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Showing papers on "Non-uniform discrete Fourier transform published in 2006"


Journal ArticleDOI
TL;DR: The fractional Fourier transform has been comprehensively and systematically treated from the signal processing point of view and a course from the definition to the applications is provided, especially as a reference and an introduction for researchers and interested readers.
Abstract: The fractional Fourier transform is a generalization of the classical Fourier transform, which is introduced from the mathematic aspect by Namias at first and has many applications in optics quickly. Whereas its potential appears to have remained largely unknown to the signal processing community until 1990s. The fractional Fourier transform can be viewed as the chirp-basis expansion directly from its definition, but essentially it can be interpreted as a rotation in the time-frequency plane, i.e. the unified time-frequency transform. With the order from 0 increasing to 1, the fractional Fourier transform can show the characteristics of the signal changing from the time domain to the frequency domain. In this research paper, the fractional Fourier transform has been comprehensively and systematically treated from the signal processing point of view. Our aim is to provide a course from the definition to the applications of the fractional Fourier transform, especially as a reference and an introduction for researchers and interested readers.

196 citations


Journal ArticleDOI
TL;DR: The sampling and reconstruction formulas are deduced, together with the construction methodology for the multiplicative filter in the time domain based on fast Fourier transform (FFT), which has much lower computational load than the construction method in the linear canonical domain.
Abstract: As generalization of the fractional Fourier transform (FRFT), the linear canonical transform (LCT) has been used in several areas, including optics and signal processing. Many properties for this transform are already known, but the convolution theorems, similar to the version of the Fourier transform, are still to be determined. In this paper, the authors derive the convolution theorems for the LCT, and explore the sampling theorem and multiplicative filter for the band limited signal in the linear canonical domain. Finally, the sampling and reconstruction formulas are deduced, together with the construction methodology for the above mentioned multiplicative filter in the time domain based on fast Fourier transform (FFT), which has much lower computational load than the construction method in the linear canonical domain.

133 citations


Journal ArticleDOI
TL;DR: This work shows the application of new approach to the 3D HNCO spectrum acquired for protein sample with radial and spiral time domain sampling and enables one to Fourier transform arbitrarily sampled time domain and thus allows for analysis of high dimensionality spectra acquired in a short time.

127 citations


Book
01 Jan 2006
TL;DR: Introduction Data Acquisition Noise Signal Averaging Real and Complex Fourier Series Continuous, Discrete, and Fast Fourier transform Fourier Transform Applications LTI systems, Convolution, Correlation, and Coherence Laplace and z-Transform.
Abstract: Introduction Data Acquisition Noise Signal Averaging Real and Complex Fourier Series Continuous, Discrete, and Fast Fourier Transform Fourier Transform Applications LTI systems, Convolution, Correlation, and Coherence Laplace and z-Transform Introduction to Filters: the RC-Circuit Filters: Analysis Filters: Specification, Bode plot, Nyquist plot Filters: Digital Filters Spike Train Analysis Wavelet Analysis: Time Domain Properties Wavelet Analysis: Frequency Domain Properties Nonlinear Techniques

112 citations


Journal ArticleDOI
TL;DR: Application of Fourier Transform for processing 3D NMR spectra with random sampling of evolution time space with general applicability and significant improvement of resolution in comparison with conventional spectra recorded in the same time is presented.
Abstract: Application of Fourier Transform for processing 3D NMR spectra with random sampling of evolution time space is presented. The 2D FT is calculated for pairs of frequencies, instead of conventional sequence of one-dimensional transforms. Signal to noise ratios and linewidths for different random distributions were investigated by simulations and experiments. The experimental examples include 3D HNCA, HNCACB and (15)N-edited NOESY-HSQC spectra of (13)C (15)N labeled ubiquitin sample. Obtained results revealed general applicability of proposed method and the significant improvement of resolution in comparison with conventional spectra recorded in the same time.

