Showing papers on "Non-uniform discrete Fourier transform published in 1997"

Computing the discrete-time 'analytic' signal via FFT

Abstract: Starting with a real-valued N-point discrete-time signal, frequency-domain algorithms are provided for computing (1) the complex-valued standard N-point discrete time 'analytic' signal of the same sample rate, (2) the complex-valued decimated N/2-point discrete-time 'analytic' signal of half the original sample rate, and (3) the complex-valued interpolated NM-point discrete-time 'analytic' signal of M times the original sample rate. Special adjustment of transform end points is shown to generate proper discrete-time 'analytic' signals.

Improved discrete fractional Fourier transform.

Abstract: The fractional Fourier transform is a useful mathematical operation that generalizes the well-known continuous Fourier transform. Several discrete fractional Fourier transforms (DFRFT's) have been developed, but their results do not match those of the continuous case. We propose a new DFRFT. This improved DFRFT provides transforms similar to those of the continuous fractional Fourier transform and also retains the rotation properties.

Ground-roll suppression using the wavelet transform

Abstract: Low-frequency, high-amplitude ground roll is an old problem in land-based seismic field records. Current processing techniques aimed at ground-roll suppression, such as frequency filtering, f - k filtering, and f - k filtering with time-offset windowing, use the Fourier transform, a technique that assumes that the basic seismic signal is stationary. A new alternative to the Fourier transform is the wavelet transform, which decomposes a function using basis functions that, unlike the Fourier transform, have finite extent in both frequency and time. Application of a filter based on the wavelet transform to land seismic shot records suppresses ground roll in a time-frequency sense; unlike the Fourier filter, this filter does not assume that the signal is stationary. The wavelet transform technique also allows more effective time-frequency analysis and filtering than current processing techniques and can be implemented using an algorithm as computationally efficient as the fast Fourier transform. This new filtering technique leads to the improvement of shot records and considerably improves the final stack quality.

Density compensation functions for spiral MRI

Abstract: In interleaved spiral MRI, an object's Fourier transform is sampled along a set of curved trajectories in the spatial frequency domain (k-space). An image of the object is then reconstructed, usually by interpolating the sampled Fourier data onto a Cartesian grid and applying the fast Fourier transform (FFT) algorithm. To obtain accurate results, it is necessary to account for the nonuniform density with which k-space is sampled. An analytic density compensation function (DCF) for spiral MRI, based on the Jacobian determinant for the transformation between Cartesian coordinates and the spiral sampling parameters of time and interleaf rotation angle, is derived in this paper, and the reconstruction accuracy achieved using this function is compared with that obtained using several previously published expressions. Various non-ideal conditions, including intersecting trajectories, are considered. The new DCF eliminated intensity cupping that was encountered in images reconstructed with other functions, and significantly reduced the level of artifact observed when unevenly spaced sampling trajectories, such as those achieved with trapezoidal gradient waveforms, were employed. Modified forms of this function were found to provide similar improvements when intersecting trajectories made the spiral-Cartesian transformation noninvertible, and when the shape of the spiral trajectory varied between interleaves.

Fractional wavelet transform.

Abstract: The wavelet transform, which has had a growing importance in signal and image processing, has been generalized by association with both the wavelet transform and the fractional Fourier transform. Possible implementations of the new transformation are in image compression, image transmission, transient signal processing, etc. Computer simulations demonstrate the abilities of the novel transform. Optical implementation of this transform is briefly discussed.

Multidimensional inverse lattice problem and a uniformly sampled arithmetic Fourier transform

Abstract: The present work develops a unified and concise solution for inverse lattice problems. Also, a uniformly sampled arithmetic Fourier transform is presented in this work which uses Ramanujan's sum rule.

