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Showing papers on "Discrete-time Fourier transform published in 2007"


Journal ArticleDOI
TL;DR: Hypercomplex numbers, specifically quaternions, are used to define a Fourier transform applicable to color images, and the properties of the transform are developed, and it is shown that the transform may be computed using two standard complex fast Fourier transforms.
Abstract: Fourier transforms are a fundamental tool in signal and image processing, yet, until recently, there was no definition of a Fourier transform applicable to color images in a holistic manner. In this paper, hypercomplex numbers, specifically quaternions, are used to define a Fourier transform applicable to color images. The properties of the transform are developed, and it is shown that the transform may be computed using two standard complex fast Fourier transforms. The resulting spectrum is explained in terms of familiar phase and modulus concepts, and a new concept of hypercomplex axis. A method for visualizing the spectrum using color graphics is also presented. Finally, a convolution operational formula in the spectral domain is discussed

535 citations


Journal ArticleDOI
TL;DR: A novel random fractional Fourier transform is proposed by randomizing the transform kernel function of the conventional fractional fourier transform, which can be directly used in optical image encryption and decryption.
Abstract: We propose a novel random fractional Fourier transform by randomizing the transform kernel function of the conventional fractional Fourier transform. The random fractional Fourier transform inherits the excellent mathematical properties from the fractional Fourier transform and can be easily implemented in optics. As a primary application the random fractional Fourier transform can be directly used in optical image encryption and decryption. The double phase encoding image encryption schemes can thus be modeled with cascaded random fractional Fourier transformers.

126 citations


Journal ArticleDOI
TL;DR: The sampling theorem for OLCT signals presented here serves as a unification and generalization of previously developed sampling theorems.
Abstract: The offset linear canonical transform (OLCT) is the name of a parameterized continuum of transforms which include, as particular cases, the most widely used linear transforms in engineering such as the Fourier transform (FT), fractional Fourier transform (FRFT), Fresnel transform (FRST), frequency modulation, time shifting, time scaling, chirping and others. Therefore the OLCT provides a unified framework for studying the behavior of many practical transforms and system responses. In this paper the sampling theorem for OLCT is considered. The sampling theorem for OLCT signals presented here serves as a unification and generalization of previously developed sampling theorems.

100 citations


Journal ArticleDOI
TL;DR: Empirically evaluate a recently proposed Fast Approximate Discrete Fourier Transform (FADFT) algorithm, FADFT-2, for the first time and it is shown that FAD FT-2 not only generally outperforms F ADFT-1 on all but the sparsest signals, but is also significantly faster than FFTW 3.1 on large sparse signals.
Abstract: In this paper we empirically evaluate a recently proposed Fast Approximate Discrete Fourier Transform (FADFT) algorithm, FADFT-2, for the first time. FADFT-2 returns approximate Fourier representations for frequency-sparse signals and works by random sampling. Its implemen- tation is benchmarked against two competing methods. The first is the popular exact FFT imple- mentation FFTW Version 3.1. The second is an implementation of FADFT-2’s ancestor, FADFT-1. Experiments verify the theoretical runtimes of both FADFT-1 and FADFT-2. In doing so it is shown that FADFT-2 not only generally outperforms FADFT-1 on all but the sparsest signals, but is also significantly faster than FFTW 3.1 on large sparse signals. Furthermore, it is demonstrated that FADFT-2 is indistinguishable from FADFT-1 in terms of noise tolerance despite FADFT-2’s better execution time.

61 citations


Journal ArticleDOI
TL;DR: In this paper, a large number of functions differing from each other only by a translation parameter are observed, and the shift parameters are estimated using the Fourier transform, which enables to transform this statistical problem into a semi-parametric framework.
Abstract: We observe a large number of functions differing from each other only by a translation parameter. While the main pattern is unknown, we propose to estimate the shift parameters using $M$-estimators. Fourier transform enables to transform this statistical problem into a semi-parametric framework. We study the convergence of the estimator and provide its asymptotic behavior. Moreover, we use the method in the applied case of velocity curve forecasting.

