scispace - formally typeset
Search or ask a question

Showing papers on "Discrete-time Fourier transform published in 1995"



Proceedings ArticleDOI
09 May 1995
TL;DR: The optimal fractional Fourier domain filter is derived that minimizes the MSE for given non-stationary signal and noise statistics, and time-varying distortion kernel.
Abstract: The ordinary Fourier transform is suited best for analysis and processing of time-invariant signals and systems. When we are dealing with time-varying signals and systems, filtering in fractional Fourier domains might allow us to estimate signals with smaller minimum mean square error (MSE). We derive the optimal fractional Fourier domain filter that minimizes the MSE for given non-stationary signal and noise statistics, and time-varying distortion kernel. We present an example for which the MSE is reduced by a factor of 50 as a result of filtering in the fractional Fourier domain, as compared to filtering in the conventional Fourier or time domains. We also discuss how the fractional Fourier transformation can be computed in O(N log N) time, so that the improvement in performance is achieved with little or no increase in computational complexity.

130 citations


Journal ArticleDOI
TL;DR: In this paper, the same rules can be applied to create a new type of fractional-order Fourier transform which results in a smooth transition of a function when transformed between the real and Fourier spaces.

112 citations


Book
01 Jan 1995
TL;DR: In this article, a partial fulfillment of the requirements for the degree of Master of Science Mathematics at the University of New Mexico (UNM) has been reported for the first time.
Abstract: OF THESIS Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science Mathematics The University of New Mexico Albuquerque, New Mexico

108 citations


Journal ArticleDOI
TL;DR: A derivation of the Mellin transform given the Fourier transform that permits closed-form derivations of the temporal moments for various simple geometries is presented and it is demonstrated that the computational cost to produce the nth moment is the same as producing the first n temporal samples of the original function.
Abstract: Modeling of the full temporal behavior of photons propagating in diffusive materials is computationally costly. Rather than deriving intensity as a function of time to fine sampling, we may consider methods that derive a transform of this function. To derive the Fourier transform involves calculation in the (complex) frequency domain and relates to intensity-modulated experiments. We consider instead the Mellin transform and show that this relates to the moments of the original temporal distribution. A derivation of the Mellin transform given the Fourier transform that permits closed-form derivations of the temporal moments for various simple geometries is presented. For general geometries a finite-element method is presented, and it is demonstrated that the computational cost to produce the nth moment is the same as producing the first n temporal samples of the original function.

93 citations


BookDOI
01 Jan 1995

89 citations



Book
31 Mar 1995
TL;DR: The formal complex Fourier transform (CFT) as mentioned in this paper is an extension of the Fourier Transform (FT) and is used in signal analysis and communication theory, as well as in spectroscopy.
Abstract: Preface to the first edition Preface to the second edition 1. Physics and Fourier transforms 2. Useful properties and theorems 3. Applications I: Fraunhofer diffraction 4. Applications II: signal analysis and communication theory 5. Applications III: spectroscopy and spectral line shapes 6. Two-dimensional Fourier transforms 7. Multi-dimensional Fourier transforms 8. The formal complex Fourier transform 9. Discrete and digital Fourier transform Appendix Bibliography.

76 citations


Journal ArticleDOI
TL;DR: The fractional Fourier transform can also be helpful for lens design, especially for specifying a lens cascade, according to its role in wave propagation and signal processing.
Abstract: The fractional Fourier transform has been used in optics so far for wave propagation and for signal processing. Now we show that this new transform can also be helpful for lens design, especially for specifying a lens cascade.

63 citations


Journal ArticleDOI
TL;DR: This work shows that the original bulk-optics configuration for performing the fractional-Fourier-transform operation provides a scaled output using a fixed lens and suggests an asymmetrical setup for obtaining a non-scaled output.
Abstract: Recently two optical interpretations of the fractional Fourier transform operator were introduced. We address implementation issues of the fractional-Fourier-transform operation. We show that the original bulk-optics configuration for performing the fractional-Fourier-transform operation [J. Opt. Soc. Am. A 10, 2181 (1993)] provides a scaled output using a fixed lens. For obtaining a non-scaled output, an asymmetrical setup is suggested and tested. For comparison, computer simulations were performed. A good agreement between computer simulations and experimental results was obtained.