105 citations


Journal ArticleDOI
TL;DR: The proposed multiple-parameter discrete fractional Fourier transform (MPDFRFT) is shown to have all of the desired properties for fractional transforms and the double random phase encoding in the MPDFRFT domain significantly enhances data security.
Abstract: The discrete fractional Fourier transform (DFRFT) is a generalization of the discrete Fourier transform (DFT) with one additional order parameter. In this letter, we extend the DFRFT to have N order parameters, where N is the number of the input data points. The proposed multiple-parameter discrete fractional Fourier transform (MPDFRFT) is shown to have all of the desired properties for fractional transforms. In fact, the MPDFRFT reduces to the DFRFT when all of its order parameters are the same. To show an application example of the MPDFRFT, we exploit its multiple-parameter feature and propose the double random phase encoding in the MPDFRFT domain for encrypting digital data. The proposed encoding scheme in the MPDFRFT domain significantly enhances data security.

103 citations


Journal ArticleDOI
TL;DR: A new nearly tridiagonal matrix is proposed, which commutes with the discrete Fourier transform (DFT) matrix and is shown to be DFT eigenvectors, which are more similar to the continuous Hermite-Gaussian functions than those developed before.
Abstract: Based on discrete Hermite-Gaussian-like functions, a discrete fractional Fourier transform (DFRFT), which provides sample approximations of the continuous fractional Fourier transform, was defined and investigated recently. In this paper, we propose a new nearly tridiagonal matrix, which commutes with the discrete Fourier transform (DFT) matrix. The eigenvectors of the new nearly tridiagonal matrix are shown to be DFT eigenvectors, which are more similar to the continuous Hermite-Gaussian functions than those developed before. Rigorous discussions on the relations between the eigendecomposition of the newly proposed nearly tridiagonal matrix and the DFT matrix are described. Furthermore, by appropriately combining two linearly independent matrices that both commute with the DFT matrix, we develop a method to obtain DFT eigenvectors even more similar to the continuous Hermite-Gaussian functions (HGFs). Then, new versions of DFRFT produce their transform outputs closer to the samples of the continuous fractional Fourier transform, and their applications are described. Related computer experiments are performed to illustrate the validity of the works in this paper

78 citations


Journal ArticleDOI
TL;DR: In this article, the authors use the Fourier spectra of the x-, y-, and z-components of a one-component signal to identify the positions of the ellipses.
Abstract: SUMMARY From basic Fourier theory, a one-component signal can be expressed as a superposition of sinusoidal oscillations in time, with the Fourier amplitude and phase spectra describing the contribution of each sinusoid to the total signal. By extension, three-component signals can be thought of as superpositions of sinusoids oscillating in the x-, y-, and z-directions, which, when considered one frequency at a time, trace out elliptical motion in three-space. Thus the total three-component signal can be thought of as a superposition of ellipses. The information contained in the Fourier spectra of the x-, y-, and z-components of the signal can then be re-expressed as Fourier spectra of the elements of these ellipses, namely: the lengths of their semi-major and semi-minor axes, the strike and dip of each ellipse plane, the pitch of the major axis, and the phase of the particle motion at each frequency. The same type of reasoning can be used with windowed Fourier transforms (such as the S transform), to give time-varying spectra of the elliptical elements. These can be used to design signal-adaptive polarization filters that reject signal components with specific polarization properties. Filters of this type are not restricted to reducing the whole amplitude of any particular ellipse; for example, the ‘linear’ part of the ellipse can be retained while the ‘circular’ part is rejected. This paper describes the mathematics behind this technique, and presents three examples: an earthquake seismogram that is first separated into linear and circular parts, and is later filtered specifically to remove the Rayleigh wave; and two shot gathers, to which similar Rayleigh-wave filters have been applied on a trace-by-trace basis.

69 citations


Journal ArticleDOI
TL;DR: This paper shows that the discrete Radon transform additionally has a fast, exact (although iterative) inverse, which reproduces to machine accuracy the pixel-by-pixel values of the original image from its DRT, without artifacts or a finite point-spread function.
Abstract: Gotz, Druckmuller, and, independently, Brady have defined a discrete Radon transform (DRT) that sums an image9s pixel values along a set of aptly chosen discrete lines, complete in slope and intercept. The transform is fast, O ( N 2 log N ) for an N × N image; it uses only addition, not multiplication or interpolation, and it admits a fast, exact algorithm for the adjoint operation, namely backprojection. This paper shows that the transform additionally has a fast, exact (although iterative) inverse. The inverse reproduces to machine accuracy the pixel-by-pixel values of the original image from its DRT, without artifacts or a finite point-spread function. Fourier or fast Fourier transform methods are not used. The inverse can also be calculated from sampled sinograms and is well conditioned in the presence of noise. Also introduced are generalizations of the DRT that combine pixel values along lines by operations other than addition. For example, there is a fast transform that calculates median values along all discrete lines and is able to detect linear features at low signal-to-noise ratios in the presence of pointlike clutter features of arbitrarily large amplitude.