59 citations


Journal ArticleDOI
TL;DR: The numerical simulation and experiment have proved the validity of the multiscale windowed Fourier transform for phase extraction of fringe patterns and makes the extracted phase more precise than other methods.
Abstract: A multiscale windowed Fourier transform for phase extraction of fringe patterns is presented. A local stationary length of signal is used to control the window width of a windowed Fourier transform automatically, which is measured by an instantaneous frequency gradient. The instantaneous frequency of the fringe pattern is obtained by detecting the ridge of the wavelet transform. The numerical simulation and experiment have proved the validity of this method. The combination of the windowed Fourier transform and the wavelet transform makes the extracted phase more precise than other methods.

52 citations


Journal ArticleDOI
TL;DR: This tutorial simply reviews the DFT and FFT, with a few characteristic examples.
Abstract: Frequency analysis is an important issue in the IEEE. Using a computer in a calculation means moving into a non-physical, synthetic environment. Numerically, discrete or fast Fourier transformations (DFTs or FFTs) are used to obtain the frequency content of a time signal, and these are totally different than the mathematical definition of the Fourier transform. This tutorial simply reviews the DFT and FFT, with a few characteristic examples.

44 citations


Journal ArticleDOI
TL;DR: A method to randomize the Fourier transform, which can be applied in the field of image encryption and decryption because of the ambiguity of the eigenvalues.
Abstract: We have investigated the multiplicity and complexity in eigenvalues of the fractional Fourier transform and found that the ambiguity of the eigenvalues may indicate randomness. We have therefore proposed a method to randomize the Fourier transform. Such a random Fourier transform can be applied in the field of image encryption and decryption.

42 citations


Journal ArticleDOI
TL;DR: This work compares the fidelity in reproducing the classical harmonic motion of discrete coherent states of the N x N Fourier matrix with several options considered in the literature.
Abstract: The N×N Fourier matrix is one distinguished element within the group U(N) of all N×N unitary matrices. It has the geometric property of being a fourth root of unity and is close to the dynamics of harmonic oscillators. The dynamical correspondence is exact only in the N→∞ contraction limit for the integral Fourier transform and its fractional powers. In the finite-N case, several options have been considered in the literature. We compare their fidelity in reproducing the classical harmonic motion of discrete coherent states.

37 citations


Journal ArticleDOI
TL;DR: A relation for the circular convolution operation in the discrete sine and cosine transform domains is derived and this method is an alternative to the discrete Fourier transform method for filtering applications.
Abstract: In this paper, we derive a relation for the circular convolution operation in the discrete sine and cosine transform domains. The transform coefficients are either symmetric or asymmetric and hence we need to calculate only half of the total coefficients. Since fast algorithms are available for the computation of discrete sine and cosine transforms, the proposed method is an alternative to the discrete Fourier transform method for filtering applications.

33 citations


Journal ArticleDOI
TL;DR: In this article, a generalized convolution theorem in the fractional Fourier domain was proposed and generalized sampling expansion was shown to be a special case of the generalized Papoulis sampling expansion, and its application in the context of image superresolution was discussed.

Proceedings ArticleDOI
01 Jul 2007
TL;DR: Two composite algorithms are proposed that build upon the existing ones based on recent advances in polynomial factoring for computing the unwrapped phase of the discrete-time Fourier transform of a one-dimensional finite-length signal.
Abstract: In this paper, the computation of the unwrapped phase of the discrete-time Fourier transform (DTFT) of a one-dimensional finite-length signal is explored. The phase of the DTFT is not unique, and may contain integer multiple of 2 pi discontinuities. The unwrapped phase is the instance of the phase function chosen to ensure continuity. This paper compares existing algorithms for computing the unwrapped phase. Then, two composite algorithms are proposed that build upon the existing ones. The core of the proposed methods is based on recent advances in polynomial factoring. The proposed methods are implemented and compared to the existing ones.

