Showing papers on "Non-uniform discrete Fourier transform published in 1999"
TL;DR: The proposed DFRFT has DFT Hermite eigenvectors and retains the eigenvalue-eigenfunction relation as a continous FRFT and will provide similar transform and rotational properties as those of continuous fractional Fourier transforms.
Abstract: The continuous fractional Fourier transform (FRFT) performs a spectrum rotation of signal in the time-frequency plane, and it becomes an important tool for time-varying signal analysis. A discrete fractional Fourier transform has been developed by Santhanam and McClellan (see ibid., vol.42, p.994-98, 1996) but its results do not match those of the corresponding continuous fractional Fourier transforms. We propose a new discrete fractional Fourier transform (DFRFT). The new DFRFT has DFT Hermite eigenvectors and retains the eigenvalue-eigenfunction relation as a continous FRFT. To obtain DFT Hermite eigenvectors, two orthogonal projection methods are introduced. Thus, the new DFRFT will provide similar transform and rotational properties as those of continuous fractional Fourier transforms. Moreover, the relationship between FRFT and the proposed DFRFT has been established in the same way as the conventional DFT-to-continuous-Fourier transform.
291 citations
TL;DR: The windowed FFT is a time windowed version of the discrete time Fourier transform that may be adjusted and shifted to scan through large volumes of power quality data.
Abstract: This paper discusses the application of the windowed fast Fourier transform to electric power quality assessment. The windowed FFT is a time windowed version of the discrete time Fourier transform. The window width may be adjusted and shifted to scan through large volumes of power quality data. Narrow window widths are used for detailed analyses, and wide window widths are used to move rapidly across archived power quality data measurements. The mathematics of the method are discussed and applications are illustrated.
272 citations
15 Mar 1999
TL;DR: This definition is based on a particular set of eigenvectors of the DFT which constitutes the discrete counterpart of the set of Hermite-Gaussian functions and supports confidence that it will be accepted as the definitive definition of this transform.
Abstract: We propose and consolidate a definition of the discrete fractional Fourier transform which generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform (FRT) generalizes the continuous ordinary Fourier Transform. This definition is based on a particular set of eigenvectors of the DFT which constitutes the discrete counterpart of the set of Hermite-Gaussian functions. The fact that this definition satisfies all the desirable properties expected of the discrete FRT, supports our confidence that it will be accepted as the definitive definition of this transform.
210 citations
TL;DR: This work applies the language of the unified FT to develop FRT expressions for discrete and continuous signals, introducing a particular form of periodicity: chirp-periodicity.
Abstract: The fractional Fourier transform (FRT) is an extension of the ordinary Fourier transform (FT). Applying the language of the unified FT, we develop FRT expressions for discrete and continuous signals, introducing a particular form of periodicity: chirp-periodicity. The FRT sampling theorem is derived as an extension of its ordinary counterpart.
152 citations
TL;DR: The fractional Fourier transform (FFT) as discussed by the authors is a generalization of the ordinary FFT with an order parameter a, and it is used to interpolate between a function f(u) and its FFT F(μ).
Abstract: Publisher Summary This chapter is an introduction to the fractional Fourier transform and its applications. The fractional Fourier transform is a generalization of the ordinary Fourier transform with an order parameter a . Mathematically, the a th order fractional Fourier transform is the a th power of the Fourier transform operator. The a = 1st order fractional transform is the ordinary Fourier transform. In essence, the a th order fractional Fourier transform interpolates between a function f(u) and its Fourier transform F(μ) . The 0th order transform is simply the function itself, whereas the 1st order transform is its Fourier transform. The 0.5th transform is something in between, such that the same operation that takes us from the original function to its 0.5 th transform will take us from its 0.5th transform to its ordinary Fourier transform. More generally, index additivity is satisfied: The a 2 th transform of the a 1 th transform is equal to the ( a 2 + a 1 )th transform. The –1th transform is the inverse Fourier transform, and the – a th transform is the inverse of the a th transform.
151 citations
TL;DR: An algorithm to reconstruct a high- resolution image from multiple aliased low-resolution images, which is based on the generalized deconvolution technique, and it is shown that the artifact caused by inaccurate motion information is reduced by regular- ization.
