Showing papers on "Discrete Hartley transform published in 2004"

Properties of continuous Fourier extension of the discrete cosine transform and its multidimensional generalization

Abstract: A versatile method is described for the practical computation of the exact discrete Fourier transforms (DFT), both the direct and the inverse ones, of a continuous function g given by its values gj at the points of a uniform grid FN generated by conjugacy classes of elements of finite adjoint order N in the fundamental region F of compact semisimple Lie groups. The present implementation of the method is for the groups SU(2), when F is reduced to a one-dimensional segment, and for SU(2)×SU(2)×⋯×SU(2) in multidimensional cases. This simplest case turns out to be a version of the discrete cosine transform (DCT). Implementations, abbreviated as DGT for Discrete Group Transform, based on simple Lie groups of higher ranks, are to be considered separately. DCT is often taken to be simply a specific type of the standard DFT. Here we show that the DCT is very different from the standard DFT when the properties of the continuous extensions of the two inverse discrete transforms are studied. The following properties of the continuous extension of DCT (called CEDCT) from the discrete tj∈FN to all t∈F are proven and exemplified. Like the standard DFT, the DCT also returns the exact values of {gj} on the N+1 points of the grid. However, unlike the continuous extension of the standard DFT: (a) The CEDCT function fN(t) closely approximates g(t) between the points of the grid as well; (b) for increasing N, the derivative of fN(t) converges to the derivative of g(t); (c) for CEDCT the principle of locality is valid. In this article we also use the continuous extension of the two-dimensional (2D) DCT, SU(2)×SU(2), to illustrate its potential for interpolation as well as for the data compression of 2D images.

Generalized eigenvectors and fractionalization of offset DFTs and DCTs

Abstract: The offset discrete Fourier transform (DFT) is a discrete transform with kernel exp[-j2/spl pi/(m-a)(n-b)/N]. It is more generalized and flexible than the original DFT and has very close relations with the discrete cosine transform (DCT) of type 4 (DCT-IV), DCT-VIII, discrete sine transform (DST)-IV, DST-VIII, and discrete Hartley transform (DHT)-IV. In this paper, we derive the eigenvectors/eigenvalues of the offset DFT, especially for the case where a+b is an integer. By convolution theorem, we can derive the close form eigenvector sets of the offset DFT when a+b is an integer. We also show the general form of the eigenvectors in this case. Then, we use the eigenvectors/eigenvalues of the offset DFT to derive the eigenvectors/eigenvalues of the DCT-IV, DCT-VIII, DST-IV, DST-VIII, and DHT-IV. After the eigenvectors/eigenvalues are derived, we can use the eigenvectors-decomposition method to derive the fractional operations of the offset DFT, DCT-IV, DCT-VIII, DST-IV, DST-VIII, and DHT-IV. These fractional operations are more flexible than the original ones and can be used for filter design, data compression, encryption, and watermarking, etc.

A new transform for 2-D signal representation (MRT) and some of its properties

Abstract: A new transform (MRT) for two-dimensional signal representation and some of its properties are proposed in this paper. The transform helps to do the frequency domain analysis of two-dimensional signals without any complex operations but in terms of real additions. It is obtained by exploiting the periodicity and symmetry of the exponential term in the discrete Fourier transform (DFT) relation, and by grouping related data. Forward and inverse relations of the transform are presented. The transform coefficients show useful redundancies among themselves. These redundancies, which can be used to implement the transform, are studied. A few properties of the transform are studied and the relevant relations are presented.

Fast, prime factor, discrete Fourier transform

Abstract: In this paper it is shown that Winograds algorithm for computing convolutions and a fast, prime factor, discrete Fourier transform (DFT) algorithm can be modified to compute Fourier-like transforms of long sequences of 2 m

Iterative approximation environments for modeling the evolution of an image propagating through a physical medium in restoration and other applications

Abstract: Image processing utilizing numerical calculation of fractional exponential powers of a diagonalizable numerical transform operator for use in an iterative or other larger computational environments. In one implementation, a computation involving a similarity transformation is partitioned so that one part remains fixed and may be reused in subsequent iterations. The numerical transform operator may be a discrete Fourier transform operator, discrete fractional Fourier transform operator, centered discrete fractional Fourier transform operator, and other operators, modeling propagation through physical media. Such iterative environments for these types of numerical calculations are useful in correcting the focus of misfocused images which may originate from optical processes involving light (for example, with a lens or lens system) or from particle beams (for example, in electron microscopy or ion lithography).

