Showing papers on "Discrete sine transform published in 1998"

Quasi-discrete Hankel transform

Abstract: A quasi-discrete Hankel transform (QDHT) is presented as a new and efficient framework for numerical evaluation of the zero-order Hankel transform A discrete form of Parseval's theorem is obtained for the first time to the authors' knowledge, and the transform matrix is discussed It is shown that the S factor, defined as the products of a truncated radius, is critical to building the QDHT

Discrete fractional Hartley and Fourier transforms

Abstract: This paper is concerned with the definitions of the discrete fractional Hartley transform (DFRHT) and the discrete fractional Fourier transform (DFRFT). First, the eigenvalues and eigenvectors of the discrete Fourier and Hartley transform matrices are investigated. Then, the results of the eigendecompositions of the transform matrices are used to define DFRHT and DFRFT. Also, an important relationship between DFRHT and DFRFT is described, and numerical examples are illustrated to demonstrate that the proposed DFRFT is a better approximation to the continuous fractional Fourier transform than the conventional defined DFRFT. Finally, a filtering technique in the fractional Fourier transform domain is applied to remove chirp interference.

The phase-corrected undecimated discrete wavelet packet transform and its application to interpreting the timing of events

Abstract: This paper is concerned with the development and application of the phase–corrected maximal overlap discrete wavelet packet transform (MODWPT). The discrete cyclic filtering steps of the MODWPT are fully explained. Energy preservation is proven. With filter coefficients chosen from Daubechie's least asymmetric class, the optimum time shifts to apply to ensure approximate zero phase filtering at every level of the MODWPT are studied, and applied to the wavelet packet coefficients to give phase corrections which ensure alignment with the original time series. Also, the time series values at each time are decomposed into details associated with each frequency band, and these line up perfectly with features in the original time series since the details are shown to arise through exact zero phase filtering. The phase–corrected MODWPT is applied to a non–stationary time series of hourly averaged Southern Hemisphere solar magnetic field magnitude data acquired by the Ulysses spacecraft. The occurrence times of the shock waves previously determined via manual pattern matching on the raw data match those times in the time–frequency plot where a broadband spectrum is obtained; in other words, the phase–corrected MODWPT provides an approach to picking the location of complicated events. We demonstrate the superiority of the MODWPT in interpreting timing information over two competing methods, namely the cosine packet transform (or ‘local cosine transform’), and the short–time Fourier transform.

Orthogonal and Nonorthogonal Multiresolution Analysis, Scale Discrete and Exact Fully Discrete Wavelet Transform on the Sphere

Abstract: Based on a new definition of dilation a scale discrete version of spherical multiresolution is described, starting from a scale discrete wavelet transform on the sphere. Depending on the type of application, different families of wavelets are chosen. In particular, spherical Shannon wavelets are constructed that form an orthogonal multiresolution analysis. Finally fully discrete wavelet approximation is discussed in the case of band-limited wavelets.

Efficient moving-window DFT algorithms

Abstract: The authors deal with the real-valued, moving-window discrete Fourier transform. After reviewing the basic recursive versions appearing in the literature, additional recursive equations are presented. Then, these equations are combined so that nonrecursive expressions involving only consecutive discrete Fourier transform (DFT) sine components are obtained for both the DFT cosine component and squared harmonic amplitude. The computational complexity of this new scheme is finally studied and compared to that of existing methods, showing that, in most practical situations, a reduction in the operation count is achieved.

Computing multidimensional DFTs in FPGA

Abstract: An FPGA configured for computation of an N×N discrete Fourier transform (DFT) using polynomial transforms defined in modified rings of transforms, comprising a first buffer for ordering a set of polynomial data in a two dimensional matrix, a multiplier for multiplying each element of the two dimensional matrix by ω-n.sbsp.2 (where ω=e-jπ/N, e is a constant (ln(e)=1), j=√-1, n2 =the column index number in the matrix, and N=the transform length) to produce a premultiplication product, a polynomial transform circuit for performing a polynomial transform (PT) modulo (zN +1), size N, root z2 on the premultiplication product to produce a polynomial transform result, where z represents the unit delay operator, a reduced DFT calculator for performing N reduced DFTs of N terms on the polynomial transform result to produce a permuted output, and an address generator for reordering the permuted output to a natural order.

Reversible discrete cosine transform

Abstract: In this paper a reversible discrete cosine transform (RDCT) is presented. The N-point reversible transform is firstly presented, then the 8-point RDCT is obtained by substituting the 2 and 4-point reversible transforms for the 2 and 4-point transforms which compose the 8-point discrete cosine transform (DCT), respectively. The integer input signal can be losslessly recovered, although the transform coefficients are integer numbers. If the floor functions are ignored in RDCT, the transform is exactly the same as DCT with determinant=1. RDCT is also normalized so that we can avoid the problem that dynamic range is nonuniform. A simulation on continuous-tone still images shows that the lossless and lossy compression efficiencies of RDCT are comparable to those obtained with reversible wavelet transform.

