Showing papers on "Discrete-time 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
TL;DR: Convolution, filtering, and multiplexing of signals in fractional domains are discussed, revealing that under certain conditions one can improve on the special cases of these operations in the conventional space and frequency domains.
Abstract: A concise introduction to the concept of fractional Fourier transforms is followed by a discussion of their relation to chirp and wavelet transforms. The notion of fractional Fourier domains is developed in conjunction with the Wigner distribution of a signal. Convolution, filtering, and multiplexing of signals in fractional domains are discussed, revealing that under certain conditions one can improve on 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.
499 citations
Book•
23 Dec 1994TL;DR: In this article, the authors introduce the image plane and the Two Dimensional Fourier Transform (2DFT) for image reconstruction, and present the projection-slice theorem for image restoration.
Abstract: 1. Introduction. 2. The Image Plane. 3. Two Dimensional Impulses. 4. The Two Dimensional Fourier Transform. 5. Two Dimensional Convolution. 6. The Convolution Theorem. 7. Sampling and Interpolation in Two Dimensions. 8. Digital Operations. 9. Rotational Symmetry and the Two Dimensional Fourier Transform. 10. Imaging by Convolution. 11. Diffraction Theory of Sensors. 12. Indirect Imaging and Interferometry. 13. Restoration of Images. 14. The Projection-Slice Theorem. 15. Computed Tomography. 16. Synthetic Aperture Radar. 17. Random Images and Fractals. Index.
391 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
01 Jan 1994
TL;DR: The Fourier Transform representation for functions whose inputs are boolean has been far less studied, but it seems that it can be used to learn many classes of boolean functions.
Abstract: The importance of using the “right” representation of a function in order to “approximate” it has been widely recognized. The Fourier Transform representation of a function is a classic representation which is widely used to approximate real functions (i.e. functions whose inputs are real numbers). However, the Fourier Transform representation for functions whose inputs are boolean has been far less studied. On the other hand it seems that the Fourier Transform representation can be used to learn many classes of boolean functions.
204 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: In this article, the application of the fractional Fourier transform to optical propagation problems is discussed, and the conceptual and practical advantages of this new formulation are noted, as well as the theoretical advantages of the new formulation.
Abstract: The application of the fractional Fourier transform to optical propagation problems is discussed. As illustrative examples, diffraction in a free medium as well as propagation through optical fibres are analysed with the fractional Fourier transform formalism. The conceptual and practical advantages of this new formulation are noted.
129 citations
TL;DR: In this paper, the Fourier series and its derivative were used for analysing time series of remotely-sensed data, which allows fundamental characteristics of time series data to be quantified.
Abstract: Fourier Series and the derivative were used in this study for analysing time series of remotely-sensed data. The technique allows fundamental characteristics of time series data to be quantified. In Fourier analysis a function in space or time is broken down into sinusoidal components, or harmonics. The first and second harmonics are a function of the mono or bi-modality of the curve, demonstrated in the study on Global Vegetation Index data classified into typical mono and bi-modal vegetation index zones. The last harmonic explains close to 100 per cent of the variance in the curve. Other important parameters of the time series, such as extreme points and rate of change, can be extracted from the derivative of the Fourier Series. Fourier Series may form a basis for a quantitative approach to the problem of handling temporal sequences of remotely-sensed data.
109 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: The slope transform is presented, which provides tangential morphology with the analytical power which the Fourier tansform lends to linear signal processing, in particular: dilation becomes addition (just as under a Fourier transform, convolution becomes multiplication).
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.
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.
TL;DR: In this paper, a simple relation between the fractional Fourier transform (FRACFT) and the Green's function for the harmonic oscillator is demonstrated, which enables us to understand easily the characteristics of FRACFT.
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: A theoretical formulation of cone-beam tomography with arbitrary discrete arrays of detectors and vertices and introduces a concept called the cross-talk matrix, which measures the strength of a Fourier component in the data, while an off-diagonal element quantifies the dependence or aliasing of two different components.