68 citations


Journal ArticleDOI
TL;DR: It is shown that radially sampled data can be processed directly using Fourier transforms in polar coordinates, and an intrinsic connection between the polar Fourier transform and the filtered backprojection method that has recently been introduced to process projection-reconstruction NOESY data is described.

59 citations


Journal ArticleDOI
TL;DR: A new registration algorithm based on pseudo-log-polar Fourier transform (PLPFT) for estimating large translations, rotations, and scalings in images is developed and the robustness and high accuracy of this algorithm is verified.
Abstract: A new registration algorithm based on pseudo-log-polar Fourier transform (PLPFT) for estimating large translations, rotations, and scalings in images is developed. The PLPFT, which is calculated at points distributed at nonlinear increased concentric squares, approximates log-polar Fourier representations of images accurately. In addition, it can be calculated quickly by utilizing the Fourier separability property and the fractional fast Fourier transform. Using the log-polar Fourier representations and cross-power spectrum method, we can estimate the rotations and scalings in images and obtain translations later. Experimental results have verified the robustness and high accuracy of this algorithm.

Journal ArticleDOI
TL;DR: An algorithm is proposed to reconstruct two-dimensional wave-front from phase differences measured by lateral shearing interferometer based on double-grating that allows large shear amount and works fast based on fast Fourier transform.
Abstract: An algorithm is proposed to reconstruct two-dimensional wave-front from phase differences measured by lateral shearing interferometer. Two one-dimensional phase profiles of object wave-front are computed using Fourier transform from phase differences, and then the two-dimensional wave-front distribution is retrieved by use of least-square fitting. The algorithm allows large shear amount and works fast based on fast Fourier transform. Investigations into reconstruction accuracy and reliability are carried out by numerical experiments, in which effects of different shear amounts and noises on reconstruction accuracy are evaluated. Optical measurement is made in a lateral shearing interferometer based on double-grating.

01 Jan 2006
TL;DR: The definition of LCT and some special cases are given at first, followed by its properties as listed, and the discrete linear canonical transform is introduced.
Abstract: As an emerging tool for signal processing,the linear canonical transform(LCT) proves itself to be more general and flexible than the Fourier transform as well as the fractional Fourier transform.So it can slove problems that can't be dealt with well by the latter.In this paper,the definition of LCT and some special cases are given at first,followed by its properties as listed.Besides,the discrete linear canonical transform is introduced.The implication of LCT is illustrated finally,displaying(LCT's) potentials and capabilities in the field of signal processing.

Journal ArticleDOI
TL;DR: A 2D discrete Fourier transform can be implemented in polar coordinates to obtain directly a frequency domain spectrum and will permit to investigate better compromises in terms of experimental time and lack of artifacts.
Abstract: In order to reduce the acquisition time of multidimensional NMR spectra of biological macromolecules, projected spectra (or in other words, spectra sampled in polar coordinates) can be used. Their standard processing involves a regular FFT of the projections followed by a reconstruction, i.e. a non-linear process. In this communication, we show that a 2D discrete Fourier transform can be implemented in polar coordinates to obtain directly a frequency domain spectrum. Aliasing due to local violations of the Nyquist sampling theorem gives rise to base line ridges but the peak line-shapes are not distorted as in most reconstruction methods. The sampling scheme is not linear and the data points in the time domain should thus be weighted accordingly in the polar FT; however, artifacts can be reduced by additional data weighting of the undersampled regions. This processing does not require any parameter tuning and is straightforward to use. The algorithm written for polar sampling can be adapted to any sampling scheme and will permit to investigate better compromises in terms of experimental time and lack of artifacts.

Journal ArticleDOI
TL;DR: A scheme for signal compression based on the combination of discrete FRFT (DFRFT) and set partitioning in hierarchical tree (SPIHT) and application to different types of signals demonstrates significant reduction in bits leading to high signal compression ratio.