Abstract: While high-resolution images are required for various applica- tions, aliased low-resolution images are only available due to the physi- cal limitations of sensors. We propose an algorithm to reconstruct a high- resolution image from multiple aliased low-resolution images, which is based on the generalized deconvolution technique. The conventional approaches are based on the discrete Fourier transform (DFT) since the aliasing effect is easily analyzed in the frequency domain. However, the useful solution may not be available in many cases, i.e., the underdeter- mined cases or the insufficient subpixel information cases. To compen- sate for such ill-posedness, the generalized regularization is adopted in the spatial domain. Furthermore, the usage of the discrete cosine trans- form (DCT) instead of the DFT leads to a computationally efficient recon- struction algorithm. The validity of the proposed algorithm is both theo- retically and experimentally demonstrated. It is also shown that the artifact caused by inaccurate motion information is reduced by regular- ization. © 1999 Society of Photo-Optical Instrumentation Engineers. (S0091-3286(99)00508-5)
142 citations
TL;DR: The goal of this article is to develop two absent schemes of fractional Fourier analysis methods that are the generalizations of Fourier series and discrete-time Fourier transform (DTFT), respectively.
Abstract: Conventional Fourier analysis has many schemes for different types of signals. They are Fourier transform (FT), Fourier series (FS), discrete-time Fourier transform (DTFT), and discrete Fourier transform (DFT). The goal of this article is to develop two absent schemes of fractional Fourier analysis methods. The proposed methods are fractional Fourier series (FRFS) and discrete-time fractional Fourier transform (DTFRFT), and they are the generalizations of Fourier series (FS) and discrete-time Fourier transform (DTFT), respectively.
128 citations
TL;DR: Two new sampling formulae for reconstructing signals that are band limited or time limited in the fractional Fourier transform sense are obtained, each taken at half the Nyquist rate.
70 citations
24 Oct 1999
TL;DR: Simulation results on motion estimation using the DCT/DFT for motion modeling are presented and results are comparable to the results from a wavelet-based approach.
Abstract: This paper presents the concept of using a motion transform for finding the motion field between two images. A motion transform is a representation for modeling the motion field in the transform domain. Compared to other parametric motion models, e.g., affine, projective, etc., a motion transform offers a considerable advantage by its capability to model any motion field, including one with motion discontinuities. It also offers the flexibility of dynamically choosing the significant time-frequency components used to model the underlying motion. Simulation results on motion estimation using the DCT/DFT for motion modeling are presented. These results are comparable to the results from a wavelet-based approach.
53 citations
TL;DR: The resulting generalized signal flow graphs for the computation of different versions of the GDFT represent simple and compact unified approach to the fast discrete sinusoidal transforms computation.
52 citations
TL;DR: A more efficient algorithm is presented, based on the properties of the Radon transform and the two-dimensional (2-D) fast Fourier transform, which can sacrifice little performance for significant computational savings.
Abstract: In this work, we describe a frequency domain technique for the estimation of multiple superimposed motions in an image sequence. The least-squares optimum approach involves the computation of the three-dimensional (3-D) Fourier transform of the sequence, followed by the detection of one or more planes in this domain with high energy concentration. We present a more efficient algorithm, based on the properties of the Radon transform and the two-dimensional (2-D) fast Fourier transform, which can sacrifice little performance for significant computational savings. We accomplish the motion detection and estimation by designing appropriate matched filters. The performance is demonstrated on two image sequences.
TL;DR: The introduction of this new virtual instrument for time-frequency analysis may be of help to the scientists and practitioners in signal analysis.
Abstract: A virtual instrument for time-frequency analysis is presented. Its realization is based on an order recursive approach to the time-frequency signal analysis. Starting from the short time Fourier transform and using the S-method, a distribution having the auto-terms concentrated as high as in the Wigner distribution, without cross-terms, may be obtained. The same relation is used in a recursive manner to produce higher order time-frequency representations without cross-terms. Thus, the introduction of this new virtual instrument for time-frequency analysis may be of help to the scientists and practitioners in signal analysis. Application of the instrument is demonstrated on several simulated and real data examples.
01 May 1999
TL;DR: It is shown that the distribution sampled after a Fourier transform over Zp can be efficiently approximated by transforming over Z, for any q in a large range, which places no restrictions on the superposition to be transformed.
