Showing papers on "Discrete-time Fourier transform published in 1984"

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.

Elastic wave calculations by the Fourier method

Abstract: We introduce a two-dimensional forward modeling algorithim based on a Fourier method. In order to be able to handle the free surface boundary condition with the Fourier method, a new set of wave equations are derived which contain the stresses as unknowns instead of the displacements. The solution algorithm includes a discretization in both space and time. Spatial derivatives are approximated with the use of the Fast Fourier Transform, whereas temporal derivatives are calculated with second order differencing. The numerical method is tested against the analytic solution for Lamb's problem in two dimensions.

Fourier Coding of Image Boundaries

Abstract: A transform coding scheme for closed image boundaries on a plane is described The given boundary is approximated by a series of straight line segments Depending on the shape, the boundary is represented by the (x-y) coordinates of the endpoints of the line segments or by the magnitude of the successive radii vectors that are equispaced in angle around the given boundary Due to the circularity present in the data, the discrete Fourier transform is used to exactly decorrelate the finite boundary data By fitting a Gaussian circular autoregressive model to represent the boundary data, estimates of the variances of the Fourier coefficients are obtained Using the variances of the Fourier coefficients and the MAX quantizer, the coding scheme is implemented The scheme is illustrated by an example

High-Accuracy Spectrum Analysis of Sampled Discrete Frequency Signals by Analytical Leakage Compensation

Abstract: A method is presented which estimates the spectrum of a uniform sampled signal, which is sinusoidal, periodic, or composed of sinusoids of arbitrary frequencies. The proposed algorithm uses the Fast Fourier Transform algorithm. If frequency resolution is sufficient to distinguish different tones, the algorithm eliminates leakage and gives unbiased and highly accurate estimates for the amplitudes, phases, and frequencies.

Recursive cyclotomic factorization--A new algorithm for calculating the discrete Fourier transform

Abstract: In this paper, a new recursive algorithm for calculating the discrete Fourier transformation is presented. This new, so-called recursive cyclotomic factorization algorithm (RCFA) is more efficient than the fast Fourier transformation (FFT) algorithm. Moreover, due to its recursive nature, the RCFA can also be easily implemented, using only a limited number of different computation cells.

Fourier transform methods for calculating action variables and semiclassical eigenvalues for coupled oscillator systems

Abstract: Two methods for calculating the good action variables and semiclassical eigenvalues for coupled oscillator systems are presented, both of which relate the actions to the coefficients appearing in the Fourier representation of the normal coordinates and momenta. The two methods differ in that one is based on the exact expression for the actions together with the EBK semiclassical quantization condition while the other is derived from the Sorbie–Handy (SH) approximation to the actions. However, they are also very similar in that the actions in both methods are related to the same set of Fourier coefficients and both require determining the perturbed frequencies in calculating actions. These frequencies are also determined from the Fourier representations, which means that the actions in both methods are determined from information entirely contained in the Fourier expansion of the coordinates and momenta. We show how these expansions can very conveniently be obtained from fast Fourier transform (FFT) methods and that numerical filtering methods can be used to remove spurious Fourier components associated with the finite trajectory integration duration. In the case of the SH based method, we find that the use of filtering enables us to relax the usual periodicity requirement on the calculated trajectory. Application to two standard Henon–Heiles models is considered and both are shown to give semiclassical eigenvalues in good agreement with previous calculations for nondegenerate and 1:1 resonant systems. In comparing the two methods, we find that although the exact method is quite general in its ability to be used for systems exhibiting complex resonant behavior, it converges more slowly with increasing trajectory integration duration and is more sensitive to the algorithm for choosing perturbed frequencies than the SH based method. The SH based method is less straightforward to use in studying resonant systems, but good results are obtained for 1:1 resonant systems using actions defined in terms of the complex coordinates Q1±iQ2. The SH based method is also shown to be remarkably accurate in determining high energy eigenvalues (about three‐quarters of the dissociation energy).

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."

Computation of fourier transform integrals using Chebyshev series expansions

Abstract: In this paper, we extend the Clenshaw-Curtis integration method for the computation of Fourier transform integrals. In particular, we examine the numerical stability of a recurrence relation occurring in this method.

A note on iterative fourier transform phase reconstruction from magnitude

Abstract: In this correspondence, a well-known iterative procedure for Fourier transform phase reconstruction from magnitude (usually referred to as the Gerchberg-Saxton-Fienup procedure) is revisited. The convergence of this algorithm has raised some controversy in the recent literature. In this correspondence we first point out a potential source of error in the numerical implementation of the algorithm. Then, we present a conjecture which would explain why this algorithm sometimes fails.

Image reconstruction from frequency domain data on arbitrary contours

Abstract: This paper presents a new view of the problem of image reconstruction in the spatial Fourier domain based on the generalized projections of a two-dimensional image. The algorithms that are presented are derived from the exact continuous relations between the Fourier transforms of a spatially limited image and those of its generalized projections. A Fourier domain reconstruction formula is derived that is a natural consequence of these relations giving values of the two-dimensional transform on a regular grid useful for inverse FFT. This approach is contrasted with the more ad hoc methods of Fourier domain interpolation and with the spatial domain approaches such as the backprojection and backpropagation methods.