Journal ArticleDOI
TL;DR: The framework for signal processing on a spatial, or undirected, 2-D hexagonal lattice for both an infinite and a finite array of signal samples is developed, which includes the proper notions of z-transform, boundary conditions, filtering or convolution, spectrum, frequency response, and Fourier transform.
Abstract: We develop the framework for signal processing on a spatial, or undirected, 2-D hexagonal lattice for both an infinite and a finite array of signal samples. This framework includes the proper notions of z-transform, boundary conditions, filtering or convolution, spectrum, frequency response, and Fourier transform. In the finite case, the Fourier transform is called discrete triangle transform. Like the hexagonal lattice, this transform is nonseparable. The derivation of the framework makes it a natural extension of the algebraic signal processing theory that we recently introduced. Namely, we construct the proper signal models, given by polynomial algebras, bottom-up from a suitable definition of hexagonal space shifts using a procedure provided by the algebraic theory. These signal models, in turn, then provide all the basic signal processing concepts. The framework developed in this paper is related to Mersereau's early work on hexagonal lattices in the same way as the discrete cosine and sine transforms are related to the discrete Fourier transform-a fact that will be made rigorous in this paper

Journal ArticleDOI
TL;DR: The implicitly restarted generalized minimum residual method (IRGMRES) combined with the fast Fourier transform (FFT) technique is developed for solving three-dimensional weak-form volume electric field integral equation of electromagnetic scattering problems.
Abstract: The implicitly restarted generalized minimum residual method (IRGMRES) combined with the fast Fourier transform (FFT) technique is developed for solving three-dimensional (3-D) weak-form volume electric field integral equation of electromagnetic scattering problems. On several electromagnetic scattering problems, the resulted IRGMRES-FFT method converges two-three times faster than the conventional biconjugate gradient (BCG)-FFT method. Comparison with other Krylov-subspace iterative fast Fourier transforms methods also demonstrates the efficiency of the IRGMRES-FFT method.

Proceedings ArticleDOI
16 Apr 2007
TL;DR: This paper will summarize the Discrete Fourier Transform method and explain a further enhancement, variable window length, that may become more popular as processor capabilities increase.
Abstract: As the speed and memory capabilities of microprocessors have increased, it has become more popular for the signal conditioning of knock sensor outputs to be performed entirely within the microprocessor. One method of this signal conditioning process utilizes the Discrete Fourier Transform (DFT). It is common for systems that use this method to limit the knock detection window to one length across all RPM and load points to reduce computation and memory constraints on the processor. This paper will summarize this method and explain a further enhancement, variable window length, that may become more popular as processor capabilities increase.

Patent
Marilynn Green1
25 Oct 2007
TL;DR: In this paper, an example embodiment of an apparatus for use in a wireless transmitter may include a continuous phase modulation (CPM) sample generator configured to generate a group of constant envelope CPM modulated signal samples, a Fourier transform block configured to perform an inverse Fourier transformation on the expanded group of Fourier coefficients to map the constant envelope time-domain samples onto a group OFC subcarriers for transmission.
Abstract: Various example embodiments are disclosed herein. According to an example embodiment, an apparatus for use in a wireless transmitter may include a continuous phase modulation (CPM) sample generator configured to generate a group of constant envelope CPM modulated signal samples, a Fourier transform block configured to perform a Fourier transform on the group of constant envelope signal samples to generate an initial group of Fourier coefficients, a zero insertion block configured to generate an expanded group of Fourier coefficients by inserting one or more zeros in the initial group of Fourier coefficients, and an inverse Fourier transform block configured to perform an inverse Fourier transform on the expanded group of Fourier coefficients to generate a group of constant envelope time-domain samples and to map the constant envelope time-domain samples onto a group of orthogonal subcarriers for transmission.

Journal ArticleDOI
TL;DR: An alternative methodology to tune the transition bandwidth of window-based FIR filters, based on FRFT, is developed and introduces a comparative ease in tuning by eliminating the need to re-compute the impulse response coefficients.

