Showing papers on "Harmonic wavelet transform published in 1975"

A new algorithm in spectral analysis and band-limited extrapolation

Abstract: If only a segment of a function f (t) is given, then its Fourier spectrum F(\omega) is estimated either as the transform of the product of f(t) with a time-limited window w(t) , or by certain techniques based on various a priori assumptions. In the following, a new algorithm is proposed for computing the transform of a band-limited function. The algorithm is a simple iteration involving only the fast Fourier transform (FFT). The effect of noise and the error due to aliasing are determined and it is shown that they can be controlled by early termination of the iteration. The proposed method can also be used to extrapolate bandlimited functions.

Improved digital filters for evaluating Fourier and Hankel transform integrals

Abstract: -~-------;----------------------------------------------... Acknowledgment---------------------------------------------------Introduction--------------------------------------~--------------Filter design----------------------------------------------------Algorithms-------------------------------------------------------Filter tests-----------------------------------------------------Discussion-------------------------------------------------------References---------------------------~---------------------------Appendix 1.----Source listing of subprograms---------------------2.----Test results---------------------------------------

A 500-point fourier transform using charge-coupled devices

Abstract: A CCD transversal filter chip, which performs a 500-point discrete Fourier transform using the chirp z-transform algorithm, will be described. Performance characteristics will be demonstrated, new operational modes presented, and system applications discussed.

A Parallel Radix-4 Fast Fourier Transform Computer

Abstract: The organization and functional design of a parallel radix-4 fast Fourier transform (FFT) computer for real-time signal processing of wide-band signals is introduced.

Generalized harmonic analysis for distributions

Abstract: Generalized harmonic analysis in the sense of Wiener is extended to the framework of Schwartz distributions. The approach seems mathematically and physically more transparent than the classical scheme, since every distribution possesses a Fourier transform so that the use of integrated Fourier transforms is avoided. A generalized Wiener-Khintchine representation is given which agrees well with the intuitive concept of the power spectrum. The latter is shown to be a tempered measure, in general, whose support is contained in the support of the Fourier transform of the signal. The correlation functional and power spectrum of filtered distributional signals is derived for a class of generalized filter impulse responses, which includes those that have bounded support or correspond to stable rational transfer functions. As an illustration, the form of the correlation functional and power spectrum for periodic and almost-periodic distributions and for delta-pulse trains occurring in sampled-data systems is given, and a deterministic white noise signal is constructed.

Apparatus for the generation of a two-dimensional discrete fourier transform

Abstract: An apparatus for the generation of a two-dimensional discrete Fourier transform of an input signal developed from data within a data block, or field, having a size of N1 by N2 data points, N1 and N2 being relatively prime with respect to each other, the transforms being in a form suitable for subsequent electronic processing. The apparatus comprises an input scan apparatus connectable to the N1 by N2 two-dimensional data field comprising the input signal, such as a television viewing field. The apparatus receives and scans in proper order, or sequence, the N1 by N2 input data so that the subsequently generated one-dimensional Fourier transform of the length N1 N2 serial data string is identical to an N1 by N2 two-dimensional discrete Fourier transform of the N1 by N2 input data samples. A serial-access one-dimensional discrete Fourier transform (DFT) device, connected to the input scan generator, generates a one-dimensional discrete Fourier transform of the length N1 N2 serial data string.

Programable filter using chirp-Z transform

Abstract: A transform structure employing surface wave devices performs Fourier transform operations on a time-limited substantially band limited signal. The output signal of the transform structure is related to the Fourier transform of the input signal in a manner which permits forming the product of this output with a similar output of a second transform structure. This product is inverted by a third transform structure so as to provide the convolution of the two original input functions. The overall structure comprises a filter apparatus readily programable in either the time or frequency domain.

Fast Fourier Transform

Abstract: The main objective of this chapter is to develop a fast algorithm for efficient computation of the DFT. This algorithm, called the fast Fourier transform (FFT), significantly reduces the number of arithmetic operations and memory required to compute the DFT (or its inverse). Consequently, it has accelerated the application of Fourier techniques in digital signal processing in a number of diverse areas. A detailed development of the FFT is followed by some numerical examples which illustrate its applications.

Pseudonoise test signals and the fast Fourier transform

Abstract: Recent published work has indicated a growing interest in the combined use of periodic pseudonoise (p.n.) test signals and the fast Fourier transform (f.f.t.). However, it has been noted by Barker and Davy that the period of pseudonoise sequences are such that it is not possible to use the most efficient form of the f.f.t. This letter points out ways round the problem by drawing attention to a previously published result in this area. In addition, a little known algorithm that directly exploits the f.f.t. to generate pseudonoise is given.

Generalized transforms

Abstract: A recent class of continuous transforms is generalized to give an infinite number of new transforms. A frequency interpretation of the transform base functions gives fast generalized transform (FGT) matrices using standard digit reversal techniques. Examples of FGT factorizations are given. Advantages of the generalized transforms include continuous transform properties for system design and analysis and FGT algorithms for practical applications using digital hardware.

