Showing papers on "Non-uniform discrete Fourier transform published in 2004"

Detection and parameter estimation of multicomponent LFM signal based on the fractional Fourier transform

Abstract: This paper presents a new method for the detection and parameter estimation of multicomponent LFM signals based on the fractional Fourier transform. For the optimization in the fractional Fourier domain, an algorithm based on Quasi-Newton method is proposed which consists of two steps of searching, leading to a reduction in computation without loss of accuracy. And for multicomponent signals, we further propose a signal separation technique in the fractional Fourier domain which can effectively suppress the interferences on the detection of the weak components brought by the stronger components. The statistical analysis of the estimate errors is also performed which perfects the method theoretically, and finally, simulation results are provided to show the validity of our method.

The truncated fourier transform and applications

Abstract: In this paper, we present a truncated version of the classical Fast Fourier Transform. When applied to polynomial multiplication, this algorithm has the nice property of eliminating the "jumps" in the complexity at powers of two. When applied to the multiplication of multivariate polynomials or truncated multivariate power series, we gain a logarithmic factor with respect to the best previously known algorithms.

Algorithms for Spectral Analysis of Irregularly Sampled Time Series

Abstract: In this paper, we present a spectral analysis method based upon least square approximation. Our method deals with nonuniform sampling. It provides meaningful phase information that varies in a predictable way as the samples are shifted in time. We compare least square approximations of real and complex series, analyze their properties for sample count towards infinity as well as estimator behaviour, and show the equivalence to the discrete Fourier transform applied onto uniformly sampled data as a special case. We propose a way to deal with the undesirable side effects of nonuniform sampling in the presence of constant offsets. By using weighted least square approximation, we introduce an analogue to the Morlet wavelet transform for nonuniformly sampled data. Asymptotically fast divide-and-conquer schemes for the computation of the variants of the proposed method are presented. The usefulness is demonstrated in some relevant applications.

Dilating Gabor transform for the fringe analysis of 3-D shape measurement

Abstract: In order to overcome the limitations of conventional Fourier transform and Gabor transform analyzing nonstationary signals, dilating Gabor transform is applied to analyze the optical fringes of 3-D shape measurement. The dilating Gabor transformation is introduced by using a changeable window of Gaussian function in a conventional Gabor transform. This phase analysis method provides more accurate results than Fourier transform and Gabor transform. Simulation and experimental results are presented that demonstrate the validity of the principle.

Windowed Fourier transform method for demodulation of carrier fringes

Abstract: The Fourier transform method for demodulation of carrier fringes has been extensively developed and widely used in optical metrology. However, the Fourier transform being a global operation, it has poor ability to localize the signal properties and hence the result of FTM is not ideal. A windowed Fourier transform method is thus proposed, with advantages of signal localization and noise filtering. An example demonstrates the improved result compared to the traditional Fourier transform.

OFDM signal receiving apparatus and OFDM signal receiving method

Abstract: An OFDM signal receiver reduces frequency response estimation error, and reduces the circuit scale needed for a hardware implementation and the number of operations performed in a software implementation. A first Fourier transform circuit converts an OFDM signal to the frequency domain by a Fourier transform. A first divider divides the pilot signal contained in the frequency domain OFDM signal by a specified pilot signal. A zero insertion means then inserts zero signals in the first divider output. A window function multiplying means multiplies the zero insertion means output by a window function, and an inverse Fourier transform means applies an inverse Fourier transform to the multiplier output. A coring means then cores the inverse Fourier transform output, and truncation means truncates the coring means output at a specified data length. A second Fourier transform circuit applies another Fourier transform to the truncated result. A window function dividing means then divides the Fourier transform result by the window function, and a second dividing means divides the output of the first Fourier transform means by the output of the window function dividing means.

Sub-pixel registration and estimation of local shifts directly in the Fourier domain

Abstract: In this paper, we propose a new approach to sub-pixel registration, and the estimation of local shifts between a pair of images, directly in the Fourier domain. For this purpose, we establish the exact relationship between the continuous and the discrete Fourier phase of two shifted images or their subregions. In particular, we show that the discrete phase difference of two shifted images is a 2-dimensional sawtooth signal, with the exact shifts determined to sub-pixel accuracy by the number of periods of the phase difference along each frequency axis. The sub-pixel portions of the shifts are determined by the non-integer fraction of a period of the phase difference.