61 citations


Book ChapterDOI
01 Sep 1995
TL;DR: By implementing the FFT algorithm on a custom computing machine (CCM) called Splash-2, a computation speed of 180 Mflops and a speed-up of 23 times over a Sparc-10 workstation is achieved.
Abstract: The two dimensional fast Fourier transform (2-D FFT) is an indispensable operation in many digital signal processing applications but yet is deemed computationally expensive when performed on a conventional general purpose processors This paper presents the implementation and performance figures for the Fourier transform on a FPGA-based custom computer The computation of a 2-D FFT requires O(N2log2N) floating point arithmetic operations for an NxN image By implementing the FFT algorithm on a custom computing machine (CCM) called Splash-2, a computation speed of 180 Mflops and a speed-up of 23 times over a Sparc-10 workstation is achieved

Journal ArticleDOI
TL;DR: Based on the fractional Fourier-transform operation, a new space-frequency chart definition is introduced, and by the application of various geometric operations on this new chart, optical systems may be designed or analyzed.
Abstract: The fractional Fourier transform is a mathematical operation that generalizes the well-known Fourier transform. This operation has been shown to have physical and optical fundamental meanings, and it has been experimentally implemented by relatively simple optical setups. Based on the fractional Fourier-transform operation, a new space-frequency chart definition is introduced. By the application of various geometric operations on this new chart, such as radial and angular shearing and rotation, optical systems may be designed or analyzed. The field distribution, as well as full information about the spectrum and the space–bandwidth product, can be easily obtained in all the stages of the optical system.

Book
01 May 1995
TL;DR: In this paper, Fourier Spectra for Non-Periodic Functions One-Dimensional Fourier series and Spectra in One-Dimension for Functions of Finite Period One Dimensional.
Abstract: Partial table of contents: Some of the How and Why of Fourier Analysis. Fourier Series and Spectra in One--Dimension for Functions of Finite Period. Fourier Series and Spectra for Functions of Infinite Period One Dimensional. Fourier Spectra for Non--Periodic Functions One--Dimensional. The Diffraction of Light and Fourier Transforms in Two Dimensions. A Brief Summary of Linear Systems Theory Applied to Optical Imaging. Fourier Optical Transformations by Computer. Apodization and Super--Resolution, Phase from Shift, and Multiple Apertures. Complex Apertures. Operations in the Fourier Transform Plane. Other Interesting and Related Topics. References. A Selected Bibliography. Index.

Journal ArticleDOI
TL;DR: It is demonstrated that the definition of a fractional-order Fourier transform can be extended into the complexorder regime and the beam width of a Gaussian beam subjected to the complex- order Fouriertransform is obtained analytically with the ABCD matrix approach.
Abstract: It is demonstrated that the definition of a fractional-order Fourier transform can be extended into the complexorder regime. A complex-order Fourier transform deals with the imaginary part as well as the real part of the exponential function in the integral. As a result, while the optical implementation of fractional-order Fourier transform involves gradient-index media or spherical lenses, the optical interpretation of complex-order Fourier transform is practically based on the convolution operation and Gaussian apertures. The beam width of a Gaussian beam subjected to the complex-order Fourier transform is obtained analytically with the ABCD matrix approach.

Journal ArticleDOI
TL;DR: By analysing a windowing signal with Fourier transform, the leakage-induced phase error is investigated, and the phase error distribution is indicated, and a practical approach to correct leakage in a discrete frequency signal to obtain accurate phase information is presented.