The Fractional Wave Packet Transform

Abstract: We introduce the concept of the Fractional Wave Packet Transform(FRWPT), based on the idea of the Fractional Fourier Transform(FRFT) and Wave Packet Transform(WPT). We show a version of the resolution of the identity and some properties of FRWPT connected with those of FRFT and WPT.

The quick Fourier transform: an FFT based on symmetries

Abstract: This paper looks at an approach that uses symmetric properties of the basis function to remove redundancies in the calculation of the discrete Fourier transform (DFT). We develop an algorithm called the quick Fourier transform (QFT) that reduces the number of floating-point operations necessary to compute the DFT by a factor of two or four over direct methods or Goertzel's method for prime lengths. By further application of the idea to the calculation of a DFT of length-2/sup M/, we construct a new O(NlogN) algorithm, with computational complexities comparable to the Cooley-Tukey algorithm. We show that the power-of-two QFT can be implemented in terms of discrete sine and cosine transforms. The algorithm can be easily modified to compute the DFT with only a subset of either input or output points and reduces by nearly half the number of operations when the data are real.

Fast DFT algorithms for length N=q*2/sup m/

Abstract: This paper presents a general split-radix algorithm which can flexibly compute the discrete Fourier transforms (DFT) of length q*2/sup m/ where q is an odd integer Comparisons with previously reported algorithms show that substantial savings on arithmetic operations can be made Furthermore, a wider range of choices on different sequence lengths is naturally provided

DCT/DST and Gauss-Markov fields: conditions for equivalence

Abstract: The correspondence addresses the intriguing question of which random models are equivalent to the discrete cosine transform (DCT) and discrete sine transform (DST). Common knowledge states that these transforms are asymptotically equivalent to first-order Gauss causal Markov random processes. We establish that the DCT and the DST are exactly equivalent to homogeneous one-dimensional (1-D) and two-dimensional (2-D) Gauss noncausal Markov random fields defined on finite lattices with appropriate boundary conditions.

The Discrete Fourier Transform and the Fast Fourier Transform

Abstract: The preceding chapters have made extensive mention of the Fourier transform (FT), the discrete Fourier transform (DFT), and the fast Fourier transform (FFT). This chapter examines the relationship between the FT and the DFT, discusses the FFT algorithm as a means of computing the DFT much more rapidly than can be achieved with the DFT algorithm directly, and presents some practical guidelines for using the FFT.

Fast algorithms for orthogonal and biorthogonal modulated lapped transforms

Abstract: New algorithms for the computation of orthogonal and biorthogonal modulated lapped transforms (MLTs) are presented. The new structures are obtained by combining the MLT window operators with stages from a previously introduced structure for the type-IV discrete cosine transform (DCT-IV). The net result is fewer multiplications and additions than previously reported algorithms. For the orthogonal MLT, in particular, the new structure requires the computation of a slightly modified DCT-IV and some extra additions, but no further multiplications; so it demonstrates that the multiplicative complexity of the orthogonal MLT is the same as that of the DCT-IV.

Discrete evolutionary transform for time-frequency analysis

Abstract: We propose a new transformation for discrete signals with time-varying spectra. The kernel of this transformation provides the energy density of the signal in time-frequency. With this discrete evolutionary transform we obtain a representation for the signal as well as its time-frequency energy density. To obtain the kernel of the transformation we use either the Gabor or the Malvar discrete signal representations. Signal adaptive analysis can be done using modulated or chirped bases, and implemented with either masking or image segmentation on the time-frequency plane. Different examples illustrate the implementation of the discrete evolutionary transform.

A fast frequency transformation techique for transform audio coders

Abstract: A method for coding digital audio data in which coded Fast Modified Discrete Cosine Transform (FMDCT) coefficients are computed utilising a Fast Fourier Transform (FFT) method. The described method allows a significant reduction in computations as compared to an ordinary DCT coding procedure. Also, pairs of audio channels can be combined to use a single FFT computation, where the selected transform length for the paired channels is the same. In such cases where pairing of identical transform length channels is not possible, a long transform length channel is combined with a short transform length channel and converted in two short transforms. A windowing function is also combined with a pre-processing stage to the transformation, further decreasing computational requireements.

Two dimensional transform generator

Abstract: A system that optically performs complex transforms, such as Fourier transforms. The system includes a pixel addressable spatial modulator that, in parallel, adjusts the phase of the light of each pixel. The modulator can be a reflective or transmissive type device. A transform lens, such as a Fourier lens, performs a two dimensional transform of the pixels outputs. This operation is repeated for the characteristic function (real and imaginary) of the function. The transformed outputs of the characteristic functions are sampled by a light detector and processed by a computer using simple fast operations, such as addition, into the final transform.

