Algebraic Signal Processing Theory: Foundation and 1-D Time

Püschel, Markus; Moura, J.M.F.

doi:10.1109/tsp.2008.925261

Cited by 150 publications

(199 citation statements)

References 48 publications

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“…This would be particularly powerful as it enables the application of a vast array of results from a signal processing point of view and in particular, recent novel results from algebraic signal processing [11][12][13][14][15][16] will find ground-breaking applications.…”

Section: A Backgroundmentioning

confidence: 99%

“…022120-5 Natalie Baddour AIP Advances 1, 022120 (2011) and (14) which will arise in the evaluation of inverse Fourier transforms in 3D, 2D and 1D respectively. In the preceding, j n (x) are spherical Bessel functions of the first kind of order n and J n (x) are Bessel functions of the first kind, also of order n. φ(x) is an analytic function defined on the real line that remains bounded as |x| goes to infinity, r is a positive real variable and k 2 is the wavenumber that may be real or complex.…”

Section: Motivation For the Theoremsmentioning

confidence: 99%

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Multidimensional wave field signal theory: Mathematical foundations

Baddour

2011

AIP Advances

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Many important physical phenomena are described by wave or diffusion-wave type equations. Since these equations are linear, it would be useful to be able to use tools from the theory of linear signals and systems in solving related forward or inverse problems. In particular, the transform domain signal description from linear system theory has shown concrete promise for the solution of problems that are governed by a multidimensional wave field. The aim is to develop a unified framework for the description of wavefields via multidimensional signals. However, certain preliminary mathematical results are crucial for the development of this framework. This first paper on this topic thus introduces the mathematical foundations and proves some important mathematical results. The foundation of the framework starts with the inhomogeneous Helmholtz or pseudo-Helmholtz equation, which is the mathematical basis of a large class of wavefields. Application of the appropriate multi-dimensional Fourier transform leads to a transfer function description. To return to the physical spatial domain, certain mathematical results are necessary and these are presented and proved here as six fundamental theorems. These theorems are crucial for the evaluation of a certain class of improper integrals which arise in the evaluation of inverse multi-dimensional Fourier and Hankel transforms, upon which the framework is based. Subsequently, applications of these theorems are demonstrated, in particular for the derivation of Green's functions in different coordinate systems

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Section: A Backgroundmentioning

confidence: 99%

Section: Motivation For the Theoremsmentioning

confidence: 99%

Multidimensional wave field signal theory: Mathematical foundations

Baddour

2011

AIP Advances

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“…Associated with SP is the time shift operator, abstractly defined (in discrete form) as (1) The formulas for linear convolution and the discrete-time Fourier transform for infinite-length signals or for circular convolution and the discrete Fourier transform (DFT) for finitelength signals can be derived from this definition of the shift.…”

mentioning

confidence: 99%

“…This shift operates undirected or symmetrically in contrast to the directed operation of the time shift in (1). For this reason, we call it the space shift; it is abstractly defined as (2) Manuscript received December 3, 2005; revised April 8, 2008.…”

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confidence: 99%

Algebraic Signal Processing Theory: 1-D Space

Püschel

Moura

2008

IEEE Trans. Signal Process.

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Abstract-In our paper titled "Algebraic Signal Processing Theory: Foundation and 1-D Time" appearing in this issue of the IEEE TRANSACTIONS ON SIGNAL PROCESSING, we presented the algebraic signal processing theory, an axiomatic and general framework for linear signal processing. The basic concept in this theory is the signal model defined as the triple ( 8), where is a chosen algebra of filters, an associated -module of signals, and 8 is a generalization of the -transform. Each signal model has its own associated set of basic SP concepts, including filtering, spectrum, and Fourier transform. Examples include infinite and finite discrete time where these notions take their well-known forms. In this paper, we use the algebraic theory to develop infinite and finite space signal models. These models are based on a symmetric space shift operator, which is distinct from the standard time shift. We present the space signal processing concepts of filtering or convolution, " -transform," spectrum, and Fourier transform. For finite length space signals, we obtain 16 variants of space models, which have the 16 discrete cosine and sine transforms (DCTs/DSTs) as Fourier transforms. Using this novel derivation, we provide missing signal processing concepts associated with the DCTs/DSTs, establish them as precise analogs to the DFT, get deep insight into their origin, and enable the easy derivation of many of their properties including their fast algorithms.

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“…Every polynomial algebra has an associated notion of boundary condition, signal extension, convolution, spectrum, and Fourier transform, as explained in the algebraic signal processing theory [5,4]. As running example, we use C[x]/(x n − 1), which is known to be associated with the DFT [3].…”

Section: Introductionmentioning

confidence: 99%

Alternatives to the discrete fourier transform

Balcan

Sandryhaila

Groß

et al. 2008

2008 IEEE International Conference on Acoustics, Speech and Signal Processing

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It is well-known that the discrete Fourier transform (DFT) of a finite length discrete-time signal samples the discrete-time Fourier transform of the same signal at equidistant points on the unit circle. Hence, as the signal length goes to infinity, the DFT approaches the DTFT. Associated with the DFT are circular convolution and a periodic signal extension. In this paper we identify a large class of alternatives to the DFT using the theory of polynomial algebras. Each of these Fourier transforms approaches the DTFT just as the DFT does, but has its own signal extension and notion of convolution. Further, these Fourier transforms have Vandermonde structure, which enables their fast computation. We provide a few experimental examples that confirm our theoretical results.

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Algebraic Signal Processing Theory: Foundation and 1-D Time

Abstract: Abstract-This paper introduces a general and axiomatic approach to linear signal processing (SP) that we refer to as the algebraic signal processing theory (ASP). Basic to ASP is the linear signal model defined as a triple ( 8)

Cited by 150 publications

References 48 publications

Multidimensional wave field signal theory: Mathematical foundations

Multidimensional wave field signal theory: Mathematical foundations

Algebraic Signal Processing Theory: 1-D Space

Alternatives to the discrete fourier transform

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