The centered discrete fractional Fourier transform and linear chirp signals

Abstract: The basis functions for the fractional Fourier transform are chirp signals where a precise relationship between the fractional parameter and the chirp angle can be established. The recently introduced centered discrete fractional Fourier transform, based on the Gr/spl uml/nbaum commuting matrix, has basis functions that have a sigmoidal instantaneous frequency and produces a transform that is approximately an impulse for discrete chirps. However, no such precise relation between the fractional parameter and the chirp rate of the basis functions exists in the discrete case. We study the relationship between the chirp rate and the fractional parameter in the discrete case and specifically look at two approximate expressions that relate the chirp rate and the angle for which one obtains an impulse-like transform. We study the efficacy of these estimates by applying them to the analysis of monocomponent and two component chirp signals.

An optimized algorithm for computing the modified discrete cosine transform and its inverse transform

Abstract: This paper presents an optimized algorithm for the modified discrete cosine transform (MDCT) and its inverse transform (IMDCT) computation in MPEG audio. The proposed algorithm is based on the DCT of type II (DCT-II). By extracting a common kernel form from the MDCT and IMDCT, we can obtain an optimized computation of the MDCT and the IMDCT using a unified structure. Also, our proposed structure is more symmetrical and simple than those of the existing researches. Proposed algorithm is, moreover, very useful in implementing the parallel VLSI system structure of the MDCT and IMDCT.

Reality preserving fractional transforms [signal processing applications]

Abstract: The unitarity property of transforms is useful in many applications (source compression, transmission, watermarking, to name a few). In many cases, when a transform is applied on real-valued data, it is very useful to obtain real-valued coefficients (i.e. a reality-preserving transform). In most applications, the decorrelation property of the transform is of importance and it would be very useful to control it under some transform parameter (e.g. in joint source-channel coding). This paper focuses on fractional transforms, as tools for obtaining such properties. We propose a methodology for obtaining them and obtain variants of the discrete fractional cosine (sine) transform which share real-valuedness as well as most of the properties required for a fractional transform matrix. As shown in (I. Venturini et al. IEEE Trans. Signal Proc.), such matrices cannot be symmetric.

A new fast bit-reversal permutation algorithm based on a symmetry

Abstract: This correspondence describes a new bit-reversal permutation algorithm based on a trivial symmetry that has not been exploited until now. According to timing experiments, this algorithm outperforms the fastest algorithms known to the author. This is of interest for applications using intensive fast Fourier transforms (or fast Hartley transforms) of constant length, such as transform domain adaptive filtering.

A novel algorithm for computing the 1-D discrete Hartley transform

Abstract: An efficient split algorithm for calculating the one-dimensional discrete Hartley transforms, by using a special partitioning in the frequency domain, is introduced. The partition determines a fast paired transform that splits the 2/sup r/-point unitary Hartley transform into a set of 2/sup r-n/-point odd-frequency Hartley transforms, n=1:r. A proposed method of calculation of the 2/sup r/-point Hartley transform requires 2/sup r-1/(r-3)+2 multiplications and 2/sup r-1/(r+9)-r/sup 2/-3r-6 additions.

A new split-radix FHT algorithm for length-q*2/sup m/DHTs

Abstract: In this paper, a new split-radix fast Hartley transform (FHT) algorithm is proposed for computing the discrete Hartley transform (DHT) of an arbitrary length N=q*2/sup m/, where q is an odd integer. The basic idea behind the proposed FHT algorithm is that a mixture of radix-2 and radix-8 index maps is used in the decomposition of the DHT. This idea and the use of an efficient indexing process lead to a new decomposition different from that of the existing split-radix FHT algorithms, since the existing ones are all based on the use of a mixture of radix-2 and radix-4 index maps. The proposed algorithm reduces substantially the operations such as data transfer, address generation, and twiddle factor evaluation or access to the lookup table, which contribute significantly to the execution time of FHT algorithms. It is shown that the arithmetic complexity (multiplications+additions) of the proposed algorithm is, in almost all cases, the same as that of the existing split-radix FHT algorithm for length- q*2/sup m/ DHTs. Since the proposed algorithm is expressed in a simple matrix form, it facilitates an easy implementation of the algorithm, and allows for an extension to the multidimensional case.

Some links between continuous and discrete Radon transform

Abstract: The Filtered BackProjection is still questionable since many discrete versions have been derived from the continuous Radon formalism. From a continuous point of view, a previous work has made a link between continuous and discrete FBP versions denoted as Spline 0-FBP model leading to a regularization of the infinite Ramp filter by the Fourier transform of a trapezoidal shape. However, projections have to be oversampled (compared to the pixel size) to retrieve the theoretical properties of Sobolev and Spline spaces. Here we obtain a novel version of the Spline 0 FBP algorithm with a complete continuous/discrete correspondence using a specific discrete Radon transform, the Mojette transform. From a discrete point of view, the links toward the FBP algorithm are shaped with the morphological description and the extended use of discrete projection angles. The resulting equivalent FBP scheme uses a selected set of angles which covers all the possible discrete Katz's directions issued from the pixels of the (square) shape under reconstruction: this is implemented using the corresponding Farey's series. We present a new version of a discrete FBP method using a finite number of projections derived from discrete geometry considerations. This paper makes links between these two approaches.

