Showing papers on "Discrete-time Fourier transform published in 2002"
Book•
06 Nov 2002
TL;DR: The Breadth and Depth of DSP Statistics, Probability and Noise ADC and DAC DSP Software Linear Systems Convolution Properties of Convolution The Discrete Fourier Transform Applications of the DFT Fourier transform Properties Fouriertransform Pairs The Fast Fouriers Transform Continuous Signal Processing Introduction to Digital Filters
Abstract: The Breadth and Depth of DSP Statistics, Probability and Noise ADC and DAC DSP Software Linear Systems Convolution Properties of Convolution The Discrete Fourier Transform Applications of the DFT Fourier Transform Properties Fourier Transform Pairs The Fast Fourier Transform Continuous Signal Processing Introduction to Digital Filters Moving Average Filters Windowed-Sinc Filters Custom Filters FFT Convolution Recursive Filters Chebyshev Filters Filter Comparison Audio Processing Image Formation and Display Linear Image Processing Special Imaging Techniques Neural Networks (and more!) Data Compression Digital Signal Processors Getting Started with DSPs Complex Numbers The Complex Fourier Transform The Laplace Transform The z-Transform Index
594 citations
TL;DR: In this article, the authors present an assortment of both standard and advanced Fourier techniques that are useful in the analysis of astrophysical time series of very long duration, where the observation time is much greater than the time resolution of the individual data points.
Abstract: We present an assortment of both standard and advanced Fourier techniques that are useful in the analysis of astrophysical time series of very long duration—where the observation time is much greater than the time resolution of the individual data points. We begin by reviewing the operational characteristics of Fourier transforms of time-series data, including power-spectral statistics, discussing some of the differences between analyses of binned data, sampled data, and event data, and we briefly discuss algorithms for calculating discrete Fourier transforms (DFTs) of very long time series. We then discuss the response of DFTs to periodic signals and present techniques to recover Fourier amplitude "lost" during simple traditional analyses if the periodicities change frequency during the observation. These techniques include Fourier interpolation, which allows us to correct the response for signals that occur between Fourier frequency bins. We then present techniques for estimating additional signal properties such as the signal's centroid and duration in time, the first and second derivatives of the frequency, the pulsed fraction, and an overall estimate of the significance of a detection. Finally, we present a recipe for a basic but thorough Fourier analysis of a time series for well-behaved pulsations.
371 citations
TL;DR: In this article, the DC-FFT algorithm was used to analyze the contact stresses in an elastic body under pressure and shear tractions for high efficiency and accuracy, and a set of general formulas of the frequency response function for the elastic field was derived and verified.
Abstract: The knowledge of contact stresses is critical to the design of a tribological element. It is necessary to keep improving contact models and develop efficient numerical methods for contact studies, particularly for the analysis involving coated bodies with rough surfaces. The fast Fourier Transform technique is likely to play an important role in contact analyses. It has been shown that the accuracy in an algorithm with the fast Fourier Transform is closely related to the convolution theorem employed. The algorithm of the discrete convolution and fast Fourier Transform, named the DC-FFT algorithm includes two routes of problem solving: DC-FFT/Influence coefficients/Green's, function for the cases with known Green's functions and DC-FFT/Influence coefficient/conversion, if frequency response functions are known. This paper explores the method for the accurate conversion for influence coefficients from frequency response functions, further improves the DC- FFT algorithm, and applies this algorithm to analyze the contact stresses in an elastic body under pressure and shear tractions for high efficiency and accuracy. A set of general formulas of the frequency response function for the elastic field is derived and verified. Application examples are presented and discussed.
265 citations
01 Jan 2002
TL;DR: This report tries to give a practical overview about the estimation of power spectra/power spectral densities using the DFT/FFT and includes a detailed list of common and useful window functions, among them the often neglected flat-top windows.