Abstract: We isolate and generalize a technique implicit in many quantum algorithms, including Shor’s algorithms for factoring and discrete log. In particular, we show that the distribution sampled after a Fourier transform over Zp can be efficiently approximated by transforming over Z, for any q in a large range. Our result places no restrictions on the superposition to be transformed, generalizing previous applications. In addition, our proof easily generalizes to multi-dimensional transforms for any constant number of dimensions.
TL;DR: An efficient technique is presented to combine two distinct types of semisystolic arrays into one truly systolic array to perform the multidimensional discrete Fourier transform (DFT).
Abstract: This paper presents an efficient technique for using a multidimensional systolic array to perform the multidimensional discrete Fourier transform (DFT). Extensions of the multidimensional systolic array suitable for fast Fourier transform (FFT) computations such as the prime-factor computation or the 2/sup n/-point decomposed computation of the one-dimensional (1-D) discrete Fourier transform are also presented. The essence of our technique is to combine two distinct types of semisystolic arrays into one truly systolic array. The resulting systolic array accepts streams of input data (i.e., it does not require any preloading), and it produces output data streams at the boundary of the array. No networks for intermediate spectrum transposition between constituent transforms are required. The systolic array has regular processing elements that contain a complex multiplier accumulator and a few registers and multiplexers. Simple and regular connections are required between the PEs.
TL;DR: In this paper, an eigendecomposition of the discrete Fourier transform (DFT) matrix is derived by sampling the Hermite Gauss functions, which are eigenfunctions of the continuous Fourier Transform and by performing a novel error-removal procedure.
Abstract: This paper is concerned with the definition of the discrete fractional Fourier transform (DFRFT). First, an eigendecomposition of the discrete Fourier transform (DFT) matrix is derived by sampling the Hermite Gauss functions, which are eigenfunctions of the continuous Fourier transform and by performing a novel error-removal procedure. Then, the result of the eigendecomposition of the DFT matrix is used to define a new DFRFT. Finally, several numerical examples are illustrated to demonstrate that the proposed DFRFT is a better approximation to the continuous fractional Fourier transform than the conventional defined DFRFT.
15 Mar 1999
TL;DR: A new uncertainty measure, H/sub p/, is used that predicts the compactness of digital signal representations to determine a good (non-orthogonal) set of basis vectors and indicates that a mixture of sinusoidal and impulsive or "blocky" basis functions may be best for compactly representing signals.
Abstract: We use a new uncertainty measure, H/sub p/, that predicts the compactness of digital signal representations to determine a good (non-orthogonal) set of basis vectors. The measure uses the entropy of the signal and its Fourier transform in a manner that is similar to the use of the signal and its Fourier transform in the Heisenberg uncertainty principle. The measure explains why the level of discretization of continuous basis signals can be very important to the compactness of representation. Our use of the measure indicates that a mixture of (non-orthogonal) sinusoidal and impulsive or "blocky" basis functions may be best for compactly representing signals.
TL;DR: In this article, the authors evaluate the performance and statistical accuracy of the fast Fourier transform method for unconditional and conditional simulation, applied under difficult but realistic circumstances of a large field (1001 by 1001 points) with abundant conditioning criteria and a band limited, anisotropic, fractal-based statistical characterization.
Abstract: We evaluate the performance and statistical accuracy of the fast Fourier transform method for unconditional and conditional simulation. The method is applied under difficult but realistic circumstances of a large field (1001 by 1001 points) with abundant conditioning criteria and a band limited, anisotropic, fractal-based statistical characterization (the von Karman model). The simple Fourier unconditional simulation is conducted by Fourier transform of the amplitude spectrum model, sampled on a discrete grid, multiplied by a random phase spectrum. Although computationally efficient, this method failed to adequately match the intended statistical model at small scales because of sinc-function convolution. Attempts to alleviate this problem through the “covariance” method (computing the amplitude spectrum by taking the square root of the discrete Fourier transform of the covariance function) created artifacts and spurious high wavenumber content. A modified Fourier method, consisting of pre-aliasing the wavenumber spectrum, satisfactorily remedies sinc smoothing. Conditional simulations using Fourier-based methods require several processing stages, including a smooth interpolation of the differential between conditioning data and an unconditional simulation. Although kriging is the ideal method for this step, it can take prohibitively long where the number of conditions is large. Here we develop a fast, approximate kriging methodology, consisting of coarse kriging followed by faster methods of interpolation. Though less accurate than full kriging, this fast kriging does not produce visually evident artifacts or adversely affect the a posteriori statistics of the Fourier conditional simulation.