Journal ArticleDOI
Olivier Adam1
TL;DR: The Hilbert Huang transform (HHT) is introduced as an efficient means for analysis of bioacoustical signals and shows that HHT is a viable alternative to the wavelet transform.
Abstract: While marine mammals emit variant signals (in time and frequency), the Fourier spectrogram appears to be the most widely used spectral estimator. In certain cases, this approach is suboptimal, particularly for odontocete click analysis and when the signal-to-noise ratio varies during the continuous recordings. We introduce the Hilbert Huang transform (HHT) as an efficient means for analysis of bioacoustical signals. To evaluate this method, we compare results obtained from three time-frequency representations: the Fourier spectrogram, the wavelet transform, and the Hilbert Huang transform. The results show that HHT is a viable alternative to the wavelet transform. The chosen examples illustrate certain advantages. (1) This method requires the calculation of the Hilbert transform; the time-frequency resolution is not restricted by the uncertainty principle; the frequency resolution is finer than with the Fourier spectrogram. (2) The original signal decomposition into successive modes is complete. If we were to multiply some of these modes, this would contribute to attenuate the presence of noise in the original signal and to being able to select pertinent information. (3) Frequency evolution for each mode can be analyzed as one-dimensional (1D) signal. We not need a complex 2D post-treatment as is usually required for feature extraction.

Journal ArticleDOI
TL;DR: For a regular twistor D-module and for a given function f, the authors compare the nearby cycles at f = ∞ and the nearby or vanishing cycles at τ = 0 for its partial Fourier-Laplace transform relative to the kernel e−τf.
Abstract: For a regular twistor D-module and for a given function f , we compare the nearby cycles at f = ∞ and the nearby or vanishing cycles at τ = 0 for its partial Fourier-Laplace transform relative to the kernel e−τf .

Journal ArticleDOI
Emily G. Allen1
TL;DR: In this article, the windowed short-time Fourier transform (STFT) was used to quantify ammonoid suture shape, which accommodates the characteristics of complex curves, such as the first and last sampled points are not equivalent, the folds are non-stationary and a position along a horizontal reference axis may map to multiple amplitudes along the suture path.
Abstract: Attempts to use Fourier methods to quantify ammonoid suture shape have failed to yield robust, repeatable results because sutures are complex curves that violate the assumptions of Fourier mathematics In particular, sampled sutures are artificially truncated such that the first and last sampled points are not equivalent, the folds are non-stationary, and a position along a horizontal reference axis may map to multiple amplitudes along the suture path Here I introduce an alternative Fourier method—the windowed short-time Fourier transform (STFT)—that accommodates these characteristics of complex curves For each suture, digitized landmarks were parameterized using a tangent angle function and then smoothed by convolving with an apodization function Piece-wise Fourier transforms were then calculated and averaged, resulting in a robust, unique quantification of line morphology STFT coefficients and estimated power spectra describing the relative weights of harmonics were generated for 576 Paleozoic-basal Triassic ammonoid genera, representative of the range of suture morphotypes While insensitive to major episodes of taxonomic turnover (Frasnian/Famennian, end-Devonian, and Permian/Triassic extinctions), the summed power data support the previously observed trend toward increasing suture complexity through time Moreover, partitioning the summed power statistic into harmonic ranges allows novel insight into Paleozoic suture evolution In particular, the data show significant shifts in the dominant morphotypes during periods of rebound and radiation and suggest that basal Triassic ammonoids possessed unique suture morphotypes when compared with those of the Paleozoic

Journal ArticleDOI
TL;DR: The proposed computation method for the discrete Fourier transform is based on factorizing the transform matrix into a product of a binary block circulant matrix and a diagonal block circULant matrix.
Abstract: The discrete Fourier transform over a finite field finds applications in algebraic coding theory. The proposed computation method for the discrete Fourier transform is based on factorizing the transform matrix into a product of a binary block circulant matrix and a diagonal block circulant matrix.

