Showing papers on "Non-uniform discrete Fourier transform published in 1994"
TL;DR: Convolution theorems generalizing well known and useful results from the abelian case are used to develop a sampling theorem on the sphere, which reduces the calculation of Fourier transforms and convolutions of band-limited functions to discrete computations.
937 citations
Book•
13 Dec 1994
TL;DR: The Discrete Fourier Transform (DFT) as mentioned in this paper is a Fourier transform based on the Fourier Integral Transform (FIFO) and is used as an estimator of the FFT.
Abstract: CONTINUOUS FOURIER ANALYSIS. Background. Fourier Series for Periodic Functions. The Fourier Integral. Fourier Transforms of Some Important Functions. The Method of Successive Differentiation. Frequency-Domain Analysis. Time-Domain Analysis. The Properties. The Sampling Theorems. DISCRETE FOURIER ANALYSIS. The Discrete Fourier Transform. Inside the Fast Fourier Transform. The Discrete Fourier Transform as an Estimator. The Errors in Fast Fourier Transform Estimation. The Four Kinds of Convolution. Emulating Dirac Deltas and Differentiation on the Fast Fourier Transform. THE USER'S MANUAL FOR THE ACCOMPANYING DISKS. Appendices. Answers to the Exercises. Index.
309 citations
TL;DR: Presents a method to analyze and filter digital signals of finite duration by means of a time-frequency representation, and proposes orthogonal and periodic basic discrete wavelets to get a correct invertibility of this procedure.
Abstract: Presents a method to analyze and filter digital signals of finite duration by means of a time-frequency representation. This is done by defining a purely invertible discrete transform, representing a signal either in the time or in the time-frequency domain, as simply as possible with the conventional discrete Fourier transform between the time and the frequency domains. The wavelet concept has been used to build this transform. To get a correct invertibility of this procedure, the authors have proposed orthogonal and periodic basic discrete wavelets. The properties of such a transform are described, and examples on brain-evoked potential signals are given to illustrate the time-frequency filtering possibilities. >
194 citations
TL;DR: The “fractional Fourier transform,” previously developed by the authors, is applied to this problem with a substantial savings in computation.
Abstract: The fast Fourier transform (FFT) is often used to compute numerical approximations to continuous Fourier and Laplace transforms. However, a straightforward application of the FFT to these problems often requires a large FFT to be performed, even though most of the input data to this FFT may be zero and only a small fraction of the output data may be of interest. In this note, the “fractional Fourier transform,” previously developed by the authors, is applied to this problem with a substantial savings in computation.
141 citations
TL;DR: A new complete and convergent set of invariant features under planar similarities is proposed using the Analytical Fourier-Mellin Transform (AFMT), which gives a distance between the shapes which is invariant under similarities.
131 citations
TL;DR: In this article, a general discrete Bessel transform based on the Bessel functions of the first kind for any integer or half-integer order ν is presented. But this transform is not applicable to the harmonic oscillator.
Abstract: We present a general discrete Bessel transform based on the Bessel functions of the first kind for any integer or half‐integer order ν. This discrete Bessel transform shares a number of similitudes with the discrete Fourier transform in that we have discretized both the coordinate and momentum continuums, and since the discrete transform of order 1/2 exactly specializes to the discrete sine Fourier transform. We demonstrate that our discrete Bessel transform is comparable to the discrete Fourier transform in terms of both the accuracy and the efficiency. Indeed, our discretization procedure provides an optimal sampling grid for Bessel functions of the first kind, and the accuracy of the transform converges exponentially as the number of grid points is increased. We successfully apply the optimally discretized Bessel methodology to the harmonic oscillator in both cylindrical and spherical coordinates.
111 citations
Patent•
12 May 1994
TL;DR: In this article, a frequency analysis method comprises using a window function to evaluate aemporal input signal present in the form of discrete sampled values, which are subsequently subjected to Fourier transformation for the purpose of generating a set of coefficients.
Abstract: A frequency analysis method comprises using a window function to evaluate aemporal input signal present in the form of discrete sampled values. The windowed input signal is subsequently subjected to Fourier transformation for the purpose of generating a set of coefficients. In order to develop such a method so that the characteristics of the human ear are simulated not only with respect to the spectral projection in the frequency range, but also with respect to the resolution in the temporal range, a set of different window functions is used to evaluate a block of the input signal in order to generate a set of blocks, weighted with the respective window functions, of sampled values whose Fourier transforms have different bandwidths, before each of the simultaneously generated blocks of sampled values is subjected to a dedicated Fourier transformation in such a way that for each window function at least respectively one coefficient is calculated which is assigned the bandwidth of the Fourier transforms of this window function, and that the coefficients are chosen such that the frequency bands assigned to them essentially adjoin one another.