Abstract: Sufficiency conditions for cone-beam data are well known for the case of continuous data collection along a cone-vertex curve with continuous detectors. These continuous conditions are inadequate for real-world data where discrete vertex geometries and discrete detector arrays are used. The authors present a theoretical formulation of cone-beam tomography with arbitrary discrete arrays of detectors and vertices. The theory models the imaging system as a linear continuous-to-discrete mapping and represents the continuous object exactly as a Fourier series. The reconstruction problem is posed as the estimation of some subset of the Fourier coefficients. The main goal of the theory is to determine which Fourier coefficients can be reliably determined from the data delivered by a specific discrete design. A Fourier component will be well determined by the data if it satisfies two conditions: it makes a strong contribution to the data, and this contribution is relatively independent of the contribution of other Fourier components. To make these considerations precise, the authors introduce a concept called the cross-talk matrix. A diagonal element of this matrix measures the strength of a Fourier component in the data, while an off-diagonal element quantifies the dependence or aliasing of two different components. One reasonable approach to system design is to attempt to make the diagonal elements of this matrix large and the off-diagonal elements small for some set of Fourier components. If this goal can be achieved, simple linear reconstruction algorithms are available for estimating the Fourier coefficients. To illustrate the usefulness of this approach, numerical results on the cross-talk matrix are presented for different discrete geometries derived from a continuous helical vertex orbit, and simulated images reconstructed with two linear algorithms are presented.
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: In this article, a modified method is proposed based on a property of the discrete Fourier transform of the logarithmic potential, and error estimates for the modified method are given.
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. >
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.
Patent•
23 Jun 1994TL;DR: In this article, the authors proposed an adaptive weak signal identification system with a simple implementation which is capable of rapidly tracking weak signals with time varying frequencies in the presence of a strong interference signal.
Abstract: An adaptive weak signal identification system having a simple implementation which is capable of rapidly tracking weak signals with time varying frequencies in the presence of a strong interference signal. The system includes a first Fast Fourier Transform circuit for performing a Fast Fourier Transform on a discrete block of data points of an input data signal. A filter coefficient generator is coupled to the output of the Fast Fourier Transform circuit, and identifies the frequency of the strong interference signal, and then based thereon generates filter coefficients for a notch filter. A notch filter receives the generated filter coefficients, and further has the input data signal as an input, on which it performs a notch filtering operation to dramatically reduce the intensity of the interference signal. A second Fast Fourier Transform circuit then performs a Fast Fourier Transform on the output passed by the notch filter, and the output of the Fast Fourier Transform circuit is analyzed to identify the frequency of the weak signal of interest.
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: In this article, the Fourier transformation is applied to evaluate the translational kinetic energy operator and two alternative treatments are possible depending on whether the transformation is utilized to calculate the kinetic energy matrix elements in coordinate space, or whether it is applied on the wave function itself to switch between coordinate and momentum representations.
Abstract: In the context of wave packet methodology we show how to take advantage of the diffractive scattering symmetry arising when the incident beam is normal to the surface or to a surface principal axis. This may lead to a reduction in dimensionality being up to a factor of 8. The Fourier transformation is applied to evaluate the translational kinetic energy operator. Two alternative treatments are possible depending on whether the transformation is utilized to calculate the kinetic energy matrix elements in coordinate space, or whether it is applied to the wave function itself to switch between coordinate and momentum representations. The first approach is similar to the discrete variable representation treatment in the spirit of Light and co‐workers whereas the second one enables the use of the fast Fourier transform (FFT) scheme of Kosloff and Kosloff. We provide a detailed comparison between the two approaches as a function of the size of the grid, with and without the presence of symmetry in the diffracti...
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. >
Patent•
07 Nov 1994TL;DR: In this article, a Fast Fourier Transform (FFT) dedicated processor includes a scrambler SM scrambling a real input data sequence x(i) and thereby providing two scrambled data subsequences a(i and b(i), respectively.
Abstract: A Fast Fourier Transform (FFT) dedicated processor includes a scrambler SM scrambling a real input data sequence x(i) and thereby providing two scrambled data subsequences a(i) and b(i). A data generation circuit GC coupled to SM provides a complex data sequence y(i) whose real and imaginary parts equal the scrambled data subsequences a(i) and b(i) respectively. y(i) is applied to an arithmetic unit AU, which under the control of a control unit CoM, is successively converted to an arithmetic means AM, a data regeneration circuit RC and a combinatorial means CM. AM generates an intermediate Fast Fourier Transform series Y(i) of y(i). RC splits up Y(i) into Fast Fourier Transform series A(i) and B(i) of a(i) and b(i) respectively and CM executes a final traditional Fast Fourier Transform combinatorial step and produces the Fast Fourier Transform sequence X(i) of the real input data sequence x(i).
TL;DR: This paper proposes a bit-reversal algorithm that reduces the computational effort to an extent that it becomes negligible compared with the data swapping operation for which the bit- reversal is required.
Abstract: The necessity for an efficient bit-reversal routine in the implementation of fast discrete Fourier transform algorithms is well known. In this paper, we propose a bit-reversal algorithm that reduces the computational effort to an extent that it becomes negligible compared with the data swapping operation for which the bit-reversal is required. >
TL;DR: In this article, the problem of defining the extension of f on all of Lpv is posed and solved for several important cases, and the necessary conditions and closure theorems are inextricably related to this program.