Proceedings ArticleDOI
04 Sep 2006
TL;DR: A new spectral analysis technique is devised to combine the features of both uniform and non-uniform signal processing chains in order to obtain a good spectrum quality with low computational complexity.
Abstract: This work is a part of a drastic revolution in the classical signal processing chain required in mobile systems. The system must be low power as it is powered by a battery. Thus a signal driven sampling scheme based on level crossing is adopted, delivering non-uniformly spaced out in time sampled points. In order to analyse the non-uniformly sampled signal obtained at the output of this sampling scheme a new spectral analysis technique is devised. The idea is to combine the features of both uniform and non-uniform signal processing chains in order to obtain a good spectrum quality with low computational complexity. The comparison of the proposed technique with General Discrete Fourier transform and Lomb's algorithm shows significant improvements in terms of spectrum quality and computational complexity.

Journal ArticleDOI
TL;DR: 3D discrete Hartley transform is applied for the compression of two medical modalities, namely, magnetic resonance images and X-ray angiograms and the performance results are compared with those of 3-D discrete cosine and Fourier transforms using the parameters such as PSNR and bit rate.

Journal Article
TL;DR: In this paper, it is shown that it is possible to obtain a good quality approximate inverse to the Constant Q transform provided that the signal to be inverted has a sparse representation in the Discrete Fourier Transform domain.
Abstract: The Constant Q transform has found use in the analysis of musical signals due to its logarithmic frequency resolution. Unfortunately, a considerable drawback of the Constant Q transform is that there is no inverse transform. Here we show it is possible to obtain a good quality approximate inverse to the Constant Q transform provided that the signal to be inverted has a sparse representation in the Discrete Fourier Transform domain. This inverse is obtained through the use of `0 and `1 minimisation approaches to project the signal from the constant Q domain back to the Discrete Fourier Transform domain. Once the signal has been projected back to the Discrete Fourier Transform domain, the signal can be recovered by performing an inverse Discrete Fourier Transform. 1. THE CONSTANT Q TRANSFORM The Constant Q transform (CQT) was derived by Brown as a means of creating a log-frequency resolution spectrogram [1]. This has considerable advantages for the analysis of musical signals, as the frequency resolution can be set to match that of the equal tempered scale used in western music, where the frequencies are geometrically spaced, as opposed to the linear spacing that occurs in the discrete Fourier transform (DFT). The frequency components of the CQT have a constant ratio of center frequency to resolution, as opposed to the constant frequency difference and constant resolution of the DFT. This constant ratio results in a constant pattern for the spectral components making up notes played on a given instrument, and this has been used to attempt sound source separation of pitched instruments from both single channel and multi-channel mixtures of instruments[2],[3]. Given an inital minimum frequency f0 for the CQT, the center frequencies for each band can be obtained from: fk = f02 k b (k = 0, 1, ...) (1) where b is the number of bins per octave. The fixed ratio of center frequency to bandwidth is then given by Q = ( 2 1 b − 1 )−1 (2) The desired bandwidth of each frequency band is FitzGerald et al. Towards an ICQT then obtained by choosing a window of length

Journal ArticleDOI
TL;DR: The approximate analytical results provide more convenience for studying the propagation and transformation of truncated EGBs than the usual way by using the integral formula directly, and the efficiency of numerical calculation is significantly improved.
Abstract: Based on the fact that a hard-edged elliptical aperture can be expanded approximately as a finite sum of complex Gaussian functions in tensor form, an analytical expression for an elliptical Gaussian beam (EGB) truncated by an elliptical aperture and passing through a fractional Fourier transform system is derived by use of vector integration. The approximate analytical results provide more convenience for studying the propagation and transformation of truncated EGBs than the usual way by using the integral formula directly, and the efficiency of numerical calculation is significantly improved.

Journal ArticleDOI
TL;DR: An adaptive windowed Fourier transform method in 3-D measurement based on a wavelet transform is proposed, in which, by applying aWavelet ridge, a series of scaling factors are calculated to determine the series of prime windows needed in the windowed Fresnel transform method.
Abstract: An adaptive windowed Fourier transform method in 3-D measurement based on a wavelet transform is proposed, in which, by applying a wavelet ridge, a series of scaling factors are calculated to determine the series of prime windows needed in the windowed Fourier transform method. Because the spectrum of each local fringe is simpler than that of the whole fringe, even though there is frequency aliasing as far as the whole fringe is concerned, the fundamental spectrum may separate into components in each local fringe. It is easy to filter out one of the fundamental frequency components from the local spectra. Adding these local fundamental components, the full fundamental component can be obtained correctly. The advantage of the method is that it not only eliminates the frequency aliasing, but also obtains the modulation distribution function to guide phase unwrapping.