105 citations
TL;DR: The complex amplitude distributions on two spherical reference surfaces of given curvature and spacing are simply related by a fractional Fourier transform, providing new insight into wave propagation and spherical mirror resonators.
Abstract: The complex amplitude distributions on two spherical reference surfaces of given curvature and spacing are simply related by a fractional Fourier transform. The order of the fractional Fourier transform is proportional to the Gouy phase shift between the two surfaces. This result provides new insight into wave propagation and spherical mirror resonators as well as the possibility of exploiting the fractional Fourier transform as a mathematical tool in analyzing such systems.
90 citations
TL;DR: Fractional Fourier transforms of an arbitrary degree can be implemented by refractive lenses as mentioned in this paper, and perfect imaging systems and correlators may be implemented using cascading fractional-fractional transformer units of the same family and fractional degree.
83 citations
TL;DR: It is claimed that the discretized version of the thinplate spline may profitably be used in place of the Discrete Fourier Transform in a variety of image processing applications besides spline smoothing.
Abstract: SUMMARY This paper describes a fast method of computation for a discretized version of the thinplate spline for image data. This method uses the Discrete Cosine Transform and is contrasted with a similar approach based on the Discrete Fourier Transform. The two methods are similar from the point of view of speed, but the errors introduced near the edge of the image by use of the Discrete Fourier Transform are significantly reduced when the Discrete Cosine Transform is used. This is because, while the Discrete Fourier Transform implicitly assumes periodic boundary conditions, the Discrete Cosine Transform uses reflective boundary conditions. It is claimed that the Discrete Cosine Transform may profitably be used in place of the Discrete Fourier Transform in a variety of image processing applications besides spline smoothing.
73 citations
21 Aug 1994
TL;DR: A method for analyzing the linear complexity of nonlinear filterings of PN-sequences that is based on the Discrete Fourier Transform is presented, which makes use of "Blahut's theorem", which relates thelinear complexity of an N-periodic sequence in GF(q)N and the Hamming weight of its frequency-domain associate.
Abstract: A method for analyzing the linear complexity of nonlinear filterings of PN-sequences that is based on the Discrete Fourier Transform is presented. The method makes use of "Blahut's theorem", which relates the linear complexity of an N-periodic sequence in GF(q)N and the Hamming weight of its frequency-domain associate. To illustrate the power of this approach, simple proofs are given of Key's bound on linear complexity and of a generalization of a condition of Groth and Key for which equality holds in this bound.
01 Mar 1994
TL;DR: Generalizations of the Fourier transform kernel lead to a number of novel transforms, in particular, special discrete cosine, discrete sine, and real discrete Fourier transforms, which have already found use in anumber of applications.
Abstract: Major continuous-time, discrete-time, and discrete Fourier-related transforms as well as Fourier-related series are discussed both with real and complex kernels. The complex Fourier transforms, Fourier series, cosine, sine, Hartley, Mellin, Laplace transforms, and z-transforms are covered on a comparative basis. Generalizations of the Fourier transform kernel lead to a number of novel transforms, in particular, special discrete cosine, discrete sine, and real discrete Fourier transforms, which have already found use in a number of applications. The fast algorithms for the real discrete Fourier transform provide a unified approach for the optimal fast computation of all discrete Fourier-related transforms. The short-time Fourier-related transforms are discussed for applications involving nonstationary signals. The one-dimensional transforms discussed are also extended to the two-dimensional transforms. >
TL;DR: In this article, the authors derived the linear and parabolic [tau]-p transform formulas for the continuous function domain and showed that the derived formulas are identical to the DRT equations obtained by other researchers.
Abstract: New derivations for the conventional linear and parabolic [tau]-p transforms in the classic continuous function domain provide useful insight into the discrete [tau]-p transformations. For the filtering of unwanted waves such as multiples, the derivation of the [tau]-p transform should define the inverse transform first, and then compute the forward transform. The forward transform usually requires a p-direction deconvolution to improve the resolution in that direction. It aids the wave filtering by improving the separation of events in the [tau]-p domain. The p-direction deconvolution is required for both the linear and curvilinear [tau]-p transformations for aperture-limited data. It essentially compensates for the finite length of the array. For the parabolic [tau]-p transform, the deconvolution is required even if the input data have an infinite aperture. For sampled data, the derived [tau]-p transform formulas are identical to the DRT equations obtained by other researchers. Numerical examples are presented to demonstrate event focusing in [tau]-p space after deconvolution.
TL;DR: It is shown how it is possible to implement the fractional Fourier transform on time signals by using optoelectronic modulators and optical fibers with suitable dispersion and a fractional-Fourier-transform-based photonic signal-processing system could be composed.