Abstract: This report tries to give a practical overview about the estimation of power spectra/power spectral densities using the DFT/FFT. One point that is emphasized is the relationship between estimates of power spectra and power spectral densities which is given by the effective noise bandwidth (ENBW). Included is a detailed list of common and useful window functions, among them the often neglected flat-top windows. Special highlights are a procedure to test new programs, a table of comprehensive graphs for each window and the introduction of a whole family of new flat-top windows that feature sidelobe suppression levels of up to −248dB, as compared with −90dB of the best flat-top windows available until now.
262 citations
Posted Content•
TL;DR: In this article, an approximate Fourier transform on L$L$ elements is defined, which is computationally attractive in a certain setting, and which may find application to the problem of factoring integers with a quantum computer.
Abstract: We define an approximate version of the Fourier transform on $2^L$ elements, which is computationally attractive in a certain setting, and which may find application to the problem of factoring integers with a quantum computer as is currently under investigation by Peter Shor. (1994 IBM Internal Report)
193 citations
TL;DR: A new architecture is proposed that encodes a primary image to white noise based on iterative fractional Fourier transform that can provide additional keys for encryption to make the code more difficult to break.
174 citations
TL;DR: In this article, a concept of integer fast Fourier transform (IntFFT) for approximating the discrete Fourier Transform (DFT) is introduced, where the lifting scheme is used to approximate complex multiplications appearing in the FFT lattice structures.
Abstract: A concept of integer fast Fourier transform (IntFFT) for approximating the discrete Fourier transform is introduced. Unlike the fixed-point fast Fourier transform (FxpFFT), the new transform has the properties that it is an integer-to-integer mapping, is power adaptable and is reversible. The lifting scheme is used to approximate complex multiplications appearing in the FFT lattice structures where the dynamic range of the lifting coefficients can be controlled by proper choices of lifting factorizations. Split-radix FFT is used to illustrate the approach for the case of 2/sup N/-point FFT, in which case, an upper bound of the minimal dynamic range of the internal nodes, which is required by the reversibility of the transform, is presented and confirmed by a simulation. The transform can be implemented by using only bit shifts and additions but no multiplication. A method for minimizing the number of additions required is presented. While preserving the reversibility, the IntFFT is shown experimentally to yield the same accuracy as the FxpFFT when their coefficients are quantized to a certain number of bits. Complexity of the IntFFT is shown to be much lower than that of the FxpFFT in terms of the numbers of additions and shifts. Finally, they are applied to noise reduction applications, where the IntFFT provides significantly improvement over the FxpFFT at low power and maintains similar results at high power.
165 citations
TL;DR: Wave-front reconstruction with the use of the fast Fourier transform (FFT) and spatial filtering is shown to be computationally tractable and sufficiently accurate for use in large Shack-Hartmann-based adaptive optics systems.
Abstract: Wave-front reconstruction with the use of the fast Fourier transform (FFT) and spatial filtering is shown to be computationally tractable and sufficiently accurate for use in large Shack–Hartmann-based adaptive optics systems (up to at least 10,000 actuators). This method is significantly faster than, and can have noise propagation comparable with that of, traditional vector–matrix-multiply reconstructors. The boundary problem that prevented the accurate reconstruction of phase in circular apertures by means of square-grid Fourier transforms (FTs) is identified and solved. The methods are adapted for use on the Fried geometry. Detailed performance analysis of mean squared error and noise propagation for FT methods is presented with the use of both theory and simulation.
158 citations
Book•
08 Mar 2002
TL;DR: In this article, the authors present a mathematical representation of a sum of sinusoidal signals and their properties, including the Fourier Transform and the Spectrum, as well as its properties and properties.