TL;DR: In this paper, a review of the properties of the fractional Fourier transform, which is used in information processing, is presented in connection with the symplectic tomography transform of optical signals.
Abstract: A review of the properties of the fractional Fourier transform, which is used in information processing, is presented in connection with the symplectic tomography transform of optical signals The relationship between the Green function of the quantum harmonic oscillator and the fractional Fourier transform is elucidated An analysis of electromagnetic signals which uses an invertible map of analytic signal onto the tomographic probability distribution is made The formal connection of the analysis with the tomography method of measuring quantum states is considered The relation to other methods of time-frequency quasidistributions (for example, the Ville-Wigner quasidistribution) characterizing a signal is studied
TL;DR: Two techniques to accelerate the Fourier fringe analysis technique are described, one of which proposes a reduction in the number of pixels calculated by the Fouriers transform, the other automates the filtering operation to avoid user interaction.
TL;DR: A novel adaptive harmonic fractional Fourier transform is proposed for analysis of voiced speech signals that provides a higher concentration and avoids the cross interference components produced by the Wigner-Ville distribution and other bilinear representations.
Abstract: A novel adaptive harmonic fractional Fourier transform is proposed for analysis of voiced speech signals. It provides a higher concentration than the short time Fourier transform (STFT) and avoids the cross interference components produced by the Wigner-Ville distribution and other bilinear representations. The proposed method rotates the base tone and harmonics in time-frequency domain. After the rotation, base tone and harmonics become in parallel to the time axis in time-frequency domain so that a high concentration can be achieved.
01 Jan 1999
TL;DR: The DFT over general commutative rings is introduced and Blahut's Theorem, which relates the DFT to linear complexity, is shown to hold unchanged in general commutation rings.
Abstract: | Some applications of the Discrete Fourier Transform (DFT) in coding and in cryptography are described. The DFT over general commutative rings is introduced and the condition for its existence given. Blahut's Theorem, which relates the DFT to linear complexity, is shown to hold unchanged in general commutative rings. I. The (Usual) Discrete Fourier Transform Let be a primitive N th root of unity in a eld F , i.e., N = 1 but i 6= 1 for 1 i < N . The (usual) Discrete Fourier Transform (DFT) of length N generated by is the mapping DFT ( ) from F N to F de ned by B =DFT (b) in the manner
TL;DR: In this article, a Fourier transform technique is proposed for use in multitone harmonic balance (HB) simulations, which is especially useful when the number of input tones is very large, such as spectral regrowth and noise-power ratio simulations.
Abstract: A novel Fourier transform technique is proposed for use in multitone harmonic-balance (HB) simulations. It is shown that computations of multitone distorted spectra reduce to efficient one-dimensional fast Fourier transform operations when certain relationships exist between the sampling rate and frequency components of the signal. The algorithm requires minimal initialization time and is readily incorporated into existing HE tools. It is especially useful when the number of input tones is very large, such as spectral regrowth and noise-power ratio simulations. The method is demonstrated on the example of a 5-GHz MESFET amplifier driven by a quadrature phase shift-keying modulated carrier.
TL;DR: In this paper, a matrix multiplication procedure for evaluating the pixelated version of the near-field pattern of a discrete, one- or two-dimensional input is described, where the phase matrix is evaluated at ϵ=1.
Abstract: We describe a matrix multiplication procedure for evaluating the pixelated version of the near-field pattern of a discrete, one- or two-dimensional input. We show that for an input with N×N pixels, in an area d×d, it is necessary to evaluate the Fresnel diffraction pattern at distances z⩾d2/λN. Our numerical algorithm is also useful for evaluating the fractional Fourier transform by multiplying by a special phase matrix with fractional parameter ϵ. If the phase matrix is evaluated at ϵ=1, we find the discrete Fourier transform matrix.
Patent•
02 Mar 1999TL;DR: In this paper, an improved method for decreasing the probability of an unacceptably high peak-to-average power ratio in a signal to be transmitted by a FDM system, such as a discrete multitone (DMT) system, is described.