Abstract: The family of fractional Fourier transforms permits presentation of a temporal signal not only as a function of time or as a pure frequency function but also as a mixed time and frequency function with a continuous degree of emphasis on time or on frequency features. We show how it is possible to implement the fractional Fourier transform on time signals by using optoelectronic modulators and optical fibers with suitable dispersion. We also show how a fractional-Fourier-transform-based photonic signal-processing system could be composed.
TL;DR: Different styles of implementations of fast inverse DCTs designed especially for sparse data and compares them on workstation processors are described.
Abstract: The discrete cosine transform (DCT) is often applied to image compression to decorrelate picture data before quantization. This decorrelation results in many of the quantized transform coefficients equaling zero, hence the compression gain. At the decoder, very few nonzero quantized transform coefficients are received, so the input to the inverse DCT is sparse, greatly reducing the required computation. This paper describes different styles of implementations of fast inverse DCTs designed especially for sparse data and compares them on workstation processors.
TL;DR: The discrete Gabor (1946) transform algorithm is introduced that provides an efficient method of calculating the complete set of discreteGabor coefficients of a finite-duration discrete signal from finite summations and to reconstruct the original signal exactly from the computed expansion coefficients.
Abstract: The discrete Gabor (1946) transform algorithm is introduced that provides an efficient method of calculating the complete set of discrete Gabor coefficients of a finite-duration discrete signal from finite summations and to reconstruct the original signal exactly from the computed expansion coefficients. The similarity of the formulas between the discrete Gabor transform and the discrete Fourier transform enables one to employ the FFT algorithms in the computation. The discrete 1-D Gabor transform algorithm can be extended to 2-D as well. >
TL;DR: The concept of filtering of signals in fractional domains is developed, revealing that under certain conditions one can improve upon the special cases of these operations in the conventional space and frequency domains.
Abstract: Fractional Fourier transforms, which are related to chirp and wavelet transforms, lead to the notion of fractional Fourier domains. The concept of filtering of signals in fractional domains is developed, revealing that under certain conditions one can improve upon the special cases of these operations in the conventional space and frequency domains. Because of the ease of performing the fractional Fourier transform optically, these operations are relevant for optical information processing.
TL;DR: It is shown that one of the best substitutions for the Gaussian function in the Fourier domain is a squared sinusoid function that can form a biorthogonal windowfunction in the time domain.
Abstract: We discuss the semicontinuous short-time Fourier transform (STFT) and the semicontinual wavelet transform (WT) with Fourier-domain processing, which is suitable for optical implementation. We also systematically analyze the selection of the window functions, especially those based on the biorthogonality and the orthogonality constraints for perfect signal reconstruction. We show that one of the best substitutions for the Gaussian function in the Fourier domain is a squared sinusoid function that can form a biorthogonal window function in the time domain. The merit of a biorthogonal window is that it could simplify the inverse STFT and the inverse WT. A couple of optical architectures based on Fourier-domain processing for the STFT and the WT, by which real-time signal processing can be realized, are proposed.
TL;DR: A joint power spectrum based processor called joint transform correlator, which is shift invariant and can produce at the output well-defined peaks identifying the presence and the location of the targets.
TL;DR: The authors present an algorithm that calculates the minimal polynomial of s, assuming that a period of s is known, and generalises both the discrete Fourier transform and the Games-Chan algorithm.
Abstract: Let s be a periodic sequence whose elements lie in a finite field. The authors present an algorithm that calculates the minimal polynomial of s, assuming that a period of s is known. The algorithm generalises both the discrete Fourier transform and the Games-Chan algorithm. >
19 Apr 1994
TL;DR: An algorithm is developed, called the quick Fourier transform (QFT), that will reduce the number of floating point operations necessary to compute the DFT by a factor of two or four over direct methods or Goertzel's method for prime lengths.
Abstract: This paper will look at an approach that uses symmetric properties of the basis function to remove redundancies in the calculation of discrete Fourier transform (DFT). We will develop an algorithm, called the quick Fourier transform (QFT), that will reduce the number of floating point operations necessary to compute the DFT by a factor of two or four over direct methods or Goertzel's method for prime lengths. Further by applying the idea to the calculation of a DFT of length-2/sup M/, we construct a new O(N log N) algorithm. The algorithm can be easily modified to compute the DFT with only a subset of input points, and it will significantly reduce the number of operations when the data are real. The simple structure of the algorithm and the fact that it is well suited for DFTs on real data should lead to efficient implementations and to a wide range of applications. >
Patent•
04 Oct 1994
TL;DR: In this article, a phase encoded filter is positioned at the transform plane and a second filter is tandemly positioned with respect to the first filter, the second filter having a transmittance which is statistically similar to the reciprocal spatial frequency spectrum of the Fourier transform of the distortion function.
