Mathematical analysis / Theory of signals Best bases for signal spaces Bases optimales pour des espaces de signaux (original) (raw)

2016

Article history: Received 2 October 2016 Accepted 3 October 2016 Available online xxxx Presented by Haïm Brézis We discuss the topic of selecting optimal orthonormal bases for representing classes of signals defined either through statistics or via some deterministic characterizations, or combinations of the two. In all cases, the best bases result from spectral analysis of a Hermitian matrix that summarizes the prior information we have on the signals we want to represent, achieving optimal progressive approximations. We also provide uniqueness proofs for the discrete cases. © 2016 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.

Modelling with Orthonormal Basis Functions

The decomposing description of linear time-invariant infinite-dimensional dynamics in terms of an orthonormal basis is an important part of modern Systems Theory and has a long history in modelling and identification of dynamical systems dating back to the classical work of Lee [19] and Wiener [36]. This approach is greatest utility when accurate system descriptions are achieved with a small number of basis functions. The development of suitable basis functions that reflect the dominant charecteristics of the system has attracted considerable interest [26, 28, 30, 31, 32, 33, 34, 35, 22, 23, 17, 3, 5, 4, 7]. In particular, in the areas of control theory, signal processing and system identification, there has long been interest in the use of the finite-impulse response, the Laguerre, and the two-parameter Kautz functions to model stable linear dynamical systems [19, 18, 16]. The Laguerre and the Kautz models are special cases of ...

Using nonstandard basis functions in description of signals and systems

2002

The subject of the thesis is the application of nonstandard basis functions in description of signals and systems. After an introduction the the general concept of representations, the frequency domain representations of discrete-time signals is elaborated, and a system of functions, called generalized orthogonal basis (GOB) functions, that forma an orthonormal basis in the space H2(D) is introduced. The signal representations based upon this system are applied in the detection and identification field in association with signals and systems emerging in application areas, such as industry, engineering, energy production, vehicles control, etc. The main motivation to apply representations upon nonstandard bases is the ability of these constructions to incorporate a priori information upon the characteristics of the system to be analyzed, which can result in high sensitivity on that characteristics themselves. The a priori information involved into the methods studied in the thesis is...

On Denoising and Signal Representation

2002

The problem of signal denoising using an orthogonal basis is considered. The framework of previous solutions converts the denoising problem into one of finding a threshold for estimates of basis coefficients. In this paper a new solution to the denoising problem is proposed. The method is based on calculation of the coefficient estimation error in each subspace of the basis. For each subspace, we estimate such criterion and suggest to choose the subspace for which this quantity is minimized. An information theoretic interpretation of the proposed approach introduces a new minimum description length (MDL) method of denoising. By comparison of the MDL of families of bases we can find the basis which minimizes this criterion. This offers a new method of best basis search for representation of the noisy data.

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