Some mathematical properties of the uniformly sampled quadratic phase function and associated issues in digital Fresnel diffraction simulations

Abstract: The quadratic phase function is fundamental in describing and computing wave-propagation-related phenomena under the Fresnel approximation; it is also frequently used in many signal processing algorithms. This function has interesting properties and Fourier transform relations. For example, the Fourier transform of the sampled chirp is also a sampled chirp for some sampling rates. These properties are essential in interpreting the aliasing and its effects as a consequence of sampling of the quadratic phase function, and lead to interesting and efficient algorithms to simulate Fresnel diffraction. For example, it is possible to construct discrete Fourier transform (DFT)-based algorithms to compute exact continuous Fresnel diffraction patterns of continuous, not necessarily bandlimited, periodic masks at some specific distances.

Identification and location of hot spots in proteins using the short-time discrete Fourier transform

Abstract: A new approach for the identification and location of hot spots in proteins based on the short-time discrete Fourier transform (DFT) is proposed. In the new approach the short-time DFT of the protein numerical sequence is first computed and its columns are then multiplied by the DFT coefficients. By performing this step, the hot spot locations can be clearly identified by distinct peaks in the spectrum, thus achieving good localization in the amino acid domain.

Lomb-Wech periodogram for non-uniform sampling

Abstract: The Lomb-Scargle transform has been proposed for the direct evaluation, namely without interpolation, of non-uniformly sampled signals. In its current form, it is suitable only for single transform evaluation due to the implicit normalization. Enhancements of this transform are proposed to allow the evaluation of shorter transforms, combined with windows and averaging of overlapped records. This requires a de-normalization of the transform by a factor of 2(sigma)/sup 2//N, the use of equal time duration records, and multiplication by windows sampled at corresponding non-uniform time instances. This results in a Welch-like periodogram for non-uniform sampling.

Properties of continuous Fourier extension of the discrete cosine transform and its multidimensional generalization

Abstract: A versatile method is described for the practical computation of the exact discrete Fourier transforms (DFT), both the direct and the inverse ones, of a continuous function g given by its values gj at the points of a uniform grid FN generated by conjugacy classes of elements of finite adjoint order N in the fundamental region F of compact semisimple Lie groups. The present implementation of the method is for the groups SU(2), when F is reduced to a one-dimensional segment, and for SU(2)×SU(2)×⋯×SU(2) in multidimensional cases. This simplest case turns out to be a version of the discrete cosine transform (DCT). Implementations, abbreviated as DGT for Discrete Group Transform, based on simple Lie groups of higher ranks, are to be considered separately. DCT is often taken to be simply a specific type of the standard DFT. Here we show that the DCT is very different from the standard DFT when the properties of the continuous extensions of the two inverse discrete transforms are studied. The following properties of the continuous extension of DCT (called CEDCT) from the discrete tj∈FN to all t∈F are proven and exemplified. Like the standard DFT, the DCT also returns the exact values of {gj} on the N+1 points of the grid. However, unlike the continuous extension of the standard DFT: (a) The CEDCT function fN(t) closely approximates g(t) between the points of the grid as well; (b) for increasing N, the derivative of fN(t) converges to the derivative of g(t); (c) for CEDCT the principle of locality is valid. In this article we also use the continuous extension of the two-dimensional (2D) DCT, SU(2)×SU(2), to illustrate its potential for interpolation as well as for the data compression of 2D images.

Fourier analysis of the sampling characteristics of the phase-shifting algorithm

Abstract: In this paper, we analyze the phase-shifting algorithm by utilizing Fourier transformation theory. In fact, the phase-shifting algorithm is corresponding to a discrete Fourier transformation (DFT), the image capturing operation is a temporal sampling procedure, and the purpose of phase-shifting technique is to retrieve the phase of one frequency component. According to the sampling theory, if the number of phase steps is too less that means a too low sampling frequency is adopted, the frequency of interest will be mixed with high order spectra. By mathematical analysis, the relationship between the number of phase steps and the effect of harmonics is deduced, and the criterion of selecting phase step number is discussed. The applications of the Fourier analysis in digital fringe projection profilometry are described.

Fast Fourier transform for discontinuous functions

Abstract: In computational electromagnetics and other areas of computational science and engineering, Fourier transforms of discontinuous functions are often required. We present a fast algorithm for the evaluation of the Fourier transform of piecewise smooth functions with uniformly or nonuniformly sampled data by using a double interpolation procedure combined with the fast Fourier transform (FFT) algorithm. We call this the discontinuous FFT algorithm. For N sample points, the complexity of the algorithm is O(/spl nu/Np+/spl nu/Nlog(N)) where p is the interpolation order and /spl nu/ is the oversampling factor. The method also provides a new nonuniform FFT algorithm for continuous functions. Numerical experiments demonstrate the high efficiency and accuracy of this discontinuous FFT algorithm.

