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Real-Valued Discrete Fractional Hadamard Transform: fast algorithms and Implementations

IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-REGULAR PAPERS 2025年

作者： Fan, Zi-Chen Li, Di Rahardja, Susanto Northwestern Polytech Univ Sch Marine Sci & Technol Xian 710072 Peoples R China Singapore Inst Technol Singapore 138683 Singapore

This paper introduces a new real-valued discrete fractional Hadamard transform (RFHT) designed to address the issues of high computational complexity and large storage demands found in the traditional discrete fractional Hadamard transform (FHT) and its modifications. Additionally, a fast algorithm for the RFHT is developed, and the corresponding computational complexity analysis demonstrates that for sizes ranging from N=2 to 1024, the proposed fast algorithm can reduce the number of multiplications and additions by up to 50.0% and 83.3%, respectively, compared to state-of-the-art fast algorithms. The comparison results show that the RFHT also has lower execution time and power consumption. Furthermore, due to its real-valued property, the RFHT has been applied in an image watermarking system and implemented on real iOS devices, demonstrating enhanced information security. Lower execution and power consumption, reduced storage and transmission requirements, and superior information protection make the RFHT a superior candidate compared to WHT, FHT, and its modifications.

关键词： Transforms Computational complexity Watermarking Power demand Error correction codes Error correction Eigenvalues and eigenfunctions Fans Protection Image coding Real-valued discrete fractional Hadamard transform (RFHT) fast algorithms computational complexity reduction power consumption iOS implementation

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fast algorithms for Jacobi expansions via nonoscillatory phase functions

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IMA JOURNAL OF NUMERICAL ANALYSIS 2020年第3期40卷 2019-2051页

作者： Bremer, James Yang, Haizhao Univ Calif Davis Dept Math Davis CA 95616 USA Natl Univ Singapore Dept Math Singapore 119076 Singapore

We describe a suite of fast algorithms for evaluating Jacobi polynomials, applying the corresponding discrete Sturm-Liouville eigentransforms and calculating Gauss-Jacobi quadrature rules. Our approach, which applies in the case in which both of the parameters alpha and beta in Jacobi's differential equation are of magnitude less than 1/2, is based on the well-known fact that in this regime Jacobi's differential equation admits a nonoscillatory phase function that can be loosely approximated via an affine function over much of its domain. We illustrate this with several numerical experiments, the source code for which is publicly available.

关键词： fast algorithms fast special function transforms butterfly algorithms special functions nonoscillatory phase functions asymptotic methods

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fast algorithms for Low-Delay SBR Filterbanks in MPEG-4 AAC-ELD

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IEEE TRANSACTIONS ON AUDIO SPEECH AND LANGUAGE PROCESSING 2012年第3期20卷 1022-1031页

作者： Chivukula, Ravi K. Reznik, Yuriy A. Devarajan, Venkat Jayendra-Lakshman, Mythreya Qualcomm Inc San Diego CA 92121 USA InterDigital Inc San Diego CA 92121 USA Univ Texas Arlington Dept Elect Engn Arlington TX 76019 USA

The MPEG committee has recently completed development of a new audio coding standard "MPEG-4 Advanced Audio Coding-Enhanced Low Delay" (AAC-ELD). AAC-ELD is targeted towards high-quality, full-duplex communication applications such as audio and video conferencing. AAC-ELD uses low delay spectral band replication (LD-SBR) technology together with a low delay AAC core encoder to achieve high coding efficiency and low algorithmic delays. In this paper, we present fast algorithms for computing LD-SBR filterbanks in AAC-ELD. The proposed algorithms map complex exponential modulation portion of the filterbanks to discrete cosine transforms of types IV and II. Our proposed mapping also allows to merge some multiplications with the windowing stage that precedes or succeeds the modulation step. This further reduces computational complexity. Our presentation includes detailed explanation and flow-graphs of the algorithms, complexity analysis, and comparisons with alternative implementations.