Use of Fourier and Karhunen-Loeve decomposition for fast pattern matching with a large set of templates

Abstract: We present a fast pattern matching algorithm with a large set of templates. The algorithm is based on the typical template matching speeded up by the dual decomposition; the Fourier transform and the Karhunen-Loeve transform. The proposed algorithm is appropriate for the search of an object with unknown distortion within a short period. Patterns with different distortion differ slightly from each other and are highly correlated. The image vector subspace required for effective representation can be defined by a small number of eigenvectors derived by the Karhunen-Loeve transform. A vector subspace spanned by the eigenvectors is generated, and any image vector in the subspace is considered as a pattern to be recognized. The pattern matching of objects with unknown distortion is formulated as the process to extract the portion of the input image, find the pattern most similar to the extracted portion in the subspace, compute normalized correlation between them at each location in the input image, and find the location with the best score. Searching for objects with unknown distortion requires vast computation. The formulation above makes it possible to decompose highly correlated reference images into eigenvectors, as well as to decompose images in frequency domain, and to speed up the process significantly.

Discrete-dipole approximation for scattering by features on surfaces by means of a two-dimensional fast Fourier transform technique

Abstract: A two-dimensional fast Fourier transform technique is proposed for accelerating the computation of scattering characteristics of features on surfaces by using the discrete-dipole approximation. The two-dimensional fast Fourier transform reduces the CPU execution time dependence on the number of dipoles N from O(N2) to O(N log N). The capabilities and flexibility of a discrete-dipole code implementing the technique are demonstrated with scattering results from circuit features on surfaces.

High Accuracy Evaluation of the Finite Fourier Transform Using Sampled Data

Abstract: Many system identification and signal processing procedures can be done advantageously in the frequency domain. A required preliminary step for this approach is the transformation of sampled time domain data into the frequency domain. The analytical tool used for this transformation is the finite Fourier transform. Inaccuracy in the transformation can degrade system identification and signal processing results. This work presents a method for evaluating the finite Fourier transform using cubic interpolation of sampled time domain data for high accuracy, and the chirp z-transform for arbitrary frequency resolution. The accuracy of the technique is demonstrated in example cases where the transformation can be evaluated analytically. Arbitrary frequency resolution is shown to be important for capturing details of the data in the frequency domain. The technique is demonstrated using flight test data from a longitudinal maneuver of the F-18 High Alpha Research Vehicle.

Extraction of spectral peak parameters using a short-time Fourier transform modeling and no sidelobe windows

Abstract: A new method which improves the estimation of frequency, amplitude and phase of the partials of a sound is presented. It allows the reduction of the analysis-window size from four periods to two periods. It therefore gives better accuracy in parameter determination, and has proved to remain efficient at low signal-to-noise ratios. The basic idea consists of using a parametric modeling of the short-time Fourier transform. The method alternately estimates the complex amplitudes and the frequencies starting from the result of the classical analysis method. It uses the least-square procedure and a first-order limited expansion of the model around previous estimations. This method leads us to design new windows which do not have any sidelobes in order to help the convergence. Finally an analysis algorithm which has been built according to the observed behavior of the method for various kinds of sound is presented.

Uniform time-sampling Fourier transform spectroscopy

Abstract: We show that uniform time sampling of both the reference and the target channels in a continuous scanning Fourier transform spectrometer is a simple and versatile way of extending the Nyquist limit shorter than the wavelength of the reference channel. We also discuss the benefits of recording the reference channel when intensity calibrating the target data.

Modified Fourier transform method for interferogram fringe pattern analysis.

Abstract: A modified Fourier transform method for interferogram fringe pattern analysis is proposed. While it retains most of the advantages of the Fourier transform method, the new method overcomes some drawbacks of the previous method. It eliminates the assumptions of slowly varying phase variation in the test section and the constant spatial carrier frequency. It also extends the frequency bandwidth and avoids phase distortion caused by discreteness of the sampling frequency. Both numerical simulation and experimental examination are performed to evaluate the performance of the method.

Three-dimensional optical Fourier transform and correlation

Abstract: Optical implementation of a three-dimensional (3-D) Fourier transform is proposed and demonstrated. A spatial 3-D object, as seen from the paraxial zone, is transformed to the 3-D spatial frequency space. Based on the new procedure, a 3-D joint transform correlator is described that is capable of recognizing targets in the 3-D space.

Adaptive short-time Fourier analysis

Abstract: This article presents a method of adaptively adjusting the window length used in short-time Fourier analysis, related to our earlier work in which we developed a means of adaptively optimizing the performance of the cone kernel distribution (CKD). The optimal CKD cone length is, by definition, a measure of the interval over which the signal has constant or slowly changing frequency structure. The article shows that this length can also be used to compute a time-varying short-time Fourier transform (STFT). The resulting adaptive STFT shares many desirable properties with the adaptive CKD, such as the ability to adapt to transient as well as long-term signal components. The optimization requires O(N) operations per step, less than the fast Fourier transform (FFT) used in computing each time slice, making it competitive in complexity with nonadaptive time-frequency algorithms.

