .tar files contain additional files which require software as well. MATLAB® software is required to run the .m files.Postscript viewer software, such as Ghostscript/Ghostview, can be used to view the .ps files.Ghostscript/Ghostview, Adobe Photoshop, and Adobe Illustrator are among the software tools that can be used to view the .ep IEEE-standard, iterative and direct linear system solution methods, eigendecomposition and model-order reduction, fast Fourier transforms, multigrid, wavelets and other multiresolution methods, matrix sparsification. Nonlinear root finding (Newton's method). Numerical interpolation and extrapolation. Quadrature.Technical RequirementsFile decompression software, such as Winzip or StuffIt, is required to open the .tar files found on this course site. The .tar files contain additional files which require software as well. MATLAB® software is required to run the .m files.Postscript viewer software, such as Ghostscript/Ghostview, can be used to view the .ps files.Ghostscript/Ghostview, Adobe Photoshop, and Adobe Illustrator are among the software tools that can be used to view the .ep

Subjects

IEEE-standard | IEEE-standard | iterative and direct linear system solution methods | iterative and direct linear system solution methods | eigendecomposition and model-order reduction | eigendecomposition and model-order reduction | fast Fourier transforms | fast Fourier transforms | multigrid | multigrid | wavelets | wavelets | other multiresolution methods | other multiresolution methods | matrix sparsification | matrix sparsification | Nonlinear root finding (Newton's method) | Nonlinear root finding (Newton's method) | Numerical interpolation | Numerical interpolation | Numerical extrapolation | Numerical extrapolation | Quadrature | Quadrature | 18.335 | 18.335 | 6.337 | 6.337

License

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1.060 Engineering Mechanics II (MIT) 1.060 Engineering Mechanics II (MIT)

Description

This subject provides an introduction to fluid mechanics. Students are introduced to and become familiar with all relevant physical properties and fundamental laws governing the behavior of fluids and learn how to solve a variety of problems of interest to civil and environmental engineers. While there is a chance to put skills from calculus and differential equations to use in this subject, the emphasis is on physical understanding of why a fluid behaves the way it does. The aim is to make the students think as a fluid. In addition to relating a working knowledge of fluid mechanics, the subject prepares students for higher-level subjects in fluid dynamics. This subject provides an introduction to fluid mechanics. Students are introduced to and become familiar with all relevant physical properties and fundamental laws governing the behavior of fluids and learn how to solve a variety of problems of interest to civil and environmental engineers. While there is a chance to put skills from calculus and differential equations to use in this subject, the emphasis is on physical understanding of why a fluid behaves the way it does. The aim is to make the students think as a fluid. In addition to relating a working knowledge of fluid mechanics, the subject prepares students for higher-level subjects in fluid dynamics.

Subjects

fluid mechanics | fluid mechanics | fluids | fluids | civil and environmental engineering | civil and environmental engineering | differential equations | differential equations | calculus | calculus | flow | flow | movement | movement | wave forms | wave forms | Bernoulli's theorem | Bernoulli's theorem | wavelets | wavelets | mechanics | mechanics | solids | solids | hydrostatics | hydrostatics | mass | mass | momentum | momentum | energy | energy | flow nets | flow nets | velocity | velocity | laminar flow | laminar flow | turbulent flow | turbulent flow | groundwater | groundwater | hydraulics | hydraulics | backwater curves | backwater curves

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm

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18.337J Parallel Computing (MIT)

Description

This is an advanced interdisciplinary introduction to applied parallel computing on modern supercomputers. It has a hands-on emphasis on understanding the realities and myths of what is possible on the world's fastest machines. We will make prominent use of the Julia Language, a free, open-source, high-performance dynamic programming language for technical computing.

Subjects

cloud computing | dense linear algebra | sparse linear algebra | N-body problems | multigrid | fast-multipole | wavelets | Fourier transforms | partitioning | mesh generation | applications oriented architecture | parallel programming paradigms | MPI | data parallel systems | Star-P | parallel Python | parallel Matlab | graphics processors | virtualization | caches | vector processors | VHLLs | Very High Level Languages | Julia programming language | distributed parallel execution

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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1.060 Engineering Mechanics II (MIT)

Description

This subject provides an introduction to fluid mechanics. Students are introduced to and become familiar with all relevant physical properties and fundamental laws governing the behavior of fluids and learn how to solve a variety of problems of interest to civil and environmental engineers. While there is a chance to put skills from calculus and differential equations to use in this subject, the emphasis is on physical understanding of why a fluid behaves the way it does. The aim is to make the students think as a fluid. In addition to relating a working knowledge of fluid mechanics, the subject prepares students for higher-level subjects in fluid dynamics.

Subjects

fluid mechanics | fluids | civil and environmental engineering | differential equations | calculus | flow | movement | wave forms | Bernoulli's theorem | wavelets | mechanics | solids | hydrostatics | mass | momentum | energy | flow nets | velocity | laminar flow | turbulent flow | groundwater | hydraulics | backwater curves

License

Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see https://ocw.mit.edu/terms/index.htm

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18.335J Numerical Methods of Applied Mathematics I (MIT)

Description

IEEE-standard, iterative and direct linear system solution methods, eigendecomposition and model-order reduction, fast Fourier transforms, multigrid, wavelets and other multiresolution methods, matrix sparsification. Nonlinear root finding (Newton's method). Numerical interpolation and extrapolation. Quadrature.Technical RequirementsFile decompression software, such as Winzip or StuffIt, is required to open the .tar files found on this course site. The .tar files contain additional files which require software as well. MATLAB® software is required to run the .m files.Postscript viewer software, such as Ghostscript/Ghostview, can be used to view the .ps files.Ghostscript/Ghostview, Adobe Photoshop, and Adobe Illustrator are among the software tools that can be used to view the .ep

Subjects

IEEE-standard | iterative and direct linear system solution methods | eigendecomposition and model-order reduction | fast Fourier transforms | multigrid | wavelets | other multiresolution methods | matrix sparsification | Nonlinear root finding (Newton's method) | Numerical interpolation | Numerical extrapolation | Quadrature | 18.335 | 6.337

License

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