In signal processing, signal subspace methods are empirical linear methods for dimensionality reduction and noise reduction. These approaches have attracted significant interest and investigation recently in the context of speech enhancement, speech modeling, and speech classification research. The signal subspace is also used in radio direction finding using the MUSIC (algorithm). Essentially the methods represent the application of a principal components analysis (PCA) approach to ensembles of observed time-series obtained by sampling, for example sampling an audio signal. Such samples can be viewed as vectors in a high-dimensional vector space over the real numbers. PCA is used to identify a set of orthogonal basis vectors (basis signals) which capture as much as possible of the energy in the ensemble of observed samples. The vector space spanned by the basis vectors identified by the analysis is then the signal subspace. The underlying assumption is that information in speech signals is almost completely contained in a small linear subspace of the overall space of possible sample vectors, whereas additive noise is typically distributed through the larger space isotropically (for example when it is white noise). By on a signal subspace, that is, keeping only the component of the sample that is in the signal subspace defined by linear combinations of the first few most energized basis vectors, and throwing away the rest of the sample, which is in the remainder of the space orthogonal to this subspace, a certain amount of noise filtering is then obtained. Signal subspace noise-reduction can be compared to Wiener filter methods. There are two main differences: * The basis signals used in Wiener filtering are usually harmonic sine waves, into which a signal can be decomposed by Fourier transform. In contrast, the basis signals used to construct the signal subspace are identified empirically, and may for example be chirps, or particular characteristic shapes of transients after particular triggering events, rather than pure sinusoids. * The Wiener filter grades smoothly between linear components that are dominated by signal, and linear components that are dominated by noise. The noise components are filtered out, but not quite completely; the signal components are retained, but not quite completely; and there is a transition zone which is partly accepted. In contrast, the signal subspace approach represents a sharp cut-off: an orthogonal component either lies within the signal subspace, in which case it is 100% accepted, or orthogonal to it, in which case it is 100% rejected. This reduction in dimensionality, abstracting the signal into a much shorter vector, can be a particularly desired feature of the method. In the simplest case signal subspace methods assume white noise, but extensions of the approach to colored noise removal and the evaluation of the subspace-based speech enhancement for robust speech recognition have also been reported. (Wikipedia).
Notation and Basic Signal Properties
http://AllSignalProcessing.com for free e-book on frequency relationships and more great signal processing content, including concept/screenshot files, quizzes, MATLAB and data files. Signals as functions, discrete- and continuous-time signals, sampling, images, periodic signals, displayi
From playlist Introduction and Background
Introduction to Signal Processing
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From playlist Introduction and Background
http://AllSignalProcessing.com for more great signal processing content, including concept/screenshot files, quizzes, MATLAB and data files. Introduces three pervasive problems in signal processing: filtering, equalization, and system identification.
From playlist Introduction and Background
Determining Signal Similarities
Get a Free Trial: https://goo.gl/C2Y9A5 Get Pricing Info: https://goo.gl/kDvGHt Ready to Buy: https://goo.gl/vsIeA5 Find a signal of interest within another signal, and align signals by determining the delay between them using Signal Processing Toolbox™. For more on Signal Processing To
From playlist Signal Processing and Communications
What's a subspace of a vector space? How do we check if a subset is a subspace?
From playlist Linear Algebra
Frequency Domain Interpretation of Sampling
http://AllSignalProcessing.com for more great signal-processing content: ad-free videos, concept/screenshot files, quizzes, MATLAB and data files. Analysis of the effect of sampling a continuous-time signal in the frequency domain through use of the Fourier transform.
From playlist Sampling and Reconstruction of Signals
Reconstruction and the Sampling Theorem
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From playlist Sampling and Reconstruction of Signals
A discrete signal has to be reconstructed to get back into the continuous domain.
From playlist Discrete
Parsimonious Representations in data science - Dr Armin Eftekhari, University of Edinburgh
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From playlist Data science classes
Lecture 18 | Introduction to Linear Dynamical Systems
Professor Stephen Boyd, of the Electrical Engineering department at Stanford University, lectures on the applications of SVD, controllability, and state transfer in electrical engineering for the course, Introduction to Linear Dynamical Systems (EE263). Introduction to applied linear al
From playlist Lecture Collection | Linear Dynamical Systems
A matrix of coefficients, when viewed in column form, is used to create a column space. This is simply the space created by a linear combination of the column vectors. A resulting vector, b, that does not lie in this space will not result in a solution to the linear system. A set of vec
From playlist Introducing linear algebra
Lecture with Ole Christensen. Kapitler: 00:00 - Reaching The Goal; 05:00 - Problem With The Fourier Transform; 13:45 - Where Does The Fourier Transform Map Into?; 16:45 - Is F Bounded?; 20:00 - Fourier Transform On L2; 30:00 - Using The Extension Theorem;
From playlist DTU: Mathematics 4 Real Analysis | CosmoLearning.org Math
Covariant LEAst-square Re-fitting for Image Restoration - Papadakis - Workshop 1 - CEB T1 2019
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From playlist 2019 - T1 - The Mathematics of Imaging
Lp Spaces On The Real Line part 2
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From playlist DTU: Mathematics 4 Real Analysis | CosmoLearning.org Math
Stanford Seminar - Towards theories of single-trial high dimensional neural data analysis
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From playlist Stanford EE380-Colloquium on Computer Systems - Seminar Series
Ke Wang: Random perturbation of low-rank matrices
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From playlist Probability and Statistics
Lecture with Ole Christensen. Kapitler: 00:00 - Introduction; 06:45 - Vector Spaces; 07:15 - Example 1; 12:00 - Mathematical Tool - Fourier Transform; 17:00 - Example 2; 20:00 - Example 3; 23:00 - New Concept - Norm; 27:45 - Lemma 2.1.2 - The Opposite Triangle Inequality; 35:15 - Convergen
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Adaptive Estimation via Optimal Decision Trees by Subhajit Goswami
Program Advances in Applied Probability II (ONLINE) ORGANIZERS: Vivek S Borkar (IIT Bombay, India), Sandeep Juneja (TIFR Mumbai, India), Kavita Ramanan (Brown University, Rhode Island), Devavrat Shah (MIT, US) and Piyush Srivastava (TIFR Mumbai, India) DATE: 04 January 2021 to 08 Januar
From playlist Advances in Applied Probability II (Online)
Characterization of Random, Multivariate Signals
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