Space-variant Fourier analysis: the exponential chirp transform

Abstract: Space-variant (or foveating) vision architectures are of importance in both machine and biological vision. In this paper, we focus on a particular space-variant map, the log-polar map, which approximates the primate visual map, and which has been applied in machine vision by a number of investigators during the past two decades. Associated with the log-polar map, we define a new linear integral transform, which we call the exponential chirp transform. This transform provides frequency domain image processing for space-variant image formats, while preserving the major aspects of the shift-invariant properties of the usual Fourier transform. We then show that a log-polar coordinate transform in frequency provides a fast exponential chirp transform. This provides size and rotation, in addition to shift, invariant properties in the transformed space. Finally, we demonstrate the use of the fast exponential chirp algorithm on a database of images in a template matching task, and also demonstrate its uses for spatial filtering.

Equivalence of two Fourier methods for biological sequences

Abstract: Two methods for defining Fourier power spectra for DNA sequences or other biological sequences are compared. The first method uses indicator sequences for each letter. The second method by Silverman and Linsker assigns to each letter a vertex of a regular tetrahedron in space, and this can be generalized to any dimension. While giving different Fourier transforms, it is shown that the power spectra of the two methods are essentially the same. This is also true if one replaces the Fourier transform in both methods with another linear transform, such as the Walsh transform.

Demodulation using a time domain guard interval with an overlapped transform

Abstract: A system and method for demodulation of an RF signal on a transmission channel is provided. The RF signal is demodulated to baseband as an in-phase (I) data signal and a quadrature (Q) data signal. A first block of I data is captured and a first block of Q data is captured. A time domain guard interval is provided in the captured first blocks of I and Q data. A complex discrete Fourier transform is performed on the captured first I and Q data blocks. An inverse frequency response for the transmission channel is determined. The inverse frequency response is multiplied by the complex discrete Fourier transform of the guard-interval protected first I and Q data blocks to generate a frequency domain product signal. An inverse Fourier transform on the product of the multiplying step is performed to generate a first equalized time domain signal. In a preferred embodiment, the method also includes using an overlapped Fourier transform and discarding a first portion of each equalized time domain signal.

Numerical investigation of phase retrieval in a fractional Fourier transform

Abstract: Recently the combination of the Gerchberg–Saxton (GS) algorithm and a fractional Fourier transform was proposed to implement beam shaping in the fractional Fourier domain [ Zalevsky , Opt. Lett.21, 842 (1996)]. We generalize this idea to deal with the problem of phase retrieval from two intensity measurements in a fractional Fourier transform system. The relevant equations for determining the unknown phases are derived, based on the general theory of amplitude–phase retrieval in an optical system. The unitarity condition of the fractional Fourier transform in a practical optical system with finite aperture is discussed. For different fractional orders P, the phase retrieval of several typical model images is studied in detail. A comparison of the GS and our algorithms is given, based on numerical simulations. It follows that our algorithm can offer the desired phase in all cases considered. However, the GS algorithm may fail when the transform system is nonunitary.

The Time Variant Discrete Fourier Transform as an Order Tracking Method

Abstract: Present order tracking methods for solving noise and vibration problems are reviewed, both FFT and resampling based order tracking methods. The time variant discrete Fourier transform (TVDFT) is developed as an alternative order tracking method. This method contains many advantages which the current order tracking methods do not possess. This method has the advantage of being very computationally efficient as well as the ability to minimize leakage errors. The basic TVDFT method may also be extended to a more complex method through the use of an orthogonality compensation matrix (OCM) which can separate closely spaced orders as well as separate the contributions of crossing orders. The basic TVDFT is a combination of the FFT and the re-sampling based methods. This method can be formulated in several different manners, one of which will give results matching the re-sampling based methods very closely. Both analytical and experimental data are used to establish the behavioral characteristics of this new method.