关键词： Advanced audio coding (AAC) discrete cosine transform (DCT) DCT-IV factorization fast algorithms filterbank low delay audio coding Moving Picture Experts Group (MPEG) spectral band replication (SBR)

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fast algorithms FOR SOLVING TOEPLITZ-SYSTEMS OF EQUATIONS USING NUMBER-THEORETIC TRANSFORMS

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SIGNAL PROCESSING 1995年第1期44卷 89-101页

作者： HSUE, JJ YAGLE, AE UNIV MICHIGAN DEPT ELECT ENGN & COMP SCIANN ARBORMI 48109

Three fast algorithms for solving Toeplitz systems of equations are presented. The algorithms all embed the Toeplitz system in a circulant system, solve the latter using number-theoretic transforms, and then implement the Trench algorithm in reverse, also using number-theoretic transforms. Assuming all system matrix entries are rational numbers, the algorithms avoid roundoff error and any attendant conditioning problems. The first algorithm represents rational numbers with integers in a Galois field. The second algorithm computes the determinant of the system matrix and multiplies the system equation by it, producing an integer solution from which the actual rational solution may be determined. The third algorithm extends the second algorithm by using residue number systems. Numerical examples illustrate the new algorithms, and demonstrate the improvement over FFT-based algorithms in avoiding roundoff error in an ill-conditioned problem.

关键词： NUMBER-THEORETIC TRANSFORMS TOEPLITZ MATRICES fast algorithms

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fast algorithms for the Sylvester equation AX-XB^T=C

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THEORETICAL COMPUTER SCIENCE 2001年第1-2期259卷 623-638页

作者： Kirrinnis, P Univ Bonn Inst Informat 2 D-53117 Bonn Germany

For given matrices Al is an element of F-m X m, B is an element of F-n X n , and C is an element of F-m X n over an arbitrary field F, the matrix equation AX - XBT = C has a unique solution X is an element of F-m X n if and only if A and B have disjoint spectra. We describe an algorithm that computes the solution X for m, n less than or equal toN with O(N-beta . logN) arithmetic operations in F, where beta > 2 is such that M X M matrices can be multiplied with O(M-beta) arithmetic operations, e.g., beta = 2.376. It seems that before no better bound than O(m(3) . n(3)) arithmetic operations was known. The state of the art in numerical analysis is O(n(3) + m(3)) flops, but these algorithms (due to Bartels/Stewart and Golub/Nash/van Loan) involve Schur decompositions, i.e., they compute the eigenvalues of at least one of A and B, and can hence not be transferred for general F. (C) 2001 Elsevier Science B.V. AH rights reserved.

关键词： matrix equations Sylvester equation fast algorithms algebraic complexity

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fast algorithms of multidimensional discrete nonseparable K-wave transforms

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2002年第6期50卷 1496-1507页

作者： Rundblad, E Labunets, V Astola, J Egiazarian, K Tampere Univ Technol Signal Proc Lab FIN-33101 Tampere Finland Tampere Univ Technol Inst Signal Proc FIN-33101 Tampere Finland

fast algorithms for a wide class of nonseparable n-dimensional (n-D) discrete unitary K transforms (DKTs) are introduced. They need fewer 1-D DKTs than in the case of the classical radix-2 FFT-type approach. The method utilizes a decomposition of the n-D K transform into the product of a new n-D discrete Radon transform and of a set of parallel/independ 1-D K transforms. If the n-D K transform has a separable kernel (e.g., the case of the discrete Fourier transform), our approach leads to decrease of multiplicative complexity by the factor of n, compared with the classical row/column separable approach.

关键词： fast algorithms Fourier Hartley transforms multidimensional Radon Nussbaumer transform

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fast algorithms for Low-Delay TDAC Filterbanks in MPEG-4 AAC-ELD

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IEEE-ACM TRANSACTIONS ON AUDIO SPEECH AND LANGUAGE PROCESSING 2014年第12期22卷 1701-1712页

作者： Chivukula, Ravi K. Reznik, Yuriy A. Hu, Yanyan Devarajan, Venkat Jayendra-Lakshman, Mythreya Indian Sch Business Mohali 140306 Punjab India InterDigital Inc San Diego CA 92121 USA Univ Texas Arlington Dept Elect Engn Arlington TX 76019 USA Qualcomm Inc San Diego CA 92121 USA

