Showing papers on "Discrete Hartley transform published in 1982"

Eigenvectors and functions of the discrete Fourier transform

Abstract: A method is presented for computing an orthonormal set of eigenvectors for the discrete Fourier transform (DFT). The technique is based on a detailed analysis of the eigenstructure of a special matrix which commutes with the DFT. It is also shown how fractional powers of the DFT can be efficiently computed, and possible applications to multiplexing and transform coding are suggested.

Vectorizing the FFTs

Abstract: Publisher Summary This chapter provides an overview on vectorizing the FFTs. The fast Fourier transform (FFT) is the most well known of all algorithms. It is superior to the slow transform and has applications in all areas of scientific computing. The term FFT was applied to a specific algorithm for the rapid computation of the discrete complex Fourier transform; however, it has become a generic term that is applied to any one of a large number of algorithms that compute the complex as well as other Fourier transforms. Many algorithms exist for a given Fourier transform, and when they are applied to a particular sequence, the result is the same. However, the algorithms differ in the ways that intermediate results are computed and stored. It is these important differences that provide the algorithms with unique properties that make one or the other more attractive for a particular application.

A fast algorithm for the discrete sine transform implemented by the fast cosine transform

Abstract: A method of composing the discrete sine transform from the discrete cosine transform is demonstrated. As a result of this method, the fast sine transform is accomplished by the existing implementation of the fast cosine transform.

The Fast Fourier Transform

Abstract: The object of this chapter is to briefly summarize the main properties of the discrete Fourier transform (DFT) and to present various fast DFT computation techniques known collectively as the fast Fourier transform (FFT) algorithm. The DFT plays a key role in physics because it can be used as a mathematical tool to describe the relationship between the time domain and frequency domain representation of discrete signals. The use of DFT analysis methods has increased dramatically since the introduction of the FFT in 1965 because the FFT algorithm decreases by several orders of magnitude the number of arithmetic operations required for DFT computations. It has thereby provided a practical solution to many problems that otherwise would have been intractable.

Very fast computation of the radix-2 discrete Fourier transform

Abstract: This paper develops and presents a radix-2 fast Fourier transform (FFT) algorithm that reduces the computational effort (as measured by the number of multiplications) to two-thirds of the effort required by most radix-2 algorithms. The resulting algorithm is similar to one obtained by applying a principle suggested by Rader and Brenner; however, its structure is particularly appealing when transforming pure real or imaginary sequences and/or symmetric or antisymmetric sequences; furthermore, memory requirements (other than those for storing the input data) do not grow with the size of the transform.

Adaptive transform coding of video signals

Abstract: In the paper, five schemes for the adaptive transform coding of video signals are presented. Adaptation is based on adaptive quantisation and adaptive bit selection. A Max quantiser having a Laplacian density distribution is used to achieve adaptive quantisation, whereas classification according to the activity within the transform block and based on the human visual characteristics are used for adaptive bit selection. The discrete cosine transform and the Hadamard transform are used to transform three source pictures of different statistics, and results are compared to find the best scheme for each transform and picture. Photographic results are presented to see the effects of the distortions introduced by different transforms on different source pictures due to data compression. Owing to the limitations of the photographic process, the subjective quality of the pictures as perceived from the video monitor cannot be accurately represented. However they are useful for comparison purposes.

Implementation of the in-order prime factor transform for variable sizes

Abstract: This paper describes a modified version of Burrus' prime factor fast Fourier transform program. The modifications produce a general-purpose program which implements the in-place, in-order algorithm for variable transform sizes. Speed tests show the resulting program to be faster than a program using a separate reordering pass.

A derivation for the discrete cosine transform

Abstract: The purpose of this letter is to derive the discrete cosine transform (DCT) as a limiting case of the Karhunen-Loeve transform (KLT) of a first-order Markov process, as the correlation coefficient approaches 1

New Algorithm for the Slant Transform

Abstract: Two new algorithms that are more convenient for computation than existing ones for the slant transform are developed. These algorithms reveal the close relationship between the slant transform and the Walsh-Hadamard transform and demonstrate that the slant transform may be approached by a series of steps which gradually change the transform from a Hadamard or Walsh transform to a slant transform.

Generalized running discrete transforms

Abstract: This paper introduces a generalized running discrete transform with respect to arbitrary transform bases, and relates the generalized transform to the running discrete Fourier z and short-time discrete Fourier transforms. Concepts associated with the running and short-time discrete Fourier transforms such as 1) filter bank implementation, 2) synthesis of the original sequence by summation of the filter bank outputs, 3) frequency sampling, and 4) recursive implementations are all extended to the generalized transform case. A formula is obtained for computing the transform coefficients of a segment of data at time n recursively from the transform coefficients of the segment of data at time n - 1. The computational efficiency of this formula is studied, and the class of transforms requiring the minimum possible number of arithmetic operations per coefficient is described.