Journal ArticleDOI
TL;DR: A 3D windowed Fourier transform is proposed for fringe sequence analysis, which processes the joint spatial and temporal information of the fringe sequence simultaneously.
Abstract: A 3D windowed Fourier transform is proposed for fringe sequence analysis, which processes the joint spatial and temporal information of the fringe sequence simultaneously. The 2D windowed Fourier transform in the spatial domain and the 1D windowed Fourier transform in the temporal domain are two special cases of the proposed method. The principles of windowed Fourier filtering and windowed Fourier ridges are developed. Experimental verification shows encouraging results despite a longer processing time.

Journal ArticleDOI
TL;DR: A new linear time-frequency model in which the instantaneous value of each signal component is mapped to the curve functionally representing its instantaneous frequency, which is linear, uniquely defined by the signal decomposition, and satisfies linear marginal-like distribution properties.
Abstract: We describe a new linear time-frequency model in which the instantaneous value of each signal component is mapped to the curve functionally representing its instantaneous frequency. This transform is linear, uniquely defined by the signal decomposition, and satisfies linear marginal-like distribution properties. We further demonstrate the transform generated surface may be estimated from the short time Fourier transform by a concentration process based on the phase of the short-time Fourier transform (STFT), differentiated with respect to time. Interference may be identified on the concentrated STFT surface, and the signal with the interference removed may be estimated by applying the linear-time-marginal to the concentrated STFT surface from which the interference components have been removed

Journal ArticleDOI
TL;DR: In this article, a real color fractional Fourier transform hologram (FLFTH) was proposed for anti-counterfeiting, which is based on the FFTH.

Patent
03 Apr 2006
TL;DR: In this article, a measured magnitude of the Fourier transform of a two-dimensional complex transmission function is provided, and an estimated phase term of the transform is derived by applying at least one constraint to the inverse transform.
Abstract: A method processes an optical image. The method includes providing a measured magnitude of the Fourier transform of a two-dimensional complex transmission function. The method further includes providing an estimated phase term of the Fourier transform of the two-dimensional complex transmission function. The method further includes multiplying the measured magnitude and the estimated phase term to generate an estimated Fourier transform of the two-dimensional complex transmission function. The method further includes calculating an inverse Fourier transform of the estimated Fourier transform, wherein the inverse Fourier transform is a spatial function. The method further includes calculating an estimated two-dimensional complex transmission function by applying at least one constraint to the inverse Fourier transform.

Journal ArticleDOI
TL;DR: An optical implementation of iterative fractional Fourier transform algorithm with phase-shifting digital holography technique and the phase-type spatial light modulator adopted for the measurement and the modulation of complex optical fields is proposed and demonstrated.
Abstract: An optical implementation of iterative fractional Fourier transform algorithm is proposed and demonstrated. In the proposed implementation, the phase-shifting digital holography technique and the phase-type spatial light modulator are adopted for the measurement and the modulation of complex optical fields, respectively. With the devised iterative fractional Fourier transform system, we demonstrate two-dimensional intensity distribution synthesis in the fractional Fourier domain and three-dimensional intensity distribution synthesis simultaneously forming desired intensity distributions at several multi-focal planes.

Patent
28 Aug 2006
TL;DR: In this article, a method for determining an optical surface model for an optical tissue system of an eye is presented, which is well suited for employing Fourier transform in wavefront reconstruction using Zernike representation.
Abstract: Systems, methods, and devices for determining an optical surface model for an optical tissue system of an eye are provided. Techniques include inputting a Fourier transform of optical data from the optical tissue system, inputting a conjugate Fourier transform of a basis function surface, determining a Fourier domain sum of the Fourier transform and the conjugate Fourier transform, calculating an estimated basis function coefficient based on the Fourier domain sum, and determining the optical surface model based on the estimated basis function coefficient. The approach is well suited for employing Fourier transform in wavefront reconstruction using Zernike representation.