Journal ArticleDOI
TL;DR: In this paper, a numerical method for processing the fringes obtained when two waves, with a quadratic phase difference function, interfere is presented, which includes straight equispaced fringes and Newton's rings as particular cases.
Abstract: We present a practical numerical method for processing the fringes obtained when two waves, with a quadratic phase difference function, interfere. This kind of fringe includes straight equispaced fringes and Newton's rings as particular cases. The numerical method we present is based on the discrete Fresnel (Fourier) transform of the data and has the same precision as least square fitting (LSF). Compared to the LSF method, this new method is better, as it is more efficient and does not require initial approximations for the fringe parameters to be determined.

Journal ArticleDOI
TL;DR: A mathematically rigorous but simple treatment of the procedure that can be reduced to the computation of the standard 1-dimensional Discrete Fourier Transform on the hexagonal lattice is provided.
Abstract: The computation of the Discrete Fourier Transform for a general lattice in ? d can be reduced to the computation of the standard 1-dimensional Discrete Fourier Transform. We provide a mathematically rigorous but simple treatment of this procedure and apply it to the DFT on the hexagonal lattice.

Patent
22 Jun 2007
TL;DR: In this paper, the Fourier transform of the pixel values forms a spectrum, which is then analyzed to find a peak and analyze the peak to determine whether the peak is indicative of the presence of a pattern.
Abstract: A method for detecting a pattern in an image includes defining a set of pixel values in an image using a window and calculating a Fourier transform of the pixel values. In one embodiment, the Fourier transform of the pixel values forms a spectrum. The method further comprises analyzing the spectrum of the Fourier transform to find a peak and analyzing the peak to determine whether the peak is indicative of the presence of a pattern in the image.

Journal ArticleDOI
TL;DR: In this article, the authors characterize the support of the Fourier transform of the band-limited scaling function and give an approach to the construction of the scaling function based on the relations of translates of three point sets.

Proceedings ArticleDOI
23 Jul 2007
TL;DR: This talk presents the mathematical derivation for a closed- form expression of the azimuth-fractional Fourier transform of the new FrCSA with application to high resolution imaging.
Abstract: The fractional Fourier transform (FrFT), which is a generalized form of the well-known Fourier transform, has opened up the possibility of a new range of potentially promising and useful applications including radar involving the use and detection of chirp signals, pattern recognition and Synthetic Aperture Radar (SAR) image processing. The Chirp Scaling Algorithm (CSA) is one of the most important and well-known radar imaging algorithms. It is attractive because of its excellent focusing ability and implementation simplicity. Benefiting from the inherent structure of the FrFT for non- stationary digital signal processing and analysis, especially for chirped-type signals, a new version of the CSA based on the Fractional Fourier Transform (FrFT) is developed. The introduced Fractional Chirp Scaling Algorithm (FrCSA) applied the Fast Fourier Transform (FFT) instead of the fractional Fourier transform (FrFT) in the azimuth direction for the analytical development tractability purposes only as it numerically tractable in both dimensions. To demonstrate the resolution and focusing enhancement in the azimuth dimension using the FrCSA and also to be able to perform azimuth fractional filtering, noise removal and flight path nonlinearity compensation, a closed form expression for the azimuth fractional transformation is required. In this talk we present the mathematical derivation for a closed- form expression of the azimuth-fractional Fourier transform of the new FrCSA with application to high resolution imaging. Results to real SAR data images will show significantly enhanced features using the FrFT-based azimuth expression instead of the classical FFT-based one within the fractional chirp scaling algorithm or any other chirped-type SAR imaging algorithm.

Journal ArticleDOI
TL;DR: In this article, an example of two continuous maps f and g of the circle to itself with the same absolute value of the Fourier transform but with different winding numbers, answering a question of Brezis is given.
Abstract: We construct an example of two continuous maps f and g of the circle to itself with the same absolute value of the Fourier transform but with different winding numbers, answering a question of Brezis.

