The fast generalized discrete Fourier transforms: A unified approach to the discrete sinusoidal transforms computation

Britaňák, Vladimír; Rao, K. R.

doi:10.1016/s0165-1684(99)00088-2

Cited by 46 publications

(34 citation statements)

References 32 publications

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“…3 with the backwards edge weighted with , the filter matrices are skew-circulant matrices, filtering is hence skewcircular convolution, and the polynomial Fourier transform has the form DFT , with a suitable diagonal matrix [9]. In particular, this class includes the generalized DFTs from [38] and [39] defined as where . We briefly investigate the 4 special cases given by , which in [39] are called DFTs of types 1-4, written as DFT-1 DFT-4.…”

Section: Boundary Conditions and Signal Extensionmentioning

confidence: 99%

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Algebraic Signal Processing Theory: Foundation and 1-D Time

Püschel

Moura

2008

IEEE Trans. Signal Process.

150

195

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Section: Boundary Conditions and Signal Extensionmentioning

confidence: 99%

“…In particular, this class includes the generalized DFTs from [38] and [39] defined as where . We briefly investigate the 4 special cases given by , which in [39] are called DFTs of types 1-4, written as DFT-1 DFT-4. Namely,…”

Section: Boundary Conditions and Signal Extensionmentioning

confidence: 99%

Algebraic Signal Processing Theory: Foundation and 1-D Time

Püschel

Moura

2008

IEEE Trans. Signal Process.

150

195

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“…The aim of the present section is to introduce and discuss fast DHT algorithms (cf. [2,7,5]). We start with some preliminaries.…”

Section: Finally We Show That (42) Is An Orthogonal Sum Representatimentioning

confidence: 99%

“…Bracewell [6]. Modified Hartley matrices of type II -IV are studied in [22,19,7,15]. Note that Fourier matrices of type I -IV are unitary (it is a trivial check) and that Hartley matrices of type I -IV are orthogonal.…”

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confidence: 99%

Fast and numerically stable algorithms for discrete Hartley transforms and applications to preconditioning

Aricò

Serra‐Capizzano

Tasche

2005

Communications in Information and Systems

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Abstract. The discrete Hartley transforms (DHT) of types I -IV and the related matrix algebras are discussed. We prove that any of these DHTs of length N = 2 t can be factorized by means of a divide-and-conquer strategy into a product of sparse, orthogonal matrices where in this context sparse means at most two nonzero entries per row and column. The sparsity joint with orthogonality of the matrix factors is the key for proving that these new algorithms have low arithmetic costs equal to 5 2 N log 2 (N )+ O(N ) arithmetic operations and an excellent normwise numerical stability. Further, we consider the optimal Frobenius approximation of a given symmetric Toeplitz matrix generated by an integrable symbol in a Hartley matrix algebra. We give explicit formulas for computing these optimal approximations and discuss the related preconditioned conjugate gradient (PCG) iterations. By using the matrix approximation theory, we prove the strong clustering at unity of the preconditioned matrix sequences under the sole assumption of continuity and positivity of the generating function. The multilevel case is also briefly treated. Some numerical experiments concerning DHT preconditioning are included.

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“…The Fourier series decomposes the given input signals into a sum of sinusoids. By removing the high frequency terms(noise) of Fourier series and then adding the remaining terms can yield better signals [4].…”

Section: Introductionmentioning

confidence: 99%