The MPEG committee has completed development of a new audio coding standard called "MPEG-4 advanced audio coding-enhanced low delay" (AAC-ELD). AAC-ELD uses low delay spectral band replication (LD-SBR) technology together with a low delay time domain alias cancellation (LD TDAC) filterbank in the encoder to achieve both high coding efficiency and low algorithmic delay. In this paper, we present fast algorithms for implementing LD-TDAC filterbanks in AAC-ELD. Two types of fast algorithms are presented. In the first, we map LD-TDAC analysis and synthesis filterbanks to modified discrete cosine transform (MDCT) and inverse modified discrete cosine transform (IMDCT), respectively. Since MDCT/IMDCT are already extensively used in AAC and they have many fast algorithms, this mapping not only provides a fast implementation but also allows a common implementation of the filterbanks in AAC Low Complexity (AAC-LC), AAC Low Delay (AAC-LD) and AAC-ELD codecs. In the second algorithm, we provide a mapping to discrete Cosine transform of type II. The mapping to DCT-II allows the merger of the matrix operations with the windowing stage that precedes or follows them. This further reduces the number of multiplications and leads to an algorithm with the lowest known arithmetic complexity. For filterbanks of lengths 1024 and 960, we also present a new fast factorization of 15-point DCT-II that requires only 14 irrational multiplications, 3 dyadic rational multiplications and 67 additions.

关键词： AAC audio coding DCT factorization fast algorithms filterbanks low delay MDCT MPEG speech coding time domain alias cancellation

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fast algorithms for polynomial time-frequency transforms of real-valued sequences

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2008年第5期56卷 1905-1915页

作者： Bi, Guoan Ju, Yingtuo Li, Xiumei Nanyang Technol Univ Sch Elect & Elect Engn Singapore 639798 Singapore

This paper presents fast algorithms for computing the polynomial time-frequency transform that deals with a real-valued sequence of length-a(P)b, where a, b and p are positive integers. In particular, it shows that the polynomial time-frequency transform has a conjugate symmetric property, similar to that of the discrete Fourier transform, if the input sequence is real-valued. The computational complexities needed by these proposed algorithms are analyzed in terms of the numbers of real additions and real multiplications. When a = 3, 4, and 8, comparisons show that the computational complexities required by the proposed algorithms are less than 60% of those needed by the fast algorithms for complex-valued sequences.

关键词： fast algorithms Fourier transforms polynomial phase signal polynomial time-frequency transform

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fast algorithms for convolution quadrature of Riemann-Liouville fractional derivative

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APPLIED NUMERICAL MATHEMATICS 2019年 145卷 384-410页

作者： Sun, Jing Nie, Daxin Deng, Weihua Lanzhou Univ Sch Math & Stat Gansu Key Lab Appl Math & Complex Syst Lanzhou 730000 Gansu Peoples R China

Recently, the numerical schemes of the Fokker-Planck equations describing anomalous diffusion with two internal states have been proposed in Nie et al., [23], which use convolution quadrature to approximate the Riemann-Liouville fractional derivative;and the schemes need huge storage and computational cost because of the non-locality of fractional derivative and the large scale of the system. This paper first provides fast algorithms for computing the Riemann-Liouville derivative based on convolution quadrature with the generating function given by the backward Euler and second-order backward difference methods;the algorithms don't require the assumption of the regularity of the solution in time, while the computation time and the total memory requirement are greatly reduced. Then we apply fast algorithms to solve the homogeneous fractional Fokker-Planck equations with two internal states for nonsmooth data and get the first- and second-order accuracy in time. Lastly, numerical examples are presented to verify the convergence and the effectiveness of fast algorithms. (C) 2019 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Convolution quadrature fast algorithms Riemann-Liouville derivative Fractional Fokker-Planck equations Error estimates

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fast algorithms for spectral collocation with non-periodic boundary conditions

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JOURNAL OF COMPUTATIONAL PHYSICS 2005年第1期207卷 173-191页

作者： Lyons, W Ceniceros, HD Chandrasekaran, S Gu, M Univ Calif Santa Barbara Dept Math Santa Barbara CA 93106 USA Univ Calif Santa Barbara Dept Elect & Comp Engn Santa Barbara CA 93106 USA Univ Calif Berkeley Dept Math Berkeley CA 94720 USA

We present a method for the numerical solution of partial differential equations using spectral collocation. By employing a structured representation of linear operators we are able to use fast algorithms without being restricted to periodic boundary conditions. The underlying ideas are introduced and developed in the context of linearly implicit methods for stiff equations. We show how different boundary conditions may be applied and illustrate the technique on the Allen-Cahn equation and the diffusion equation. © 2005 Elsevier Inc. All rights reserved.

关键词： pseudospectral methods Chebyshev collocation fast algorithms fast direct solver semi-separable boundary conditions

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