Abstract: We describe an improved method for decreasing the probability of an unacceptably high peak-to-average power ratio in a signal to be transmitted by a Frequency Division Multiplexing (FDM) system, such as a discrete multitone (DMT) system. The method involves generating at least two alternative signal sequences, computing Fourier transforms of the respective alternative signal sequences, and selecting for transmission one of these sequences, based on the Fourier transform computations. More specifically, the selection of one sequence may be based, e.g., on the determination that the Fourier transform of that sequence has an acceptable peak power. Alternatively, a comparison may be made among the Fourier transforms of the respective signal sequences, and selection made of that sequence whose Fourier transform exhibits the lowest peak power.
Dissertation•
01 Jan 1999
TL;DR: This dissertation first describes an important property of real-valued time sequences in the frequency domain, i.e. symmetry, and presents an algorithm that uses this property to improve the performance of a multidimensional index built on a sequence data set by more than a factor of two.
Abstract: Fourier-Transform Based Techniques in Efficient Retrieval of Similar Tirne Sequences Davood Rafiei Doctor of Philosophy Graduate Department of Computer Science University of Toronto 1999 The idea of posing queries in terms of similarity of objects, rather than equality or inequality, is of growing importance in new database applications, such as data mining or data warehousing. In this dissertation, the notion of similarity is defined in terms of a distance function and a set of linear transformations. This turns out to be a proper notion of similarity for time series data since it can eliminate seasonal effects and shortterm fluctuations before aligning them. The focus of this dissertation is on efficiently processing s i rn i l~ i ty queries on time series data. The dissertation first describes an important property of real-valued time sequences in the frequency domain, i.e. symmetry, and presents an algorithm that uses this property to improve the performance of a multidimensional index built on a sequence data set by more than a factor of two. This improvement is confirmed both analytically and
TL;DR: The algorithms developed in this paper update the DFT to reflect the modified window contents, using less computation than directly evaluating the modified transform via the FFT algorithm, which reduces the computational order by a factor of log 2 N for both the 1-D and 2-D cases.
TL;DR: The newly developed, to the authors' knowledge, nonuniform fast Fourier transform algorithm is used for the fast computation of Hankel transforms on a set of non uniformly spaced sample points.
Abstract: We investigate the fast computation of Hankel transforms on a set of nonuniformly spaced sample points. Previous fast Hankel transform (FHT) algorithms require the sample points to distribute uniformly in a logarithmic scale. This limitation is removed here by the use of the newly developed, to our knowledge, nonuniform fast Fourier transform algorithm. The resulting nonuniform FHT algorithm has a much higher accuracy than the previous FHT algorithms and provides important flexibility in many applications.
15 Mar 1999
TL;DR: The new lapped transform is real-valued, and at the same time allows unambiguous detection of spatial orientation, and its performance in spectral approaches to image restoration and enhancement in comparison to the DFT is investigated.
Abstract: We propose a new real-valued lapped transform for 2D-signal and image processing Lapped transforms are particularly useful in block-based processing, since their intrinsically overlapping basis functions reduce or prevent block artifacts Our transform is derived from the modulated lapped transform (MLT), which, as a real-valued and separable transform like the discrete cosine transform, does not allow to unambiguously identify oriented structures from modulus spectra This is in marked contrast to the (complex-valued) discrete Fourier transform (DFT) The new lapped transform is real-valued, and at the same time allows unambiguous detection of spatial orientation Furthermore, a fast algorithm for this transform exists As an application example, we investigate the transform's performance in spectral approaches to image restoration and enhancement in comparison to the DFT
30 May 1999
TL;DR: It is shown that this method fails on chirp-like signals, and a new strategy to denoise these types of signals is proposed, which is compared to the well-known wavelet shrinkage method and the ideal Wiener filtering method.
Abstract: Wavelet shrinkage is a simple yet powerful tool for denoising piecewise smooth signals In this paper we show that this method fails on chirp-like signals We propose a new strategy to denoise these types of signals First, we transform the signal to the Fourier domain, and denoise its Fourier transform using conventional wavelet shrinkage Then, we obtain the resulting time domain signal with the inverse Fourier transform The results are compared to the well-known wavelet shrinkage method and the ideal Wiener filtering method