An improved acoustic Fourier boundary element method formulation using fast Fourier transform integration

Abstract: Effective use of the Fourier series boundary element method (FBEM) for everyday applications is hindered by the significant numerical problems that have to be overcome for its implementation. In the FBEM formulation for acoustics, some integrals over the angle of revolution arise, which need to be calculated for every Fourier term. These integrals were formerly treated for each Fourier term separately. In this paper a new method is proposed to calculate these integrals using fast Fourier transform techniques. The advantage of this integration method is that the integrals are simultaneously computed for all Fourier terms in the boundary element formulation. The improved efficiency of the method compared to a Gaussian quadrature based integration algorithm is illustrated by some example calculations. The proposed method is not only usable for acoustic problems in particular, but for Fourier BEM in general.

Wavelet transform based fast approximate Fourier transform

Abstract: We propose an algorithm that uses the discrete wavelet transform (DWT) as a tool to compute the discrete Fourier transform (DFT). The Cooley-Tukey FFT is shown to be a special case of the proposed algorithm when the wavelets in use are trivial. If no intermediate coefficients are dropped and no approximations are made, the proposed algorithm computes the exact result, and its computational complexity is on the same order of the FFT, i.e. O(N log/sub 2/ N). The main advantage of the proposed algorithm is that the good time and frequency localization of wavelets can be exploited to approximate the Fourier transform for many classes of signals resulting in much less computation. Thus the new algorithm provides an efficient complexity vs. accuracy tradeoff. When approximations are allowed, under certain sparsity conditions, the algorithm can achieve linear complexity, i.e. O(N). The proposed algorithm also has built-in noise reduction capability.

Reducing peak spectral error in inverse Fast Fourier Transform using MMX™ technology

Abstract: A method for computing a decimation-in-time Fast Fourier Transform of a sample is provided, the method including inputting first 2B-bit values representing the sample into a radix-4 first section of the decimation-in-time Fast Fourier Transform and performing first complex 2B-bit integer additions and subtractions on the first 2B-bit values to form second 2B-bit values, without performing a multiplication. The method also includes rounding the second 2B-bit values to form B-bit values output from the radix-4 first section of the decimation-in-time Fast Fourier Transform.

Nonuniform Fast Fourier Transform

Abstract: The nonuniform discrete Fourier transform (NDFT) can be computed with a fast algorithm, referred to as the nonuniform fast Fourier transform (NFFT). In L dimensions, the NFFT requires O(N(-lne)L+(∏𝓁=1LM𝓁)∑𝓁=1LlogM𝓁) operations, where M𝓁 is the number of Fourier components along dimension 𝓁, N is the number of irregularly spaced samples, and e is the required accuracy. This is a dramatic improvement over the O(N∏𝓁=1LM𝓁) operations required for the direct evaluation (NDFT). The performance of the NFFT depends on the lowpass filter used in the algorithm. A truncated Gauss pulse, proposed in the literature, is optimized. A newly proposed filter, a Gauss pulse tapered with a Hanning window, performs better than the truncated Gauss pulse and the B-spline, also proposed in the literature. For small filter length, a numerically optimized filter shows the best results. Numerical experiments for 1-D and 2-D implementations confirm the theoretically predicted accuracy and efficiency properties of the algorithm.

Moving direction measuring device and tracking apparatus

Abstract: A moving direction measuring device includes a collation Fourier pattern data generating section, a registration Fourier pattern data generating section, a pattern processing section, and a moving direction measuring section. The collation Fourier pattern data generating section generates collation Fourier two-dimensional pattern data by performing two-dimensional discrete Fourier transform for two-dimensional pattern data input at time T n . The registration Fourier pattern data generating section generates registration Fourier two-dimensional pattern data by performing two-dimensional discrete Fourier transform for two-dimensional pattern data input at time T n+1 . The pattern processing section synthesizes these Fourier two-dimensional pattern data, and performs either two-dimensional discrete Fourier transform or two-dimensional discrete inverse Fourier transform for the resultant data. The moving direction measuring section obtains a correlation peak in a correlation component area appearing in the synthesized Fourier two-dimensional pattern data having undergone Fourier transform performed by the pattern processing section, and obtains a direction from a reference position in the correlation component area to a position of the correlation peak as a moving direction of the device.