Journal ArticleDOI
TL;DR: Fast Fourier beam-propagation methods for simulating the roles of internal refractive effects and external propagation from nonlinear media are introduced in this article, which is applied to model picosecond Z-scan measurements for the induced absorber, the dye Chloro-Aluminum Phthalocyanine, at 532 nm.
Abstract: Fast Fourier beam–propagation methods (BPM’s) for simulating the roles of internal refractive effects and external propagation from nonlinear media are introduced. These techniques are applied to model picosecond Z-scan measurements for the induced absorber, the dye Chloro-Aluminum Phthalocyanine, at 532 nm. Within the thin-sample approximation an incident Gaussian beam is taken to experience a change in phase profile on propagation through the medium but remains of Gaussian amplitude profile. Outside this approximation one must determine both the phase and the amplitude profiles at the sample exit face that are due to the influence of nonlinear refraction (and nonlinear absorption) on the beam propagating through the medium. The BPM technique allows this to be achieved efficiently, and the external propagation technique enables a single discrete fast Fourier transform to be used to describe the subsequent external propagation of the non-Gaussian-shaped beams. The analysis is especially useful for such self-enhancing nonlinearities as one would wish to exploit in optical limiting.


Journal ArticleDOI
Shutian Liu1, Jiandong Xu1, Yan Zhang1, Lixue Chen1, Chunfei Li1 
TL;DR: These architectures are demonstrated to be flexible in practical spatially variant filtering systems that employ cascaded multiple stages of fractional Fourier transforms, because of their capabilities of changing the standard focal length.
Abstract: General optical setups that implement the fractional Fourier transforms are proposed by use of the impulse response theory. These architectures are demonstrated to be flexible in practical spatially variant filtering systems that employ cascaded multiple stages of fractional Fourier transforms, because of their capabilities of changing the standard focal length.


Journal ArticleDOI
TL;DR: A lens system that is able to perform equidistant fractional Fourier transforms that cover the whole range of orders and that consist of a minimum number of modules is introduced.
Abstract: The fractional Fourier transform is a new topic in optics. To make use of the fractional Fourier transform as an experimental tool, I design a fractional Fourier transformer of variable order: I introduce a lens system that is able to perform equidistant fractional Fourier transforms that cover the whole range of orders and that consist of a minimum number of modules. By module, I mean an elementary fractional Fourier transform of certain order that consists of a lens between two free-space lengths. Because of the commutative additivity of the transform, various fractional orders can be achieved by means of different constellations of the modules. It is possible to perform a large variety of fractional Fourier transforms with a small number of modules.

Journal ArticleDOI
TL;DR: In this paper, it was shown that information of order six suffices to determine a structure uniquely, where six is the number of atoms in a set of equal atoms that can occupy positions on a circle subdivided into equally spaced markings.
Abstract: The three-dimensional configuration of crystallized structures is obtained by reading off partial information about the Fourier transform of such structures from diffraction data obtained with an X-ray source. We consider a discrete version of this problem and discuss the extent to which `intensity only' measurements as well as `higher-order invariants' can be used to settle the reconstruction problem. This discrete version is an extension of the study undertaken by Patterson in terms of `cyclotomic sets', corresponding to arrangements of equal atoms that can occupy positions on a circle subdivided into N equally spaced markings. This model comes about when the usual three-dimensional Fourier transform is replaced by a one-dimensional discrete Fourier transform. The model in this paper considers molecules made up of atoms with possibly different (integer-valued) atomic numbers. It is shown that information of order six suffices to determine a structure uniquely.

Proceedings ArticleDOI
09 May 1995
TL;DR: A discrete version of the AFT (DRFT) that represents a rotation in discrete time-frequency space and some properties of the transform that support its interpretation as a rotation are presented.
Abstract: The continuous-time angular Fourier transformation (AFT) represents a rotation in continuous time-frequency space and also serves as an orthonormal signal representation for chirp signals. We present a discrete version of the AFT (DRFT) that represents a rotation in discrete time-frequency space and some properties of the transform that support its interpretation as a rotation. The transform is a generalization of the DFT. The eigenvalue structure of the DFT is then exploited to develop an efficient algorithm for the computation of this transform.

Patent
Masanobu Miyashita1, Kenji Okajima1
24 Feb 1995
TL;DR: In this paper, the images of an object equivalent to those obtained when as object is viewed respectively by the right and left eyes are subdivided into small areas with overlaps allowed therebetween and are Fourier transformed to achieve a local Fourier transform for the images.
Abstract: The images of an object equivalent to those obtained when as object is viewed respectively by the right and left eyes are subdivided into small areas with overlaps allowed therebetween and are Fourier transformed to achieve a local Fourier transform for the images. Phase difference of Fourier transforms is obtained according to the right and left local Fourier transform images of a reference image which is stored beforehand in a storage memory. Alternatively, a Fourier transform power spectrum is obtained. Disparity is represented in terms of a phase difference or in the form of a Fourier transform power spectral pattern. According to either one of the above results, and on the basis of a local Fourier transform of an input image including the reference image, an identical object image is estimated. For the image, a local inverse Fourier transform is effected such that geometric mean values are attained between the right and left images so as to segment an image matching the reference image and, also, with respect to disparity.

