Showing papers on "Non-uniform discrete 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.

Gauss and the history of the fast fourier transform

Abstract: THE fast Fourier transform (Fm has become well known . as a very efficient algorithm for calculating the discrete Fourier Transform (Om of a sequence of N numbers. The OFT is used in many disciplines to obtain the spectrum or . frequency content of a Signal, and to facilitate the computation of discrete convolution and correlation. Indeed, published work on the FFT algorithm as a means of calculating the OFT, by J. W. Cooley and J. W. Tukey in 1965 [1], was a turning point in digital signal processing and in certain areas of numerical analysis. They showed that the OFT, which was previously thought to require N 2 arithmetic operations, could be calculated by the new FFT algorithm using only N log Noperations. This algorithm had a revolutionary effect on many digital processing methods, and remains the most Widely used method of computing Fourier transforms [2]. In their original paper, Cooley and Tukey referred only to I. J. Good's work published in 1958 [3] as having influenced their development. However, It was soon discovered there are major differences between the Cooley-Tukey FFT and the algorithm described by Good, which is now commonly referred to as the prime factor algorithm (PFA). Soon after the appearance of the CooleyTukey paper, Rudnick [4] demonstrated a similar algorithm, based on the work of Danielson and Lanczos [5] which had appeared in 1942. This discovery prompted an investigation into the history of the FFT algorithm by Cooley, Lewis, and Welch [6]. They discovered that the Oanielson-Lanczos paper referred to work by Runge published at the tu rn of the centu ry [7, 8]. The algorithm developed by Cooley and Tukey clearly had its roots in, though perhaps not a direct influence from, the early twentieth century. In a recently published history of numerical analysis [9], H. H. Goldstine attributes to Carl Friedrich Gauss, the eminent German mathematician, an algorithm similar to the FFT for the computation of the coefficients of a finite Fourier series. Gauss' treatise describing the algorithm was not published in his lifetime; it appeared only in his collected works [10] as an unpublished manuscript. The presumed year of the composition of this treatise is 1805, thereby suggesting that efficient algorithms for evaluating

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.

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."

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.

On the use of discrete cosine transform in cepstral analysis

Abstract: If the original signal is defined to be symmetrical, the discrete Fourier transform (DFT) used in cepstral analysis can be replaced by a discrete cosine transform (DCT). This principle is applied to the evaluation of the real and complex pseudocepstrum of speech signals. In both the real and complex cepstrum cases, it is found that the use of the DCT does not degrade the information contained in the cepstrum while substantially reducing the computational complexity.

Prime factor FFT algorithms for real-valued series

Abstract: This paper presents two techniques for computing a discrete transform of a vector of real-valued data using the Prime Factor Algorithm (PFA) with high-speed convolution. These techniques are applied to the Discrete Fourier Transform (DFT) and the Discrete Hartley Transform (DHT). The primary goals of these techniques are to eliminate unnecessary computations required when implementing a complex transform on a real-valued vector, to compute the transform in-place in the original length-N real vector, and to obtain the transform coefficients in-order. The two algorithms described require modification of the Winograd short-length transform modules to accommodate a real input. One technique replaces the modules in the Burrus-Eschenbacher PFA program with the modified real-input modules and constructs the complete transform in a final step of additions and subtractions after modules for each factor have been executed. The other technique uses these real-input DFT modules for part of the computation associated with each factor and requires complex input DFT modules for another part of the computation. These algorithms require exactly one half of the number of multiplications and slightly less than one half of the number of additions required by a complex-input PFA.

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 .

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.

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.

FCT -- A fact cosine transform

Abstract: A new discrete cosine transform algorithm, named Fast Cosine Transform (FCT), is introduced for the 2m-point discrete cosine transform This algorithm reduces the number of multiplications to about half of earlier results, and furthermore, it renders a simple and systematic structure for implementation

High‐resolution frequency determination of discrete signal components

Abstract: An important problem in many applications involves the determination of the frequency of a limited set of sinusoidal components in a time domain signal. The Fourier transform is used as the fundamental tool for conversion to the frequency domain. In the present investigation, an analysis is conducted of the errors inherent in the use of the Fourier transform without time domain windowing, and an algorithm is developed which makes it possible to obtain the frequency of the component signals with high accuracy. This algorithm combines the advantages of the Gaussian window with the computational efficiency of the Fast Fourier Transform (FFT).

The phase and amplitude corrected Fourier transform for the detection of small signals

Abstract: In many cases one is interested in small EXAFS signals which are present along with large signals. When the small signal has a frequency close to the dominant frequency of the EXAFS spectrum it may be fully obscured in the Fourier transform. Detection may be even more difficult when the radial structure function (RSF) of the dominant signal shows a complicated structure due to the k-dependence of the phase and amplitude functions of the dominant absorber scatterer pair. This k-dependence especially appears for high Z elements. When a suitable reference compound for the dominant signal is available, the RSF may be simplified by using a Fourier transform which eliminates the complex structure of the phase shift function and the back-scattering amplitude. This can be done by transforming Χ(k).exp(−iφj.(k))/ fj.(k) instead of Χ(k) (1–4), where φj.(k) and fj. (k) are the phase shift function and the backscattering amplitude for the jth shell of the reference compound, respectively. Such a phase and amplitude corrected Fourier transform reduces the complicated peak in the RSF to a single, symmetrical and localized peak which peaks at the correct distance and exhibits a very simple, symmetrical imaginary part. Small signals in the neighbourhood of the main peak will now appear as distortions from the symmetrical peak.

Two Extensions Of Fourier Optical Processors

Abstract: We discuss two generalizations of the continuous Fourier transform performed by coherent optical systems. The first concerns the introduction of an appropriate exponential damping factor in the input plane, which leads to a processor that evaluates a two-dimensional slice through the four-dimensional complex Laplace transform domain. By performing Laplace filtering, rather than Fourier filtering, one can in principle trade off dynamic range in the filter plane for dynamic range in the input plane. Using a Laplace transform, it is also possible to find the complex roots of polynomials. The second generalization concerns modification of the continuous Fourier transform to behave as a discrete Fourier transform. With such a modification, it is in principle possible to find (in a single step, without iterations) the eigenvalues of any circulant matrix, or any circulant approximation to a Toeplitz matrix (including correlation matrices) using a coherent optical processor. Furthermore, if a light valve having a suitable nonlinear relation between amplitude transmittance and exposure is available, it is possible to obtain the inverse of any matrix in the class described above in a single pass through a coherent optical processor.© (1984) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.