Results 1 
9 of
9
Fast Fourier transforms for nonequispaced data: A tutorial
, 2000
"... In this section, we consider approximative methods for the fast computation of multivariate discrete Fourier transforms for nonequispaced data (NDFT) in the time domain and in the frequency domain. In particular, we are interested in the approximation error as function of the arithmetic complexity o ..."
Abstract

Cited by 110 (32 self)
 Add to MetaCart
In this section, we consider approximative methods for the fast computation of multivariate discrete Fourier transforms for nonequispaced data (NDFT) in the time domain and in the frequency domain. In particular, we are interested in the approximation error as function of the arithmetic complexity of the algorithm. We discuss the robustness of NDFTalgorithms with respect to roundoff errors and apply NDFTalgorithms for the fast computation of Bessel transforms.
Fast algorithms for discrete polynomial transforms
 Math. Comput
, 1998
"... Abstract. Consider the Vandermondelike matrix P: = (Pk(cos jπ ..."
Abstract

Cited by 27 (7 self)
 Add to MetaCart
Abstract. Consider the Vandermondelike matrix P: = (Pk(cos jπ
Fast and stable algorithms for discrete spherical Fourier transforms
, 1996
"... . In this paper, we propose an algorithm for the stable and efficient computation of Fourier expansions of square integrable functions on the unit sphere S ae R 3 , as well as for the evaluation of these Fourier expansions at special knots. The heart of the algorithm is an efficient realization of ..."
Abstract

Cited by 16 (4 self)
 Add to MetaCart
. In this paper, we propose an algorithm for the stable and efficient computation of Fourier expansions of square integrable functions on the unit sphere S ae R 3 , as well as for the evaluation of these Fourier expansions at special knots. The heart of the algorithm is an efficient realization of discrete Legendre function transforms based on a modified and stabilized version of the DriscollHealy algorithm. 1991 Mathematics Subject Classification. Primary 65T99, 33C35, 33C25, 42C10 Key words and phrases. Discrete spherical Fourier transform, spherical harmonics, sampling theorem, discrete Legendre function transform, fast cosine transform, Chebyshev nodes, cascade summation 1 Introduction Fourier analysis on the sphere S ae R 3 has practical relevance in tomography, geophysics, seismology, meteorology and crystallography. It can be used in spectral methods for solving partial differential equations on the sphere (see [4], [16]). In [12], the authors utilize spherical Fourier...
CooleyTukey FFT Like Algorithms for the DCT
"... The CooleyTukey FFT algorithm decomposes a discrete Fourier transform (DFT) of size n = km into smaller DFTs of size k and m. In this paper we present a theorem that decomposes a polynomial transform into smaller polynomial transforms, and show that the FFT is obtained as a special case. Then we us ..."
Abstract

Cited by 13 (8 self)
 Add to MetaCart
The CooleyTukey FFT algorithm decomposes a discrete Fourier transform (DFT) of size n = km into smaller DFTs of size k and m. In this paper we present a theorem that decomposes a polynomial transform into smaller polynomial transforms, and show that the FFT is obtained as a special case. Then we use this theorem to derive a new class of recursive algorithms for the discrete cosine transforms (DCTs) of type II and type III. In contrast to other approaches, we manipulate polynomial algebras instead of transform matrix entries, which makes the derivation transparent, concise, and gives insight into the algorithms' structure. The derived algorithms have a regular structure and, for 2power size, minimal arithmetic cost (among known DCT algorithms).
Algebraic Signal Processing Theory: CooleyTukey Type Algorithms for DCTs and DSTs
, 2008
"... This paper presents a systematic methodology to derive and classify fast algorithms for linear transforms. The approach is based on the algebraic signal processing theory. This means that the algorithms are not derived by manipulating the entries of transform matrices, but by a stepwise decompositi ..."
Abstract

Cited by 7 (4 self)
 Add to MetaCart
This paper presents a systematic methodology to derive and classify fast algorithms for linear transforms. The approach is based on the algebraic signal processing theory. This means that the algorithms are not derived by manipulating the entries of transform matrices, but by a stepwise decomposition of the associated signal models, or polynomial algebras. This decomposition is based on two generic methods or algebraic principles that generalize the wellknown Cooley–Tukey fast Fourier transform (FFT) and make the algorithms ’ derivations concise and transparent. Application to the 16 discrete cosine and sine transforms yields a large class of fast general radix algorithms, many of which have not been found before.
SplitRadix Algorithms for Discrete Trigonometric Transforms
 Preprint, Gerhard{Mercator{ Univ. Duisburg
, 2002
"... In this paper, we derive new split{radix DCT{algorithms of radix{2 length, which are based on real factorization of the corresponding cosine matrices into products of sparse, orthogonal matrices. These algorithms use only permutations, scaling with 2, buttery operations, and plane rotations/rotat ..."
Abstract