Abstract: 1. Introduction. Mathematical Representation of Signals. Mathematical Representation of Systems. Thinking about Systems. 2. Sinusoids. Tuning Fork Experiment. Review of Sine and Cosine Functions. Sinusoidal Signals. Sampling and Plotting Sinusoids. Complex Exponentials and Phasors. Phasor Addition. Physics of the Tuning Fork. Time Signals: More Than Formulas. 3. Spectrum Representation. The Spectrum of a Sum of Sinusoids. Beat Notes. Periodic Waveforms. More Periodic Signals. Fourier Series Analysis and Synthesis. Time-Frequency Spectrum. Frequency Modulation: Chirp Signals. 4. Sampling and Aliasing. Sampling. Spectrum View of Sampling and Reconstruction. Strobe Demonstration. Discrete-to-Continuous Conversion. The Sampling Theorem. 5. FIR Filters. Discrete-Time Systems. The Running Average Filter. The General FIR Filter. Implementation of FIR Filters. Linear Time-Invariant (LTI) Systems. Convolution and LTI Systems. Cascaded LTI Systems. Example of FIR Filtering. 6. Frequency Response of FIR Filters. Sinusoidal Response of FIR Systems. Superposition and the Frequency Response. Steady State and Transient Response. Properties of the Frequency Response. Graphical Representation of the Frequency Response. Cascaded LTI Systems. Running-Average Filtering. Filtering Sampled Continuous-Time Signals. 7. z-Transforms. Definition of the z-Transform. The z-Transform and Linear Systems. Properties of the z-Transform. The z-Transform as an Operator. Convolution and the z-Transform. Relationship between the z -Domain and the w-Domain. Useful Filters. Practical Bandpass Filter Design. Properties of Linear Phase Filters. 8. IIR Filters. The General IIR Difference Equation. Time-Domain Response. System Function of an IIR Filter. Poles and Zeros. Frequency Response of an IIR Filter. Three Domains. The Inverse z-Transform and Some Applications. Steady-State Response and Stability. Second-Order Filters. Frequency Response of Second-Order IIR Filter. Example of an IIR Lowpass Filter. 9. Continuous-Time Signals and LTI Systems. Continuous-Time Signals. The Unit Impulse. Continuous-Time Systems. Linear Time-Invariant Systems. Impulse Responses of Basic LTI Systems. Convolution of Impulses. Evaluating Convolution Integrals. Properties of LTI Systems. Using Convolution to Remove Multipath Distortion. 10. The Frequency Response. The Frequency Response Function for LTI Systems. Response to Real Sinusoidal Signals. Ideal Filters. Application of Ideal Filters. Time-Domain or Frequency-Domain? 11. Continuous-Time Fourier Transform. Definition of the Fourier Transform. The Fourier Transform and the Spectrum. Existence and Convergence of the Fourier Transform. Examples of Fourier Transform Pairs. Properties of Fourier Transform Pairs. The Convolution Property. Basic LTI Systems. The Multiplication Property. Table of Fourier Transform Properties and Pairs. Using the Fourier Transform for Multipath Analysis. 12. Filtering, Modulation, and Sampling. Linear Time-Invariant Systems. Sinewave Amplitude Modulation. Sampling and Reconstruction. 13. Computing the Spectrum. Finite Fourier Sum. Too Many Fourier Transforms? Time-windowing. Analysis of a Sum of Sinusoids. Discrete Fourier Transform. Spectrum Analysis of Finite-Length Signals. Spectrum Analysis of Periodic Signals. The Spectrogram. The Fast Fourier Transform (FFT). Appendix A: Complex Numbers. Notation for Complex Numbers. Euler's Formula. Algebraic Rules for Complex Numbers. Geometric Views of complex Operations. Powers and Roots. Appendix B: Programming in MATLAB. MATLAB Help. Matrix Operations and Variables. Plots and Graphics. Programming Constructs. MATLAB Scripts. Writing a MATLAB Function. Programming Tips. Appendix C: Laboratory Projects. Introduction to MATLAB. Encoding and Decoding Touch-Tone Signals. Two Convolution GUIs. Appendix D: CD-ROM Demos. Index.
152 citations
TL;DR: In this article, a theory of Fourier coefficients for modular forms on the split exceptional group G2 over ℚ was developed, where the coefficients are derived from the Fourier coefficient theory of modular forms.
Abstract: We develop a theory of Fourier coefficients for modular forms on the split exceptional group G2 over ℚ.
118 citations
TL;DR: A simple alternative procedure to reduce leakage in the Fourier spectrum of a periodic signal is proposed and results obtained are empirically analyzed and compared with those given by an instrument with built-in FFT capabilities.