A new transform for 2-D signal representation (MRT) and some of its properties

Abstract: A new transform (MRT) for two-dimensional signal representation and some of its properties are proposed in this paper. The transform helps to do the frequency domain analysis of two-dimensional signals without any complex operations but in terms of real additions. It is obtained by exploiting the periodicity and symmetry of the exponential term in the discrete Fourier transform (DFT) relation, and by grouping related data. Forward and inverse relations of the transform are presented. The transform coefficients show useful redundancies among themselves. These redundancies, which can be used to implement the transform, are studied. A few properties of the transform are studied and the relevant relations are presented.

Comments on Hilbert Transform Based Signal Analysis

Abstract: The use of the Hilbert transform for time/frequency analysis of signals is briefly considered. While the Hilbert transform is based on arbitrary continuous signals, most practical signals are digitially sampled and time-limited. To avoid aliasing in the sampling process the signals must also be bandlimited. It is argued that it is reasonable to consider such sampled signals as periodic (this is the basis of the Discrete Fourier Transform [DFT]) since any other interpretation is inconsistent. A simple derivation of the Hilbert transform for a sampled, periodic is then given. It is shown that the instantaneous frequency can be easily computed from the Discrete Fourier Series (or, equivalently, the DFT) representation of the signal. Since this representation is exact, the Hilbert transform representation is also exact.

An algebraic approach to symmetry detection

Abstract: We present an algorithm for detecting cyclic and dihedral symmetries of an object. Both symmetry types can be detected by the special patterns they generate in the object's Fourier transform. These patterns are effectively detected and analyzed using the "angular difference function" (ADF), which measures the difference in the angular content of images. The ADF is accurately computed by using the pseudo-polar Fourier transform, which rapidly computes the Fourier transform of an object on a near-polar grid. The algorithm detects all the axes of centered and non-centered symmetries. The proposed algorithm is algebraically accurate and uses no interpolations.

Fourier fringe processing by use of an interpolated Fourier-transform technique

Abstract: Recently a powerful Fourier transform technique was introduced that was able to extract the phase from only one image. However, because the method is based on the two-dimensional Fourier transform, it inherently suffers from leakage effects. A novel procedure is proposed that does not exhibit this distortion. The procedure uses localized information and estimates both the unknown frequencies and the phases of the fringe pattern (using an interpolated fast Fourier transform method). This allows us to demodulate the fringe pattern without any distortion. The proposed technique is validated on both computer simulations and the profile measurements of a tube.

Filtering of chirped ultrasound echo signals with the fractional Fourier transform

Abstract: The fractional Fourier transform represents a generalisation of the conventional Fourier transform. Previous work (M. Bennett et al, Proc. IEEE EMBS vol. 1, pp. 882-885, 2003) has shown that the application of the fractional Fourier transform to conventional, un-coded ultrasound signals has little advantage over conventional filtering techniques such as band-pass filtering. However, the fractional Fourier transform can be 'tuned' to be sensitive to signals of a particular chirp rate (C. Capus et al, IEE Seminar on Time-Scale and Time-Freq. Analysis and Appl., 2000) and can achieve levels of pulse compression similar to those obtained using a matched filter. To this end a system was developed which could generate and transmit linear chirp coded ultrasound signals. The fractional Fourier transform was then used to process the signals received from a simple phantom arrangement. When the transform was used with the optimum transform order corresponding to the chirp rate of the signals, the transform domain signals demonstrated a degree of pulse compression similar to that given by a matched filter. Results are also presented which demonstrate that a chirp signal identified in the fractional Fourier domain may be completely recovered in the time domain through the use of the inverse transform. Matched filtering was found to give a greater degree of pulse compression, but the fraction Fourier method can be applied without a-priori knowledge of the transmitted signal. Further work will be carried out to determine the best way of extracting useful information from the fractional domain signals.

On the orthogonality of anti-leakage Fourier transform based seismic trace interpolation

Abstract: Summary Seismic data regularization, which aims to estimate the seismic traces on a spatially regular grid using the a cquired irregular sampled data, is an interpolation/extrapolation problem. Sampling theory offers the basic conditions for all the seismic data regularization implementations. In sampling theory, Fourier transform plays a crucial role in the analysis of the reconstruction/interpolation basis (interpolant); it estimates the frequency components in frequency/wave-number domain, and its inverse transform creates the seismic data on the desired regular grid. Difficulties arise from the non-orthogonality of the global Fourier basis on an irregular grid, which results in the energy from one frequency component leaks onto others. This well-known phenomenon is called “spectral leakage”. The updated Fourier transform: Anti-leakage Fourier transform (ALFT) offers to overcome the above mentioned difficulties. It estimates the spatial frequency content on a n irregularly sampled grid with significantly reduced frequency leakage. In this paper, we investigate the properties of ALFT and give an insight on how it works. The interpolants are numerically calculated and analyzed in detail. The orthogonality condition of the interpolants is discussed, which demonstrates that the ALFT data reconstruction meets the two most important interpolation conditions (e.g. orthogonal condition and unity condition). With the amplitude analysis on interpolants, the stability of ALFT algorithm is also addressed.