Journal ArticleDOI
TL;DR: The fractional Fourier transform is redefined for working with incoherent light and overcomes coherent system disadvantages such as the speckle effect and the need for incoherent-coherent conversion.
Abstract: The fractional Fourier transform is redefined for working with incoherent light. As a real transformation, the incoherent fractional Fourier transform overcomes coherent system disadvantages such as the speckle effect and the need for incoherent–coherent conversion. It also might have some applications for digital image and signal processing owing to its decreased computing complexity. An incoherent optical implementation of the new transform based on the shearing interferometer is suggested. Laboratory experimental results are given.

Journal ArticleDOI
TL;DR: A method for invariant pattern recognition of range images by means of the phase Fourier transform is introduced and an invariant representation under changes of position, scale, and orientation for the characteristic normals is defined.
Abstract: A method for invariant pattern recognition of range images by means of the phase Fourier transform is introduced. The phase Fourier transform may be used for the segmentation of connected planar and quadric surfaces. The method is generalized to nonconnected planar surfaces through the use of the concept of the characteristic normal. An invariant representation under changes of position, scale, and orientation for the characteristic normals is defined. This representation is used as the input for a feedforward neural network. Examples of applications are given, and finally the method is applied to the problems of classification and occlusion.

Journal ArticleDOI
Eugene Sorets1
TL;DR: The algorithm is based on the Lagrange interpolation formula and the Green's theorem, which are used to preprocess the data before applying the fast Fourier transform, and readily generalizes to higher dimensions and to piecewise smooth functions.

01 Nov 1995
TL;DR: The basic terminology and the main concepts of the area are introduced, as well as several application domains, providing common ground for further discussion and study.
Abstract: The Fourier transform is among the most widely used tools for transforming data sequences and functions from what is referred to as the {\it time domain} to the {\it frequency domain}. Applications of the transform range from designing filters for noise reduction in audio-signals (such as music or speech), to fast multiplication of polynomials. This report is meant to serve as a brief introduction to the Fourier transform, for readers who are not familiar with frequency domain. It introduces the basic terminology and the main concepts of the area, as well as several application domains, providing common ground for further discussion and study.

Journal ArticleDOI
TL;DR: The result is able to define a class of discrete tranformations which can be considered as a generalization of the discrete Fourier transform.
Abstract: In [2] we have considered a class of integral transforms which generalizs the classical Fourier tranform. We are able now to define a class of discrete tranformations which can be considered as a generalization of the discrete Fourier transform. Furthermore, when our result is considered in connnection with the particular kernel associated with th Fourier transform, the fast Fourier transform algorithm can be used in order to approximate the Hermite-Fourier coefficients of a class of functions *

Journal ArticleDOI
TL;DR: A simple but necessary condition for cascading the multiple stages of an optical fractional Fourier transform with different variable scales is presented and that condition permits more flexibility in the design of optical information-processing systems that use optical fractionals Fourier transforms.
Abstract: A simple but necessary condition for cascading the multiple stages of an optical fractional Fourier transform with different variable scales is presented That condition permits more flexibility in the design of optical information-processing systems that use optical fractional Fourier transforms We have expanded the optical applications of fractional Fourier transforms with adjustable variable scales

Proceedings ArticleDOI
Leon Cohen1
07 Jun 1995
TL;DR: In this paper, the authors show that there are several uncertainty principles for the short-time Fourier transform and spectrogram, and they generalize the concept of the short time Fourier transformation to arbitrary variables.
Abstract: We show that there are a number of uncertainty principles for the short-time Fourier transform and spectrogram. Explicit expressions are derived and examples given. Also, we generalize the concept of the short-time Fourier transform to arbitrary variables.