Showing papers on "Discrete sine transform published in 1984"

A new algorithm to compute the discrete cosine Transform

Abstract: A new algorithm is introduced for the 2m-point discrete cosine transform. This algorithm reduces the number of multiplications to about half of those required by the existing efficient algorithms, and it makes the system simpler.

Fast algorithms for the discrete W transform and for the discrete Fourier transform

Abstract: A systematic method of sparse matrix factorization is developed for all four versions of the discrete W transform, the discrete cosine transform, and the discrete sine transform, as well as for the discrete Fourier transform. The factorization leads to fast algorithms in which only real arithmetic is involved. A scheme for reducing multiplications and a convenient index system are introduced. This makes new algorithms more efficient than conventional algorithms for the discrete Fourier transform, the discrete cosine transform, and the discrete sine transform.

Fast decimation-in-time algorithms for a family of discrete sine and cosine transforms

Abstract: Fast decimation-in-time (DIT) algorithms for the various discrete cosine transforms (DCT) and discrete sine transforms (DST) are systematically developed, based on a radix-2 factorization of the transformation matrix. The results indicate these to be attractive alternatives to existing algorithms in terms of computational complexity and structural simplicity.

Short-space Fourier transform image processing

Abstract: The short-space Fourier transform (SSFT) is introduced as a means of describing discrete multi-dimensional signals of finite extent. It is an adaptation of the short-time Fourier transform developed for one-dimensional infinite-duration signals such as speech. By reflectively extending the finite signal segment, one can imagine an infinite duration signal which is "continuous." The proposed SSFT is the multidimensional generalization of the short-time Fourier transform operating upon the resulting infinite duration signal. Because boundary "discontinuities" are avoided, the proposed SSFT provides a transform representation free of extraneous spectral energy. An efficient algorithm for computing the SSET is described. SSFT image coding, an important application of the new transform method, provides localized spectral information without the undesirable phenomenon of "blocking effects."

Discrete Fourier transform algorithms for real valued sequences

Abstract: It is well known that the discrete Fourier transformation (DFT) of a real-valued sequence contains some redundancies. More precisely, approximately half of the DFT coefficients suffice to completely determine the DFT. In this paper, it is shown that the choice of the set of (nonredundant) DFT coefficients to be calculated affects the efficiency of the resulting algorithm. One especially interesting choice is discussed in detail for the case of mixed radix-(2, 3) DFT algorithms. Algorithms for the DFT calculation of both one and more dimensional real-valued arrays are discussed.

Differential and Discrete Spectral Problems and Their Inverses

Abstract: Publisher Summary This chapter discusses differential and discrete spectral problems and their inverses. A set of spectral data is defined and a method of solving the inverse problem—that is, that of reconstructing B from data is described. The success of the spectral transform or “inverse scattering method” in solving various non linear evolution equations is well known. The direct spectral problems are considered and the spectral data is defined. A method of solving the inverse spectral problem is explained. The practical use of the spectral transform is demonstrated in the chapter. The Boussinesq equation is solved using a differential spectral transform and the Toda lattice equations are solved using a discrete spectral transform. The spectral data consists of certain information about the singularities of the Jost functions in the complex plane.

On the use of discrete cosine transform in cepstral analysis

Abstract: If the original signal is defined to be symmetrical, the discrete Fourier transform (DFT) used in cepstral analysis can be replaced by a discrete cosine transform (DCT). This principle is applied to the evaluation of the real and complex pseudocepstrum of speech signals. In both the real and complex cepstrum cases, it is found that the use of the DCT does not degrade the information contained in the cepstrum while substantially reducing the computational complexity.

Prime factor FFT algorithms for real-valued series

Abstract: This paper presents two techniques for computing a discrete transform of a vector of real-valued data using the Prime Factor Algorithm (PFA) with high-speed convolution. These techniques are applied to the Discrete Fourier Transform (DFT) and the Discrete Hartley Transform (DHT). The primary goals of these techniques are to eliminate unnecessary computations required when implementing a complex transform on a real-valued vector, to compute the transform in-place in the original length-N real vector, and to obtain the transform coefficients in-order. The two algorithms described require modification of the Winograd short-length transform modules to accommodate a real input. One technique replaces the modules in the Burrus-Eschenbacher PFA program with the modified real-input modules and constructs the complete transform in a final step of additions and subtractions after modules for each factor have been executed. The other technique uses these real-input DFT modules for part of the computation associated with each factor and requires complex input DFT modules for another part of the computation. These algorithms require exactly one half of the number of multiplications and slightly less than one half of the number of additions required by a complex-input PFA.

Improved method for the discrete fast Fourier transform

Abstract: A new algorithm is proposed here for the discrete fast Fourier transform with greatly reduced aliasing which is known to be inherent in the conventional algorithm of Cooley and Tukey, unless the function is band limited and the sampling frequency satisfies the Nyquist condition. Like the algorithm recently proposed by Schutte and extended by Makinen in this journal, this is also based on the polynomial expansion of the function to be transformed but more general in formulation and less restrictive than theirs. Its power is demonstrated with a few non‐band‐limited functions that can be exactly transformed with chosen limits as usually met in different experimental situations. In all cases tried, this yields, in general, much improved accuracy in comparison to others at little or no corresponding increase of computation time.

Comparative performance of two different versions of the discrete cosine transform

Abstract: The performance of version I of the discrete cosine transform(DCT-I) is compared to version II of the discrete cosine transform (DCT-II) on various criteria. The results show that for a Markovian signal with correlation coefficient less than 0.8, the DCT-I performs as well as the DCT-II.