A New Split-Radix FHT Algorithm for Length- DHTs

Abstract: In this paper, a new split-radix fast Hartley transform (FHT) algorithm is proposed for computing the discrete Hartley transform (DHT) of an arbitrary length , where is an odd integer. The basic idea behind the proposed FHT algorithm is that a mixture of radix-2 and radix-8 index maps is used in the decomposition of the DHT. This idea and the use of an efficient in- dexing process lead to a new decomposition different from that of the existing split-radix FHT algorithms, since the existing ones are all based on the use of a mixture of radix-2 and radix-4 index maps. The proposed algorithm reduces substantially the operations such as data transfer, address generation, and twiddle factor evaluation or access to the lookup table, which contribute significantly to the execution time of FHT algorithms. It is shown that the arithmetic complexity (multiplications additions) of the proposed algorithm is, in almost all cases, the same as that of the existing split-radix FHT algorithm for length- DHTs. Since the proposed al- gorithm is expressed in a simple matrix form, it facilitates an easy implementation of the algorithm, and allows for an extension to the multidimensional case. Index Terms—Discrete Hartley transform (DHT), fast Hartley transform (FHT) algorithms, mixed radix, split radix.

Interspersed sinusoidal transforms for OFDM systems

Abstract: A method for using the discrete cosine transform (DCT) and discrete sine transform (DST) for orthogonal frequency division multiplexing (OFDM) wireless transmission methods is described. These transforms satisfy the cyclic convolution properties of the normally used discrete Fourier transform (DFT) when they are used with a symmetric extension. Taking advantage of properties of the DCT and DST, an interspersed transmission method is proposed. This method can lead to highly spectrally efficient OFDM when compared to the DFT.

Method for realising reversible integer type-IV discrete cosine transform

Abstract: A novel method for realising reversible integer discrete cosine transform type IV (DCT-IV) is presented. The method is derived by first constructing a transform matrix using DCT-IV matrix, then factorising the transform matrix into the product of three lifting matrices. The proposed method exhibits low rounding number and is the most accurate to the floating-point DCT-IV transform.

Quantum circuit design of 8 /spl times/ 8 discrete Hartley transform

Abstract: In this paper, the 8 /spl times/ 8 discrete Hartley transform will be implemented by using quantum elementary gates such as controlled NOT gates and Hadamard gates Two steps involved are as follows: first, the discrete Hartley transform matrix is decomposed into the product of several sparse unitary matrices using the structure of fast Hartley transform algorithm Then, each sparse matrix is implemented by elementary quantum gates and cascade them to obtain the final circuit The design results show that the proposed design method has less circuit complexity than the conventional Cybenko's method

Fast Fourier Transform Algorithms

Abstract: This chapter is devoted to the fast algorithms for various types of discrete Fourier transforms (DFTs).

Inverting the Burrows—Wheeler transform

Abstract: The Burrows–Wheeler Transform is a string-to-string transform which, when used as a preprocessing phase in compression, significantly enhances the compression rate However, it often puzzles people how the inverse transform is carried out In this pearl we to exploit simple equational reasoning to derive the inverse of the Burrows–Wheeler transform from its specification We also outline how to derive the inverse of two more general versions of the transform, one proposed by Schindler and the other by Chapin and Tate

Novel recursive discrete Fourier transform with compact architecture

Abstract: We propose the novel recursive method to compute discrete Fourier transforms (DFT). The advantages of proposed recursive structure are the reduction of the loop computing numbers and the signal to quantization noise ratio (SQNR) is greater than the well-known Goertzel's method. The compact recursive DFT applies the grouped frequency indices to accelerate the computation of the DFT transformation. By sharing the loop and output coefficients, we can implement the recursive DFT with hardware sharing architectures.

On pruning the discrete cosine and sine transforms

Abstract: In the paper it is shown that a limited set of output discrete cosine transform (DCT) samples can be computed by a modified real-valued output-pruned FFT algorithm for appropriately permuted data samples. The same is true for the discrete sine transform (DST). Analogously, when computing data contribution from few DCT or DST samples the input-pruned FFT algorithm for inverse FFT can be applied, the input-pruned algorithms for the inverse DCT or DST are obtained. The algorithms are very efficient, their complexities are O(NlogK), where N is the transform size, and K is a divisor of N equal to or greater than the number of computed transform samples, which is less than the number of computed transform samples, which is less than O(NlogN) for the full DCT or DST algorithm. The algorithms are easy to implement, too.