Cited by 6 (5 self)
 Add to MetaCart
In this paper, we derive new split{radix DCT{algorithms of radix{2 length, which are based on real factorization of the corresponding cosine matrices into products of sparse, orthogonal matrices. These algorithms use only permutations, scaling with 2, buttery operations, and plane rotations/rotation{reections. They can be seen by analogy with the well{known split{radix FFT. Our new algorithms have a very low arithmetical complexity which compares with the best known fast DCT{algorithms. Further, a detailed analysis of the roundo errors for the new split{radix DCT{algorithm shows its excellent numerical stability which outperforms the real fast DCT{algorithms based on polynomial arithmetic.
Numerical stability of fast trigonometric transforms  a worst case study
 J. Concrete Appl. Math
, 2003
"... This paper presents some new results on numerical stability for various fast trigonometric transforms. In a worst case study, we consider the numerical stability of the classical fast Fourier transform (FFT) with respect to different precomputation methods for the involved twiddle factors and show t ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
This paper presents some new results on numerical stability for various fast trigonometric transforms. In a worst case study, we consider the numerical stability of the classical fast Fourier transform (FFT) with respect to different precomputation methods for the involved twiddle factors and show the strong influence of precomputation errors on the numerical stability of the FFT. The examinations are extended to fast algorithms for the computation of discrete cosine and sine transforms and to efficient computations of discrete Fourier transforms for nonequispaced data. Numerical tests confirm the theoretical estimates of numerical stability.
Complexity Analysis of Two Permutations Used by Fast Cosine Transform Algorithms
, 1995
"... The fast cosine transform algorithms introduced in [ST91, Ste92] require fewer operations than any other known general algorithm. Similar to related fast transform algorithms (e.g., the FFT), these algorithms permute the data before, during, or after the computation of the transform. The choice of t ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
The fast cosine transform algorithms introduced in [ST91, Ste92] require fewer operations than any other known general algorithm. Similar to related fast transform algorithms (e.g., the FFT), these algorithms permute the data before, during, or after the computation of the transform. The choice of this permutation may be an important consideration in reducing the complexity of the permutation algorithm. In this paper, we derive the complexity to generate the permutation mappings used in [ST91, Ste92] for powerof2 data sets by representing them as linear index transformations and translating them into combinational circuits. Moreover, we show that the permutation used in [Ste92] not only allows efficient implementation, but is also selfinvertible, i.e., we can use the same circuit to generate the permutation mapping for both the fast cosine transform and its inverse, like the bitreversal permutation used by FFT algorithms. These results may be useful to designers of lowlevel algori...
ALGEBRAIC SIGNAL PROCESSING THEORY: COOLEYTUKEY TYPE ALGORITHMS FOR POLYNOMIAL TRANSFORMS BASED ON INDUCTION ∗
"... Abstract. A polynomial transform is the multiplication of an input vector x ∈ C n by a matrix Pb,α ∈ C n×n, whose (k,ℓ)th element is defined as pℓ(αk) for polynomials pℓ(x) ∈ C[x] from a list b = {p0(x),...,pn−1(x)} and sample points αk ∈ C from a list α = {α0,...,αn−1}. Such transforms find appli ..."
Abstract
 Add to MetaCart
Abstract. A polynomial transform is the multiplication of an input vector x ∈ C n by a matrix Pb,α ∈ C n×n, whose (k,ℓ)th element is defined as pℓ(αk) for polynomials pℓ(x) ∈ C[x] from a list b = {p0(x),...,pn−1(x)} and sample points αk ∈ C from a list α = {α0,...,αn−1}. Such transforms find applications in the areas of signal processing, data compression, and function interpolation. Important examples include the discrete Fourier and cosine transforms. In this paper we introduce a novel technique to derive fast algorithms for polynomial transforms. The technique uses the relationship between polynomial transforms and the representation theory of polynomial algebras. Specifically, we derive algorithms by decomposing the regular modules of these algebras as a stepwise induction. As an application, we derive novel O(nlogn) generalradix algorithms for the discrete Fourier transform and the discrete cosine transform of type 4.