A Fast Two-Dimensional Karhunen-Loeve Transform

Abstract: One frequently used image compression method is based on transform coding. In terms of RMS error, the best transform is the Karhunen-Loeve (Principal Components). This method is not generally used due to computational complexity. In this paper we show that under isotropicity conditions the Karhunen-Loeve is almost separable and that an approximate fast principal components transform exists. Our results indicate that the fast K-L is nearly as good as the true K-L and that it yields better results than other discrete transforms such as DLB, SLANT, or Hadamard. The approximations and errors are discussed in terms of the RMS and RMS correlated error.© (1975) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

On approximating Fourier integral transforms by their discrete counterparts in certain geophysical applications

Abstract: In the past, fast Fourier transforms (FFT's) have been used in geophysical data analysis and, with some success, in theoretical analysis. However, in the general situation when neither the function nor its transform image is of compact support, the digital transforms inevitably introduce aliasing errors in both the domain and range space. A technique to assess and minimize this error is given for a restricted set of functions.

A fast Gaussian method for Fourier transform evaluation

Abstract: A Gaussian method for fast evaluation of approximations to Fourier integral transforms is presented. This method is faster than the FFT for transforms of functions that require considerable computer time to compute. It is especially useful when transforms of high accuracy are needed.

Roundoff error in fast Fourier transforms

Abstract: The finite word length used in the computer causes round-off error in the calculation of Fourier coefficients. When the fast Fourier transform method is used, the statistical mean-square error has been previously determined [3] for the case of the decimation-infrequency algorithm. This letter treats the same problem for the decimation-in-time algorithm.

Bandlimiting effects in an optical Laplace transform computer.

Abstract: Data processing in coherent optical systems has been limited to a large extent by operations involving spectrum manipulations through the Fourier transform. By appropriate modification of the Green function kernel of the Rayleigh-Sommerfeld diffraction integral, it should be possible to obtain transform relationships between the input and output of a coherent processor, other than the Fourier transform. Cutrona has suggested a possible implementation for a Laplace transformer but has apparently not attempted a physical realization [ IRE Trans. Inf. TheoryIT-6, 386 ( 1960)]. In this study, a Laplace transform computer is described in which it is found that spatial bandlimiting is a restriction on its usefulness. To compensate for this bandlimiting, a convergence filter is described along with a method to produce this and other spatial filters physically for optical processors.

Novel hybrid optical correlator: theory and optical simulation.

Abstract: The inverse transform of the product of two Fourier transform holograms is analyzed and shown to contain the correlation of the two images from which the holograms were formed The theory, analysis, and initial experimental demonstration of the feasibility of a novel correlation scheme using this multiplied Fourier transform hologram system are presented

Computing Robust Walsh-Fourier Transform by Error Product Minimization

Abstract: A scheme for computing the Walsh-Fourier transform of noisy data is presented. The results are insensitive to isolated noise points, even when their magnitudes are large.

Fourier transforms in surface structure determination from LEED. The transform-deconvolution method

Abstract: Fourier transforms of LEED intensities contain convolution products of functions of the interatomic vectors with data truncation, lattice vibration and potential windows. The dominant structural information in the transform originates from kinematical processes. Extraction of structural parameters of the surface, can be achieved upon a proper deconvolution of the windows from the transforms of the intensities.

Direct Coherent Optical Fourier Transform of Curves

Abstract: A coherent optical Fourier transform system is studied for which masks of curves only are used as input. A general theory for this problem is discussed, resulting in the functional connection between input and output data and limitations due to thickness, curvature, and extension of the curve. The general result can be simplified for a curve which represents a signal modulated by a carrier. In this case the Fourier transform of the signal can be found optically in first-order diffraction. The method is applied to interferogram curves arising in Fourier spectroscopy.

Parallel and pipeline computation of fast unitary transforms

Abstract: The letter discusses the parallel and pipeline organization of fast-unitary-transform algorithms such as the fast Fourier transform, and points out the efficiency of a combined parallel-pipeline processor of a transform such as the Haar transform, in which (2 to the n-th power) -1 hardware 'butterflies' generate a transform of order 2 to the n-th power every computation cycle.

Numerical image reconstruction from an off-axis sound-wave hologram

Abstract: A numerical image reconstruction technique is proposed, where the images are calculated by performing a Fourier transform once instead of transforming twice as in the conventional method. This technique enables us to reconstruct images from off-axis holograms. An image reconstruction experiment was done with a sound-wave hologram of 18 kHz.

Fourier Representation of Sequences

Abstract: The Fourier representation of analog signals was discussed in the previous chapter. This representation is now extended to data sequences, and digital signals. To this end, the discrete Fourier transform (DFT) is defined and several of its properties are developed. Specifically, the convolution and correlation theorems are described and the spectral properties such as amplitude, phase, and power spectra are developed. By illustrating the 2-dimensional DFT, it is shown that the DFT can be extended to multiple dimensions. Finally, the concepts of time-varying Fourier power and phase spectra are introduced.