5,612 research outputs found
Algebraic Signal Processing Theory: Cooley-Tukey Type Algorithms for Polynomial Transforms Based on Induction
A polynomial transform is the multiplication of an input vector x\in\C^n by
a matrix \PT_{b,\alpha}\in\C^{n\times n}, whose -th element is
defined as for polynomials p_\ell(x)\in\C[x] from a list
and sample points \alpha_k\in\C from a list
. 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 general-radix
algorithms for the discrete Fourier transform and the discrete cosine transform
of type 4.Comment: 19 pages. Submitted to SIAM Journal on Matrix Analysis and
Application
New Decimation-In-Time Fast Hartley Transform Algorithm
This paper presents a new algorithm for fast calculation of the discrete Hartley transform (DHT) based on decimation-in-time (DIT) approach. The proposed radix-2^2 fast Hartley transform (FHT) DIT algorithm has a regular butterfly structure that provides flexibility of different powers-of-two transform lengths, substantially reducing the arithmetic complexity with simple bit reversing for ordering the output sequence. The algorithm is developed through the three-dimensional linear index map and by integrating two stages of the signal flow graph together into a single butterfly. The algorithm is implemented and its computational complexity has been analysed and compared with the existing FHT algorithms, showing that it is significantly reduce the structural complexity with a better indexing scheme that is suitable for efficient implementation
Representations of Toeplitz-plus-Hankel matrices using trigonometric transformations with application to fast matrix-vector multiplication
AbstractRepresentations of real Toeplitz and Toeplitz-plus-Hankel matrices are presented that involve real trigonometric transformations (DCT, DST, DHT) and diagonal matrices. These representations can be used for fast matrix-vector multiplication. In particular, it is shown that the multiplication of an n × n Toeplitz-plus-Hankel matrix by a vector requires only 4 transformations of length n plus O(n) operations
Direct split-radix algorithm for fast computation of type-II discrete Hartley transform
In this paper, a novel split-radix algorithm for fast calculation the discrete Hartley transform of type-II (DHT-II) is intoduced. The algorithm is established through the decimation in time (DIT) approach, and implementedby splitting a length N of DHT-II into one DHT-II of length N/2 for even-indexed samples and two DHTs-II of length N/4 for odd-indexed samples. The proposed algorithm possesses the desired properties such as regularity, inplace calculation and it is represented by simple closed form decomposition sleading to considerable reductions in the arithmetic complexity compared to the existing DHT-II algorithms. Additionally, the validity of the proposed algorithm has been confirmed through analysing the arithmetic complexityby calculating the number of real additions and multiplications and associating it with the existing DHT-II algorithms
New fast Walsh–Hadamard–Hartley transform algorithm
This paper presents an efficient fast Walsh–Hadamard–Hartley transform (FWHT) algorithm that incorporates the computation of the Walsh-Hadamard transform (WHT) with the discrete Hartley transform (DHT) into an orthogonal, unitary single fast transform possesses the block diagonal structure. The proposed algorithm is implemented in an integrated butterfly structure utilizing the sparse matrices factorization approach and the Kronecker (tensor) product technique, which proved a valuable and fast tool for developing and analyzing the proposed algorithm. The proposed approach was distinguished by ease of implementation and reduced computational complexity compared to previous algorithms, which were based on the concatenation of WHT and FHT by saving up to 3N-4 of real multiplication and 7.5N-10 of real addition
Matrix algebras and displacement decompositions
A class xi of algebras of symmetric nxn matrices, related to Toeplitz-plus-Hankel structures and including the well-known algebra H diagonalized by the Hartley transform, is investigated. The algebras of xi are then exploited in a general displacement decomposition of an arbitrary nxn matrix A. Any algebra of xi is a 1-space, i.e., it is spanned by n matrices having as first rows the vectors of the canonical basis. The notion of 1-space (which generalizes the previous notions of L1 space [Bevilacqua and Zellini, Linear and Multilinear Algebra, 25 (1989), pp.1-25] and Hessenberg algebra [Di Fiore and Zellini, Linear Algebra Appl., 229 (1995), pp.49-99]) finally leads to the identification in xi of three new (non-Hessenberg) matrix algebras close to H, which are shown to be associated with fast Hartley-type transforms. These algebras are also involved in new efficient centrosymmetric Toeplitz-plus-Hankel inversion formulas
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