Algebraic Signal Processing Theory: Cooley–Tukey Type Algorithms for DCTs and DSTs

Pueschel, Markus; Moura, J.M.F.

doi:10.1109/tsp.2007.907919

Cited by 81 publications

(85 citation statements)

References 67 publications

(163 reference statements)

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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%

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%

Multidimensional wave field signal theory: Mathematical foundations

Baddour

2011

AIP Advances

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“…3 (bottom), in the same way as the DFT is associated with the time shift. Third, knowing those signal models is the key to deriving and understanding the DTTs' fast algorithms [7], [9].…”

Section: Finite 1-d Space Models and Dttsmentioning

confidence: 99%

“…The polynomial DTTs will play an important role in the derivation of fast DTT algorithms [7]. Also, in some cases the polynomial DTTs have a lower complexity than the actual DTT.…”

Section: Dtt Dttmentioning

confidence: 99%

“…Second, they are necessary building blocks in the generalradix Cooley-Tukey type DTT algorithms derived in [7].…”

Section: Finite Skew -Transform and Skew Dttsmentioning

confidence: 99%

“…As expected, the finite space signals models underlying the DTTs are again built from polynomial algebras. One application of the ASP interpretation of the DTTs is the easy derivation of many of their properties and and their fast algorithms [7]- [9].…”

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confidence: 99%

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Algebraic Signal Processing Theory: 1-D Space

Püschel

Moura

2008

IEEE Trans. Signal Process.

106

113

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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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