Modified discrete wavelet transform for odd length data appropriate for image and video compression applications

Abstract: A method of producing a transform decomposition of data having an odd length the method comprising the steps of dividing the data into a portion having an even length of one element; performing a discrete wavelet transform on the even length data to produce low frequency subband data and high frequency subband data; adding the difference of the one element and an adjacent element to high frequency subband data. Preferably the transform is a Discrete Wavelet Transform utilised in the compression of image data.

The problem of defining the Fourier transform of a colour image

Abstract: A discrete two-dimensional Fourier transform based on quaternion (or hypercomplex) numbers allows colour images to be transformed as a whole, rather than as colour-separated components. The transform is reviewed and its basis functions presented with example images.

Method and apparatus for efficient computation of discrete fourier transform (dft) and inverse discrete fourier transform (idft)

Abstract: The present invention significantly reduces the number of complex computations that must be performed in computing the discrete Fourier transform (DFT) and inverse DFT (IDFT) operations. In particular, the DFT and IDFT operations are computed using the same computing device. The computation operations are substantially identical for both operations with the exception that for the IDFT operation, the data are complex conjugated before and after processing. Using the same computing device/operations, both DFT and IDFT computations are optimized for maximum efficiency. A common transform process is selectively connected to first and second data processing paths. A DFT operation is performed on an N-point sequence on the first data processing path, and an IDFT operation is performed on an N-point sequence on the second data processing path using the same N-point fast Fourier transform (FFT).

Computation reduction for discrete cosine transform

Abstract: This paper presents a method to accelerate software video encoders by reducing the number of operations for discrete cosine transform (DCT) and quantization. We present a model of DCT coefficients based on the information of motion compensated difference blocks. The relationship between the quantization level and DCT coefficients is also studied. Based on the model, we can adaptively make the decision of DCT calculations. A fast algorithm is also proposed for approximating the 4/spl times/4 low-frequency coefficients. The results show that significant computation reductions can be achieved with negligible peak signal-to-noise ratio (PSNR) degradations.

The slantlet transform

Abstract: The discrete wavelet transform (DWT) is usually carried out by filter bank iteration, however, for a fixed number of zero moments, this does not yield a discrete-time basis that is optimal with respect to time-localization. This paper discusses the implementation and properties of an orthogonal DWT, with two zero moments and with improved time-localization. The basis is not based on filter bank iteration, instead different filters are used for each scale. For coarse scales, the support of the discrete-time basis functions approaches 2/3 that of the corresponding functions obtained by filter bank iteration. This slantlet basis is piecewise linear and retains the octave-band characteristic. Closed form expressions for the filters are given and improvement in a denoising example is shown. This basis, being piecewise linear, is reminiscent of the slant transform, to which it is compared.

Recursive formulation of short-time discrete trigonometric transforms

Abstract: Recursive formulations of the moving-window discrete Fourier transform (DFT) are well known. However, recursive versions of other useful discrete transforms, like the moving-window discrete cosine transform (DCT), discrete sine transform (DST), or discrete Hartley transform (DHT), have not been developed so far. In this paper, second-order recursive expressions for the DCT, DST, and DHT, intended for real-valued windowed sequences, are presented.

Unified recursive decomposition architecture for cosine modulated filter banks

Abstract: A unified architecture for implementing the modified cosine transforms of various cosine modulated filter banks in audio compression standards comprises a permutation module and a transform computing module. A modified cosine transform is computed by a pre-permutation followed by a discrete cosine transform and an inverse modified cosine transform is computed by a discrete cosine transform followed by a post-permutation. The discrete cosine transform computed in the unified architecture is selected from the group of type-II, type-III and type-IV cosine transforms. The computation of an N point discrete cosine transform is decomposed into a permutation-add stage, a sub-transform stage for computing two N/2 point discrete cosine transforms selected from the same group, and a combination stage. The architecture results in good regularity and general applicability as well as reduces complexity.

A generalized fourier transform for boundary element methods with symmetries

Abstract: We study the solution of linear systems that typically arise in discretizations of boundary value problems on a domain with geometrical symmetries If the discretization is done in an appropriate way, then such a system commutes with a group of permutation matrices Recently, algorithms have been developed that exploit this special structure, however these methods are limited to the case that the permutations have no fixed points Here a new symmetry exploiting algorithm, based on the Fourier transform on the symmetry group, is introduced which is capable of handling fixed points The techniques developed can also be used to achieve further reductions when the right hand side of the proposed system has symmetries The approach is illustrated by the boundary element method on an equilateral triangle and on a 3-cube The reduction technique can also be applied to other solution methods, eg, finite element methods

Image coding using discrete sine transform with axis rotation

Abstract: The blocking effect is a common problem that is always present in image coding when using the blocking transform. A simple method for image coding, with low blocking effect, using pre- and post-axis rotation and the discrete sine transform, is proposed. It uses the overlap of only the sample and the total number of transform coefficients remains the same as the number of pixels in the original image. The method was implemented, and the results obtained by simulation in the image coding are compared with image coding by DCT, DST, and LOT.