Abstract: A first lens produces a Fourier transform of the wavefront distorted optical image at the Fourier transform plane. A phase encoded filter is positioned at the transform plane and a second filter is tandemly positioned with respect to the first filter, the second filter having a transmittance which is statistically similar to the reciprocal spatial frequency spectrum of the Fourier transform of the distortion function, to in turn produce an intermediate signal at the transform plane, which is now Fourier transformed by a second lens to recover the optical image having a substantially reduced degree of distortion.
TL;DR: An algorithm for high-precision phase measurement is developed by using the Fourier coefficient that corresponds to the spatial frequency of the Fizeau fringes, and methods for determining the fringe carrier frequency are described.
Abstract: The Fourier transform method is applied to analyze the initial phase of linear and equispaced Fizeau fringes We develop an algorithm for high-precision phase measurement by using the Fourier coefficient that corresponds to the spatial frequency of the Fizeau fringes, and we describe methods for determining the fringe carrier frequency Errors caused by carrier frequency fluctuation and data truncation are studied theoretically and by computer simulation To demonstrate the method we apply it to the real-time calibration of a piezoelectric transducer mirror in a Twyman–Green interferometer
TL;DR: The problem of expressing the kernel of a fractional Fourier transform in elementary functions is posed and solved.
Abstract: The problem of expressing the kernel of a fractional Fourier transform in elementary functions is posed and solved.
TL;DR: A new set of input and output index mappings for DST is presented based on the Lee algorithm for the discrete cosine transform and from the relations obtained between DST and DCT.
Abstract: Presents a fast algorithm for computing the discrete sine transform (DST) when the transform size N is decomposable into mutually prime factors. Based on the Lee algorithm for the discrete cosine transform (DCT) and from the relations obtained between DST and DCT, a new set of input and output index mappings for DST is presented. >
TL;DR: In this article, a phase retrieval algorithm that uses real-plane zero locations to generate a simple parameterization of the Fourier phase and uses knowledge about the image to estimate the phase parameters is presented.
Abstract: Locations at which the Fourier transform F(u, υ) of an image equals zero have been called real-plane zeros, since they are the intersections of the zero curves of the analytic extension of F(u, υ) with the real–real (u, υ) plane. It has been shown that real-plane zero locations have a significant effect on the Fourier phase in that they are the end points of phase branch cuts, and it has been shown that real-plane zero locations can be estimated from Fourier magnitude data. Thus real-plane zeros can be utilized in phase retrieval algorithms to help constrain the possible Fourier phases. First we show a simplified procedure for estimating real-plane zeros from the Fourier magnitude. Then we present a new phase retrieval algorithm that uses real-plane zero locations to generate a simple parameterization of the Fourier phase and uses knowledge about the image to estimate the Fourier phase parameters. We show by example that this algorithm generates improved phase retrieval results when it is used as an initial guess into existing iterative algorithms. We assume that the image is real valued.
01 May 1994
TL;DR: A new approach to Fourier analysis within the context of circuit simulation is presented that is considerably more accurate and flexible than the traditional SPICE approach and can be used to accurately compute a small number of Fourier coefficients for broad-spectrum signals.
Abstract: A new approach to Fourier analysis within the context of circuit simulation is presented that is considerably more accurate and flexible than the traditional SPICE approach. It is based on the direct computation of the Fourier integral rather than on the discrete Fourier transform and so it is not subject to aliasing. It can be used to accurately compute a small number of Fourier coefficients for broad-spectrum signals such as those generated by mixers, /spl Sigma//spl Delta/ and pulse-width modulators, DACs, and SC-filters. In addition, techniques that reduce errors and provide the ability to resolve harmonics 120 dB-140 d below the carrier are presented. >
30 May 1994
TL;DR: In this paper the sliding implementation of the other useful transforms, that can also be implemented with the order of N complexity, are worked out in detail.
Abstract: Implementation of the transform domain adaptive filters is addressed. Recent results have shown that if the input data to a radix-2 fast Fourier transform (FFT) structure is sliding one sample at a time, only N-1 butterflies need to be calculated for updating the FFT structure, after the arrival of every new data sample. This is opposed to most of the previous reports that, assume order of N log N complexity, for such implementation. In this paper the sliding implementation of the other useful transforms, that can also be implemented with the order of N complexity, are worked out in detail. >
TL;DR: A new approach to inhomogeneous layer synthesis is proposed that differs from the classical Fourier transform methods, based on the wave-number discretization of the continuous Q function, and an iterative procedure is introduced to overcome the inaccuracy connected with the approximate Q-function representation.
Abstract: A new approach to inhomogeneous layer synthesis is proposed that differs from the classical Fourier transform methods. This approach, based on the wave-number discretization of the continuous Q function, is described together with the relationship between the smoothness of the inhomogeneous layer’s refractive-index profile, the complexity of the target reflectance specification, and the total optical thickness of the layer. An iterative procedure is introduced to overcome the inaccuracy connected with the approximate Q-function representation. Some control features of the Q-function phase are studied, and numerical examples are given.