The deconvolution of Doppler-broadened positron annihilation measurements using fast Fourier transforms and power spectral analysis

Abstract: A deconvolution procedure which corrects Doppler-broadened positron annihilation spectra for instrument resolution is described. The method employs fast Fourier transforms, is model independent, and does not require iteration. The mathematical difficulties associated with the incorrectly posed first order Fredholm integral equation are overcome by using power spectral analysis to select a limited number of low frequency Fourier coefficients. The FFT/power spectrum method is then demonstrated for an irradiated high purity single crystal sapphire sample.

Evaluation of Fourier transform integrals using FFT with improved accuracy and its applications

Abstract: A new technique that significantly minimizes the aliasing error encountered in the conventional use of the fast Fourier transform (FFT) algorithms for the efficient evaluation of Fourier transforms of spatially limited functions (such as those that occur in the radiation pattern analysis of reflector antennas and planar near field to far field (NF-FF) transformation) is presented and illustrated through a typical example. Employing this technique and a discrete Fourier series (DFS) expansion for the integrand, a method for computing the radiation integrals of reflector antennas and planar NF-FF transformation integrals at arbitrary observation angles with optimum use of computer memory and time is also described.

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.

Signal reconstruction from one bit of Fourier transform phase

Abstract: In this paper, we present new results on the reconstruction of signals from one bit of Fourier transform phase, defined as the sign of the real part of the Fourier transform. Specifically, we develop a new theoretical result which shows that most two-dimensional signals can in fact be reconstructed to within a scale factor from only one bit of FF phase. Experimental results showing images reconstructed from one bit of FT phase are also presented.

An algebraic approach to discrete short-time Fourier transform analysis and synthesis

Abstract: An algebraic representation of the discrete short-time Fourier transform (DSTFT) is presented for the case in which the analysis window length N equals the transform block size M . This representation allows the application of algebraic tools for determining an optimal synthesis system which minimizes the mean square error between a given modified DSTFT (which is not necessarily a valid DSTFT sequence) and the DSTFT of the synthesized signal. If no modification is applied, the result is a unity analysis-synthesis system for any given time update R of the sliding analysis window (provided that R\leqM ). It is shown that the optimal synthesis system can be implemented by the well known weighted overlap-add (WOLA) method using an optimal synthesis window. The algebraic approach enables the extension of some recent results and the relaxation of a constraint on the analysis window. The proposed approach is found also to have a potential for solving the synthesis problem for the more general case of N>M .

The solution of the one-dimensional sign problem for Fourier transforms

Abstract: An iterative procedure for the determination of the signs of scattering amplitudes is considered. It is assumed that the scattering density is a one-dimensional antisymmetric function with a limited range of definition. The convergence of the method to a rigorous solution is proved. The stability of the procedure with respect to various experimental errors is shown in model examples. The proof can be generalized for a one-dimensional phase determination of a continuous intensity distribution.

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.

Improvement of the high frequency content of fast Fourier transforms

Abstract: A method is described of improving the high frequency components of the spectrum derived by the fast Fourier transform. It is particularly applicable to the analysis of time series which decay with time such as in stress relaxation. The method involves the addition of components at the same frequency derived from fast Fourier transforms taken over a succession of increasing time intervals. A numerical example is given.

An Inverse View of Scattering

Abstract: Born and Rytov approximations to the Helmholtz equation are described using graphical methods in the spatial Fourier domain. Angular spectra of scattered waves are graphically related to an estimate of the Fourier transform of the variations in compressibility of the object. The Fourier transform of the compressibility is also graphically related to the angular spectrum of the scattered wave.

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.

Lebesgue constants of multiple de la Vallée-Poussin sums over polyhedra

Abstract: One considers linear summation methods for the multiple Fourier series the multidimensional analogues of the de la Valle-Poussin sums. The summation of the Fourier series is carried out over the homotheties of an m-dimensional starshaped polyhedron Λ. It is shown that if Λ has rational vertices, then the Lebesgue constants of the considered methods, with the accuracy of O((p+1)−1. logm−1 (n+2)) are equal to where is the Fourier transform of the function ϕ. The exact value of the principal term of the Lebesgue constant is computed in two particular cases: 1) Λ is obtained from an m-dimensional cube by means of a linear nonsingular transformation; 2) ρ=0. Λ is an m-dimensional simplex.

On the interrelationships among a class of convolutions

Abstract: In this paper some interrelationships among a class of circular operations are investigated based on matrix formulation. It is shown that a class of convolutions representing forward/backward and convolution/correlation of two periodic sequences may be related to each other in terms of discrete transforms having the circular convolution property. The results obtained are useful in efficient realization of adaptive digital filters using fast transforms.

On window functions

Abstract: A now window function based on modified Bessel functions has been proposed. The Courier transform of the window function is obtained analytically. Computation of the Fourier transform shows a slight improvement over the Kaiser-Bessel window for suitable choice of parameters. The proposed function involves a simpler and faster computational method than that of the Kaiser-Bessel window.