On the computation of discrete fourier transform using fermat number transform

Abstract: In the paper the results of a study using Fermat number transforms (FNTs) to compute discrete Fourier transforms (DFTs) are presented. Eight basic FNT modules are suggested and used as the basic sequence lengths to compute long DFTs. The number of multiplications per point is for most cases not more than one, whereas the number of shift-adds is approximately equal to the number of additions in the Winograd-Fourier-transform algorithm and the polynomial transform. Thus the present technique is very effective in computing discrete Fourier transforms.

Relation between the Karhunen-Loeve and sine transforms

Abstract: In an earlier paper it was demonstrated that the cosine transform can be derived from the optimum (Karhunen–Loeve) transform of a first-order Markov source in the limiting case when the adjacent data element correlation coefficient tends to unity. The purpose of the letter is to show that, in the alternative limiting case when the correlation tends to zero, the sine transform results.

Symbolic network function generation via discrete Fourier transform

Abstract: A new method of symbolic network function generation is presented. The method is based upon the theory of the discrete Fourier transform and not restricted in its application to any particular type of network analysis or network configuration. It is particularly attractive when the number of symbolic variables to be handled is not large.

FCT -- A fact cosine transform

Abstract: A new discrete cosine transform algorithm, named Fast Cosine Transform (FCT), is introduced for the 2m-point discrete cosine transform This algorithm reduces the number of multiplications to about half of earlier results, and furthermore, it renders a simple and systematic structure for implementation

Recursive discrete functional expansion

Abstract: New methods are given for the general recursive and ongoing calculation of discrete functional expansions. For any such transform of order n , each computational iteration updates the previous result to give the expansion based upon the most recent n data points.

Symmetry relations in multidimensional Fourier transform pairs

Abstract: A relation between the types of symmetries that exist in signal and Fourier transform domain representations is derived for continuous as well as discrete domain signals. The symmetry is expressed by a set of parameters, and the relations derived in this paper will help to find the parameters of a symmetry in the signal or transform domain resulting from a given symmetry in the transform or signal domain respectively. A duality among the relations governing the conversion of the parameters of symmetry in the two domains is also brought to light. The application of the relations is illustrated by a number of two-dimensional examples.

Block classification image coding with combined transforms

Abstract: Methods of employing a combination of two recently reported simple transforms (High Correlation Transform and Low Correlation Transform) using block adaptive transform coding with spatial domain activity classification are examined. It is found that for small block size, (viz. 8×8), increased coding efficiency can be had by employing a combination of the two new transforms, particularly for high activity images. The performance of such systems for small block size is found to be close, and sometimes superior, to that of a more conventional block classification adaptive transform coder employing the more complex real number cosine transform.

A two-dimensional discrete convolution algorithm

Abstract: A theoretical result concerning the discrete Fourier transform is derived and used to develop a transform algorithm for computing two-dimensional convolutions. The use of this algorithm minimises the number of arithmetic operations and the memory requirements in computing a convolution of order (M×N) using a transform processor or program designed for a length No, where M≤N0 and N≤N0.It is particularly suitable for computing convolutions whose orders are not powers of two using conventional fast Fourier transform processors. Methods of implementing the algorithm are also presented.

Two-dimensional discrete Fourier transform with small multiplicative complexity using number theoretic transforms

Abstract: The conventional approach to computing the 2-D discrete Fourier transform (DFT) by row column or nesting algorithms is still computationally demanding because of the excessive number of multiplications required. It is shown that the number theoretic transform (NTT) can be used to compute the 2-D DFT very efficiently, with less than one multiplication per point. The technique makes use of index mapping for efficient calculation of convolution as a subset of transform computations.

Fast computation of zoom-DFT via the Walsh transform

Abstract: A method is introduced to compute zoom-DFT of a time aeries via its Walsh transform, which can be useful in applications where both discrete Fourier and discrete Walsh transforms of a sampled data function are required to be computed simultaneously with high resolution.

A modified discrete fourier transform algorithm: A delta rotation algorithm

Abstract: A new method for the evaluation of the Discrete Fourier Transform (DFT) is presented. This method evaluates the DFT of samples of a continuous time signal by multiplying the DFT of the difference signal at the output of a Linear Delta Modulator (LDM) by a rotation factor. The novelty of the proposed algorithm is to reduce the number of multiplication and to simplify the hardware implementation. Further modification of this algorithm does not require any multiplication at all during the DFT computation. An implementation of convolutions, chirp-z transform and the discrete Hilbert transform with the proposed technique will offer good opportunities for additional research with respect to the point of a simple hardware implementation, high-speed, and a computational simplicity. This proposed technique is, in fact, the combination of an encoding technique and a FFT algorithm. The results show a very promising and will be a viable alternative to the FFT and any other DFT algorithms.

Geometric interpretations of the Discrete Fourier Transform (DFT)

Abstract: One, two, and three dimensional Discrete Fourier Transforms (DFT) and geometric interpretations of their periodicities are presented These operators are examined for their relationship with the two sided, continuous Fourier transform Discrete or continuous transforms of real functions have certain symmetry properties The symmetries are examined for the one, two, and three dimensional cases Extension to higher dimension is straight forward

2D Discrete cosine transform computation by fast polynomial transform algorithms

Abstract: The 2D forward and inverse discrete cosine transforms (DCT) are mapped by polynomial transforms into several one-dimensional DFTs; This algorithm is more attractive for computing the "inverse" 2D DCT, it nicely eliminates N2premultiplications for the forward algorithm, and requires only N reduced real DFTs of size 2N; This inverse 2D DCT algorithm is 3 times faster than the forward one, more importantly it requires much less computation complexity than the most efficient recent algorithms.