01 Jun 2007
TL;DR: In this paper, a separable representation consisting of a Fourier Bessel series expansion for the spectral components and a conventional Fourier series expansion of the spatial dependence is proposed to reproduce the Head Related Transfer Function (HRTF) in the horizontal auditory scene.
Abstract: This paper proposes a method to reproduce the Head Related Transfer Function (HRTF) in the horizontal auditory scene The method is based on a separable representation which consists of a Fourier Bessel series expansion for the spectral components and a conventional Fourier series expansion for the spatial dependence The proposed representation can be used to predict HRTFs at any azimuth position and at any frequency sampling point from a finite number of measurements Implementation details are demonstrated in the paper Measured HRTFs from a KEMAR manikin and analytically simulated HRTFs were used to validate the fidelity and predictive capabilities of the method The average mean square error for model reconstruction is less than two percent

Journal Article
TL;DR: In this paper, a new system of multi-channel single-output joint fractional Fourier transform correlator (JFRTC) for color pattern recognition is proposed based on the conventional system of multichannel single output joint transform correlators (JTC).
Abstract: A new system of multi-channel single-output joint fractional Fourier transform correlator (JFRTC) for color pattern recognition is proposed based on the conventional system of multi-channel single-output joint transform correlator (JTC). The theoretical analysis and optical experiments are performed. With this method, one can obtain three correlation peaks at the output plane which show a pair of desired cross-correlation peaks and one auto-correlation peak. In comparison, the conventional system leads to more correlation peaks playing a noise role in color pattern recognition.

Journal ArticleDOI
TL;DR: In this article, a one-to-one correspondence between Fourier transforms of ultradistribution semigroups in the sense of Beurling and some class of pseudoresolvents characterized by conditions concerning their domains of existence and growth is established.

Proceedings ArticleDOI
16 Sep 2007
TL;DR: A computationally efficient method to extract 3D planar surface orientation from the spectral variations of a visual texture by the novel method of identifying ridges of its Fourier transform is proposed.
Abstract: Shape from texture has received much attention in the past few decades. We propose a computationally efficient method to extract 3D planar surface orientation from the spectral variations of a visual texture. Under the assumption of homogeneity, the texture is represented by the novel method of identifying ridges of its Fourier transform. Local spatial frequencies are then computed using a minimal set of selected Gabor filters. Under perspective projection, frequencies are backprojected and orientation is computed so as to minimize the variance of the frequencies’ backprojections. A comparative study with two existing methods, and experimentation on simulated and real texture images is given.

01 Jan 2007
TL;DR: Denotti et al. as discussed by the authors proposed a Fourier analysis of functions of boolean functions and showed that the Fourier representation of functions can be used to evaluate function complexity and function complexity.
Abstract: OntheFourierAnalysisofBo oleanFunctionsA.BernasconiyB.Co denottiJ.SimonzIMCB4-97-03AbstractWestudytheFourierrepresentationofBo oleanfunctions.Thegoalistolo okatfrequencydomainofBo oleanfunctionstogetcomplexityprop erties.Preliminaryresultsindicatethatthismightb efruitful.Inadditiontopresentingnewresults,wereviewsomeofthemostsigni cantorkonsub ject.IstitutodiMatematicaComputazionale,ConsiglioNazionaledelleRicerche,andDipartimentoIn-formatica,Pisa(Italy).yIstitutodiMatematicaComputazionale,ConsiglioNazionaledelleRicerche,Pisa(Italy).e-mail:co denotti@iei.pi.cnr.itzDepartmentofComputerScience,TheUniversityChicago.PortionsthisworkeredonewhilevisitingIEI-CNRinPisa,sp onsoredbyagrantfromCNR.

Journal ArticleDOI
TL;DR: The windowed Fourier transform with a SAWS is theoretically better than that with a SFWS but it is also more challenging in use.

Proceedings ArticleDOI
27 May 2007
TL;DR: An image sensor with nonuniform pixel placement that enables a highly efficient computation of the discrete cosine transform, which is the most computationally demanding step of the image compression algorithm.
Abstract: In this paper we describe an image sensor with nonuniform pixel placement that enables a highly efficient computation of the discrete cosine transform, which is the most computationally demanding step of the image compression algorithm. This technique is based on the arithmetic Fourier transform (AFT), which we show to be 5 times more computationally efficient than currently used DCT computation methods. The architecture and circuits described can be implemented in conventional CMOS processes.