Introduction of algorithms used in PIV technique

Abstract: The particle image velocimetry (PIV) is an effective and non-intrusive technique to measure the planar distribution of velocity in the fluid based on the cross-correlation of flow images. In general, cross-correlation analysis in the PIV can be implemented quickly by the fast Fourier transform, and the efficiency can be improved by considering the property of real Fourier transform. Furthermore, the higher efficiency can be obtained by using the Hartley transform of separable kernels. Three algorithms are specified in detail in the paper, and the comparison of the three algorithms is conducted in the theory and computation time in practice,which can show the advantages of the Hartley transform.

Data embedding in speech signals using perceptual masking

Abstract: In this paper, a data embedding technique for speech signals, exploiting the masking property of the human auditory system, is presented. The signal in the frequency domain is partitioned into subbands. The data embedding parameters of each subband are computed from the auditory masking threshold function and a channel noise estimate. Data embedding is performed by modifying the Discrete Hartley Transform (DHT) coefficients according to the principles of the Scalar Costa Scheme (SCS). A maximum likelihood detector is employed in the decoder for embedded-data presence detection and data-embedding quantization-step estimation. We demonstrate the proposed data embedding technique by simulation of data embedding in a speech signal transmitted over a telephone line. The demonstrated system achieves transparent data-embedding at the rate of 300 information bits/second with a bit-error-rate of approximately 10P−4. The proposed technique outperforms spread spectrum (SS) based data-embedding techniques for speech signals.

Adaptive transform scheme to reduce PAR of an OFDM signal

Abstract: One major drawback in orthogonal frequency division multiplexing (OFDM) system is its high peak-to-average power ratio (PAR). Most of the PAR reduction techniques are based on changing the signal constellation symbols in a frame before evaluating the inverse discrete Fourier transformation (IDFT). A simple way to achieve this change is to use a linear transformation in the form of orthogonal matrix. Hadamard transform of data symbols has recently been proposed to reduced PAR. Motivated by this approach, we investigate the use of Fourier and cosine transforms to reduce the PAR. We also propose an adaptive transform scheme to further reduce the PAR with minimized complexity. This approach reduces the PAR by 4 dB for an OFDM signal with 128 subcarriers. We also compare the performance and the complexity of the proposed scheme with the selected mapping (SLM) technique.

Low power, high performance transform coprocessor for video compression

Abstract: First and second integer transform matrices can be used to approximate the discrete cosine transform. An input matrix of data is multiplied by a first transform matrix of integers to produce an intermediate matrix of data. The intermediate matrix is multiplied by a second transform matrix of integers to produce a transform result matrix of data. The multiplications by the first and second transform matrices can be pipelined to increase throughput. A plurality of transform data paths can also be provided in parallel to increase throughput.

Scalable system for inverse discrete cosine transform and method thereof

Abstract: The present invention provides an input data control method and system for a data processing system. The system comprises at least one basic operation unit (BOU) and is used for transforming one input matrix X into data in a plurality of specified columns in an output matrix Y via an inverse discrete cosine transform procedure. The method generates and outputs a transform control signal together with the input matrix to at least one of the BOUs. A new transform control signal is generated according to the received transform control signal, and outputted together with the input matrix X, to other following BOUs. The step of generating the new transform control signals is repeated until each specific column of the output matrix Y is decoded by a corresponding BOU. A basic operation procedure is then performed, and the received input matrix is decoded to obtain the data in the specified columns corresponding to the transform control signal.

Sense matrix model and discrete cosine transform

Abstract: In this paper we first present a brief introduction of the Sense Matrix Model (SMM), which employs a word-sense matrix representation of text for information retrieval, and then a discussion about one of the document transform techniques introduced by the model, namely discrete cosine transform (DCT) on document matrices and vectors. A first system implementation along with its experimental results is discussed, which provides marginal to medium improvements and validates the potential of DCT.

Analog VLSI architecture for discrete cosine transform using dynamic switched capacitors

Abstract: This paper describes an analog VLSI architecture, to compute discrete cosine transform (DCT), using switched-switched capacitor blocks. The scheme operates from the general expression of DCT where input samples are multiplied by all the DCT coefficients simultaneously using an array of capacitors. These multiplied values are then switched parallely with the help of a cross-point switch, to different integrators to perform multiplication and accumulation(MAC). It can be used to compute DHT,DST and DFT and also its inverses. Proposed architecture is very simple to implement and suitable where silicon area and power issues are important with some compromise on accuracy.