DFT-Vocoder using harmonic-sieve pitch extraction

Abstract: The DFT-vocoder is based on speech analysis and synthesis using the discrete Fourier transform (DFT). Analysis is done using hopping-DFT and spectral parameters are obtained by a piece-wise constant approximation of the amplitude spectrum. The harmonic-sieve technique for pitch extraction combines very well with this scheme because it is based on hopping-DFT as well. Synthesis is achieved by convolution of the generated excitation signal with the inverse-DFT of the reconstructed piece-wise constant amplitude spectrum. In this paper a 2400 bit/s implementation of the DFT-vocoder is discussed.

Discrete Fourier transforms of nonuniformly spaced data

Abstract: Time series or spatial series of measurements taken with nonuniform spacings have failed to yield fully to analysis using the Discrete Fourier Transform (DFT). This is due to the fact that the formal DFT is the convolution of the transform of the signal with the transform of the nonuniform spacings. Two original methods are presented for deconvolving such transforms for signals containing significant noise. The first method solves a set of linear equations relating the observed data to values defined at uniform grid points, and then obtains the desired transform as the DFT of the uniform interpolates. The second method solves a set of linear equations relating the real and imaginary components of the formal DFT directly to those of the desired transform. The results of numerical experiments with noisy data are presented in order to demonstrate the capabilities and limitations of the methods.

Discrete fourier transform circuit

Abstract: In a transmultiplexer, which is an interface between a number of PCM highways and an FDM group, the conversion between PCM and FDM involves first converting the PCM words into their linear form and filtering them, applying them to a circuit which performs a discrete Fourier transform (DFT) on them followed by a second stage of filtering. An efficient implementation of a DFT is described which reduces the number of multiplications required by making use of certain symmetry properties in a DFT matrix. The whole can be used «backwards», as it were to give an inverse Fourier transform as needed for the FDM-PCM conversion.

An Approach to the Implementation of a Discrete Cosine Transform

Abstract: An approach to the implementation of a discrete cosine transform (DCT) for application to coding speech is described. The approach is oriented toward single speech channel encoding. In addition, a detailed computer simulation of an adaptive transform coder is described. The purpose of the computer simulation is to determine the internal precision at various points in the implementation required to avoid subjective degradation. Specific recommmendations are made on the required internal precision in the implementation of the discrete cosine transform. A breadboard implementation of the DCT using SSI and MSI TTL logic based on the results of the computer simulation is reported.

An adaptive interframe transform coding system for images

Abstract: An adaptive interframe coding system for monochrome pictures is described, which uses the three-dimensional discrete cosine transform. Based on three-dimensional classification the coder is matched according to the amount of detail and the predominant direction of the structure within a transform block. The distinction between temporally changed and unchanged blocks enables coding of still blocks with an alternative algorithm. This algorithm increases the reconstruction quality for still pictures, making possible a transmission of graphs with good resolution. Simulation results are presented for a moving picture at a rate of 0.4 bit per element and still grey and black-and-white pictures with increasing quality.

Recursive calculation of Fourier transform of discrete signal

Abstract: Two algorithms for the calculation of Fourier transform of a discrete signal are derived from the known recursive method for polynomial evaluation. The first algorithm processes the elements of the discrete signal in a natural order of elements and the second one in the reverse order. Both algorithms are modified for operation with real numbers only. The relation of these algorithms to the well known Goertzel algorithm and the Collatz's rule is demonstrated. Moreover, the application of the recursive algorithm to repeated DFT calculation is described.

Residual Correlation for Generalized Discrete Transforms

Abstract: We have undertaken a systematic investigation of the performance of a complete set of discrete orthogonal transforms Gr(n). The criterion of performance is that defined by Hamidi and Pearl, namely the Residual Correlation. This criterion measures the proportional correlation left in by a transform which is suboptimal, in the sense of not completely decorrelating the signal as in the case of the Karhunen-Loeve transform. The family of orthogonal transforms Gr(n), as defined by Ahmed and Rao, ranges from the discrete Walsh transform DWT, to the discrete Fourier transform DFT, as r varies from r = 0 to r = n - 1. Our study is applied to Markov-1 signals only.

A note on polynomial transform error analysis

Abstract: In this correspondence, a fixed-point error analysis is given for polynomial transforms computed using two's complement arithmetic. The results are extended for computing the mean-square error of a two-dimensional discrete Fourier transform (DFT) computed by polynomial transform technique. Also, an earlier result derived by Nussbaumer [1] for comparing the rms error/rms result of the polynomial transform and the fast Fourier transform (FFT) has been modified.