Signal-processing apparatus and method

Abstract: First and second coefficients are fed into a Fast Fourier Transform unit through real number input and imaginary number input portions thereof, respectively, to perform the Fast Fourier Transform of the entered first and second coefficients, thereby producing a frequency-domain coefficient vector. The Fast Fourier Transform of an input signal is performed to transform the input signal into a frequency-domain signal vector. Thereafter, the transformed signal vector is multiplied by the coefficient vector for each element, thereby providing a multiplication result. The Inverse Fast Fourier Transform of the multiplication result renders real number output and imaginary number output portions of the inverse transformation result as first and second series of output signals, respectively.

Fast, prime factor, discrete Fourier transform

Abstract: In this paper it is shown that Winograds algorithm for computing convolutions and a fast, prime factor, discrete Fourier transform (DFT) algorithm can be modified to compute Fourier-like transforms of long sequences of 2 m

Seismic Data Regularization with Anti-Leakage Fourier Transform

Abstract: D032 Seismic Data Regularization with Anti-leakage Fourier Transform SHENG XU and DON PHAM Abstract 1 Veritas DCG Inc. Town Park Drive Houston Tx 77072 USA In the theory of Fourier reconstruction from discrete seismic data it aims to estimate the spatial frequency content on an irregularly sampled grid. After obtaining the Fourier coefficients the data can be reconstructed on any desired grid. For this type of transform difficulties arise from the non-orthogonality of the global basis functions on an irregular grid. As a consequence energy from one Fourier coefficient leaks onto other coefficients. This well-known phenomenon is called “spectral leakage”.

Speech analysis with the fast chirp transform

Abstract: The Chirp transform is a powerful analysis tool for variable frequency signals such as speech. The computational load represents the main limitation of its original formulation, discouraging its use in real-time applications. This paper analyzes a fast implementation, based on performing time-warping on the signal under analysis, combined with the Fast Fourier Transform. The performance of the Fast Chirp transform depends on the one hand on the estimation of the time-warping operation based on the signal characteristics, and, on the other hand on the interpolation technique used for the warping. Observations from the analysis of speech signals support the method and the further lines.

The centered discrete fractional Fourier transform and linear chirp signals

Abstract: The basis functions for the fractional Fourier transform are chirp signals where a precise relationship between the fractional parameter and the chirp angle can be established. The recently introduced centered discrete fractional Fourier transform, based on the Gr/spl uml/nbaum commuting matrix, has basis functions that have a sigmoidal instantaneous frequency and produces a transform that is approximately an impulse for discrete chirps. However, no such precise relation between the fractional parameter and the chirp rate of the basis functions exists in the discrete case. We study the relationship between the chirp rate and the fractional parameter in the discrete case and specifically look at two approximate expressions that relate the chirp rate and the angle for which one obtains an impulse-like transform. We study the efficacy of these estimates by applying them to the analysis of monocomponent and two component chirp signals.

Robust adaptive local polynomial Fourier transform

Abstract: A robust form of the local polynomial Fourier transform (LPFT) is introduced. This transform can produce a highly concentrated time-frequency (TF) representation for signals embedded in an impulse noise. Calculation of the adaptive parameter in the proposed transform is based on the concentration measure. A modified form, calculated as a weighted sum of the robust LPFT, is proposed for multicomponent signals.

Quantization Error Reduction in the Measurement of Fourier Intensity for Phase Retrieval

Abstract: The quantization error in the measurement of Fourier intensity for phase retrieval is discussed and a multispectra method is proposed to reduce this error. The Fourier modulus used for phase retrieval is usually obtained by measuring Fourier intensity with a digital device. Therefore, quantization error in the measurement of Fourier intensity leads to an error in the reconstructed object when iterative Fourier transform algorithms are used. The multispectra method uses several Fourier intensity distributions for a number of measurement ranges to generate a Fourier intensity distribution with a low quantization error. Simulations show that the multispectra method is effective in retrieving objects with real or complex distributions when the iterative hybrid input-output algorithm (HIO) is used.