Algebraic study of Discrete Multidimensional Signal processing (original) (raw)
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A digital signal processing based on a representation over various algebraic systems is discussed. Theorems of spectral decomposition of a multiple-valued function are formulated. Examples of the function decomposition are given. i i a i t t f
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The vector space of the multi-indexed sequences over a field and the vector space of the sequences with finite support are dual to each other, with respect to a scalar product, which we used to define orthogonals in these spaces. The closed subspaces in the first vector space are then the orthogonals of subsets in the second space. Using power series and polynomials, we prove that the polynomial operator in the shift which U. Oberst and J. C. Willems have introduced to define time invariant discrete linear dynamical systems is the functorial adjoint of the polynomial multiplication. These results are generalized to the case of vectors of sequences and vectors of power series and polynomials. We end this paper by describing discrete linear algebraic dynamical systems.
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arXiv (Cornell University), 2012
In this paper, we use a duality between the vector space of the multi-indexed sequences over a field and the vector subspace of the sequences with finite support over this field to characterize the closed subpaces of multi-indexed sequences. Then we prove that the polynomial operator in the shift which Oberst and Willems have introduced to define time invariant discrete linear dynamical systems is the adjoint of the polynomial multiplication. We end this paper by describing these systems.
Some algebraic aspects of signal processing
Linear Algebra and its Applications, 1998
It has recently been shown in (M. Barnabei, L.B. Montefusco, Linear Algebra and applications 274 (1998) 367-388) that the algebraic-combinatorial notion of recursive matrix tan fruitfully be used to represent and easily handle the basic operations of filter theory, such as convolution, up-sampling, and down-sampling. In this Paper we show how the recursive matrix reinterpretation of two-channel FIR filter bank theory leads to a notable simplification in language and proofs, together with an easy and immediate generalization to the M-channel case. For example, in both 2-channel and M-channel cases, perfett reconstruction and alias concelation conditions tan be restated in an algebraic language, thereby obtaining an easy and constructive proof using the fundamental properties of recursive matrices. 0 1998 Elsevier Science Inc. All rights reserved.
Classifying Signals Under a Finite Abelian Group Action: The Finite Dimensional Setting
arXiv: Functional Analysis, 2019
Let GGG be a finite group acting on mathbbCN\mathbb{C}^NmathbbCN. We study the problem of identifyng the class in mathbbCN/G\mathbb{C}^N / GmathbbCN/G of a given signal: this encompasses several types of problems in signal processing. Some instances include certain generalizations of phase retrieval, image recognition, the analysis of textures, etc. In our previous work \cite{prev}, based on an algebraic approach, we constructed a Lipschitz translation invariant transform -- the case when GGG is cyclic. Here, we extend our results to include all finite Abelian groups. Moreover, we show the existence of a new transform that avoids computing high powers of the moduli of the signal entries--which can be computationally taxing. The new transform does not enjoy the algebraic structure imposed in our earlier work and is thus more flexible. Other (even lower) dimensional representations are explored, however they only provide an almost everywhere (actually generic) recovery which is not a significant drawback for applica...
Discrete linear Algebraic Dynamical Systems
arXiv (Cornell University), 2012
The vector space of the multi-indexed sequences over a field and the vector space of the sequences with finite support are dual to each other, with respect to a scalar product, which we used to define orthogonals in these spaces. The closed subspaces in the first vector space are then the orthogonals of subsets in the second space. Using power series and polynomials, we prove that the polynomial operator in the shift which U. Oberst and J. C. Willems have introduced to define time invariant discrete linear dynamical systems is the functorial adjoint of the polynomial multiplication. These results are generalized to the case of vectors of sequences and vectors of power series and polynomials. We end this paper by describing discrete linear algebraic dynamical systems.
Two-channel perfect reconstruction filter banks over commutative rings [online]
2000
The relation between ladder and lattice implementations of two-channel lter banks is discussed and it is shown that these two concepts di er in general. An elementary proof is given for the fact that over any integral domain which is not a eld there exist causal realizable perfect reconstructing lter banks that can not be implemented with causal lifting lters. A complete parametrization of lter banks with coe cients in local rings and semiperfect rings is given.
Multidimensional Systems and Signal Processing
We consider discrete behaviors with varying coefficients. Our results are new also for one-dimensional systems over the time-axis of natural numbers and for varying coefficients in a field, we derive the results, however, in much greater generality: Instead of the natural numbers we use an arbitrary submonoid N of an abelian group, for instance the standard multidimensional lattice of r-dimensional vectors of natural numbers or integers. We replace the base field by any commutative self-injective ring F, for instance a direct product of fields or a quasi-Frobenius ring or a finite factor ring of the integers. The F-module W of functions from N to F is the canonical discrete signal module and is a left module over the natural associated noncommutative ring A of difference operators with variable coefficients. Our main result states that this module is injective and therefore satisfies the fundamental principle: An inhomogeneous system of linear difference equations with variable coef...
Multi-dimensional signal processing using an algebraically extended signal representation
Lecture Notes in Computer Science, 1997
Many concepts that are used in multi{dimensional signal processing are derived from one{dimensional signal processing. As a consequence, they are only suited to multi{dimensional signals which are intrinsically one{dimensional. We claim that this restriction is due to the restricted algebraic frame used in signal processing, especially to the use of the complex numbers in the frequency domain. We propose a generalization of the two{dimensional Fourier transform which yields a quaternionic signal representation. We call this transform quaternionic Fourier transform (QFT). Based on the QFT, we generalize the conceptions of the analytic signal, Gabor lters, instantaneous and local phase to two dimensions in a novel way which is intrinsically two{dimensional. Experimental results are presented.