325 research outputs found
A fast analysis-based discrete Hankel transform using asymptotic expansions
A fast and numerically stable algorithm is described for computing the
discrete Hankel transform of order as well as evaluating Schl\"{o}milch and
Fourier--Bessel expansions in
operations. The algorithm is based on an asymptotic expansion for Bessel
functions of large arguments, the fast Fourier transform, and the Neumann
addition formula. All the algorithmic parameters are selected from error bounds
to achieve a near-optimal computational cost for any accuracy goal. Numerical
results demonstrate the efficiency of the resulting algorithm.Comment: 22 page
Multiplier-less discrete sinusoidal and lapped transforms using sum-of-powers-of-two (SOPOT) coefficients
This paper proposes a new family of multiplier-less discrete cosine and sine transforms called the SOPOT DCTs and DSTs. The fast algorithm of Wang [10] is used to parameterize all the DCTs and DSTs in terms of certain (2×2) matrices, which are then converted to SOPOT representation using a method previously proposed by the authors [7]. The forward and inverse transforms can be implemented with the same set of SOPOT coefficients. A random search algorithm is also proposed to search for these SOPOT coefficients. Experimental results show that the (2×2) basic matrix can be implemented, on the average, in 6 to 12 additions. The proposed algorithms therefore require only O(N log2N) additions, which is very attractive for VLSI implementation. Using these SOPOT DCTs/DSTs, a family of SOPOT Lapped Transforms (LT) is also developed. They have similar coding gains but much lower complexity than their real-valued counterparts.published_or_final_versio
A fast, simple, and stable Chebyshev-Legendre transform using an asymptotic formula
A fast, simple, and numerically stable transform for converting between Legendre and Chebyshev coefficients of a degree polynomial in operations is derived. The basis of the algorithm is to rewrite a well-known asymptotic formula for Legendre polynomials of large degree as a weighted linear combination of Chebyshev polynomials, which can then be evaluated by using the discrete cosine transform. Numerical results are provided to demonstrate the efficiency and numerical stability. Since the algorithm evaluates a Legendre expansion at an Chebyshev grid as an intermediate step, it also provides a fast transform between Legendre coefficients and values on a Chebyshev grid
Discrete Cosine Transforms on Quantum Computers
A classical computer does not allow to calculate a discrete cosine transform
on N points in less than linear time. This trivial lower bound is no longer
valid for a computer that takes advantage of quantum mechanical superposition,
entanglement, and interference principles. In fact, we show that it is possible
to realize the discrete cosine transforms and the discrete sine transforms of
size NxN and types I,II,III, and IV with as little as O(log^2 N) operations on
a quantum computer, whereas the known fast algorithms on a classical computer
need O(N log N) operations.Comment: 5 pages, LaTeX 2e, IEEE ISPA01, Pula, Croatia, 200
M-Channel Fast Hartley Transform Based Integer DCT for Lossy-to-Lossless Image Coding
This paper presents an M-channel (M=2n (n ∈ N)) integer discrete cosine transforms (IntDCTs) based on fast Hartley transform (FHT) for lossy-to-lossless image coding which has image quality scalability from lossy data to lossless data. Many IntDCTs with lifting structures have already been presented to achieve lossy-to-lossless image coding. Recently, an IntDCT based on direct-lifting of DCT/IDCT, which means direct use of DCT and inverse DCT (IDCT) to lifting blocks, has been proposed. Although the IntDCT shows more efficient coding performance than any conventional IntDCT, it entails many computational costs due to an extra information that is a key point to realize its direct-lifting structure. On the other hand, the almost conventional IntDCTs without an extra information cannot be easily expanded to a larger size than the standard size M=8, or the conventional IntDCT should be improved for efficient coding performance even if it realizes an arbitrary size. The proposed IntDCT does not need any extra information, can be applied to size M=2n for arbitrary n, and shows better coding performance than the conventional IntDCTs without any extra information by applying the direct-lifting to the pre- and post-processing block of DCT. Moreover, the proposed IntDCT is implemented with a half of the computational cost of the IntDCT based on direct-lifting of DCT/IDCT even though it shows the best coding performance
Novel Fourier Quadrature Transforms and Analytic Signal Representations for Nonlinear and Non-stationary Time Series Analysis
The Hilbert transform (HT) and associated Gabor analytic signal (GAS)
representation are well-known and widely used mathematical formulations for
modeling and analysis of signals in various applications. In this study, like
the HT, to obtain quadrature component of a signal, we propose the novel
discrete Fourier cosine quadrature transforms (FCQTs) and discrete Fourier sine
quadrature transforms (FSQTs), designated as Fourier quadrature transforms
(FQTs). Using these FQTs, we propose sixteen Fourier-Singh analytic signal
(FSAS) representations with following properties: (1) real part of eight FSAS
representations is the original signal and imaginary part is the FCQT of the
real part, (2) imaginary part of eight FSAS representations is the original
signal and real part is the FSQT of the real part, (3) like the GAS, Fourier
spectrum of the all FSAS representations has only positive frequencies, however
unlike the GAS, the real and imaginary parts of the proposed FSAS
representations are not orthogonal to each other. The Fourier decomposition
method (FDM) is an adaptive data analysis approach to decompose a signal into a
set of small number of Fourier intrinsic band functions which are AM-FM
components. This study also proposes a new formulation of the FDM using the
discrete cosine transform (DCT) with the GAS and FSAS representations, and
demonstrate its efficacy for improved time-frequency-energy representation and
analysis of nonlinear and non-stationary time series.Comment: 22 pages, 13 figure
- …