Abstract: The Fourier spectrum of a periodic signal may be obtained by fast Fourier transform algorithms, but, as is well known, special care must be taken to avoid severe distortions introduced by the sampling process. The main problem is the leakage generated by the truncation required to obtain a finite length sampled data. The usual procedure to reduce leakage is to multiply the sampled signal by a weighting window. Several kinds of windows have been proposed in the literature, and today they are also included in many commercial instruments. A simple alternative procedure is proposed in this paper. It is implemented with a PC compatible data acquisition board (DAQ) and consists of an algorithm that uses decimation and interpolation techniques. This algorithm is equivalent to the use of an adjustable sampling frequency and correspondingly an adjustable window size. Results obtained by this method on both harmonic and polyharmonic signals are empirically analyzed and compared with those given by an instrument with built-in FFT capabilities.
TL;DR: In this paper, the authors describe improvements in the implementation of the mixed Fourier transform, which make the method more robust and efficient and avoid potential numerical instabilities, which occasionally caused problems in the previous implementation.
Abstract: [1] A standard method for modeling electromagnetic propagation in the troposphere is the Fourier split-step algorithm for solving the parabolic wave equation. An important advance in this technique was the introduction of the mixed Fourier transform, which permitted the extension of the method from propagation over only smooth perfectly conducting surfaces to quite general surfaces with impedance boundary conditions. This paper describes improvements in the implementation of the mixed Fourier transform, which make the method more robust and efficient and avoid potential numerical instabilities, which occasionally caused problems in the previous implementation. Some examples are also presented.
Book•
01 Aug 2002
TL;DR: The z Transform and Its Properties are compared to Solving Linear Differential Equation and Eigenvalues in Digital Signal Processing, which is a very simple and straightforward way to model the dynamic response of a discrete-time system.
Abstract: Preface. 1. Introduction to Linear Systems. 1.1 Continuous and Discrete Linear Systems and Signals. 1.2 System Linearity and Time Invariance. 1.3 Mathematical Modeling of Systems. 1.4 System Classification. 1.5 MATLAB System Computer Analysis and Design. 1.6 Book Organization. 1.7 Chapter One Summary. 1.8 References. 1.9 Problems. 2. Introduction to Signals. 2.1 Common Signals in Linear Systems. 2.2 Signal Operations. 2.3 Signal Classification. 2.4 MATLAB Laboratory Experiment on Signals. 2.5 Chapter Two Summary. 2.6 References. 2.7 Problems. I. FREQUENCY DOMAIN TECHNIQUES. 3. Fourier Series and Fourier Transform. 3.1 Fourier Series. 3.2 Fourier Transform and Its Properties. 3.3 Fourier Transform in System Analysis. 3.4 Fourier Series in Systems Analysis. 3.5 From Fourier Transform to Laplace Transform. 3.6 Fourier Analysis MATLAB Laboratory Experiment. 3.7 Chapter Three Summary. 3.8 References. 3.9 Problems. 4. Laplace Transform. 4.1 Laplace Transform and Its Properties. 4.2 Inverse Laplace Transform. 4.3 Laplace Transform in Linear System Analysis. 4.4 Block Diagrams. 4.5 From Laplace to the z-Transform. 4.6 MATLAB Laboratory Experiment. 4.7 Chapter Four Summary. 4.8 References. 4.9 Problems. 5. The z Transform. 5.1 The z Transform and Its Properties. 5.2 Inverse of the z Transform. 5.3 The z Transform in Linear System Analysis. 5.4 Block Diagram. 5.5 Discrete-Time Frequency Spectra. 5.6 MATLAB Laboratory Experiment. 5.7 Chapter Five Summary. 5.8 References. 5.9 Problems. II. TIME DOMAIN TECHNIQUES. 6. Convolution. 6.1 Convolution of Continuous-Time Signals. 6.2 Convolution for Linear Continuous-Time Systems. 6.3 Convolution of Discrete-Time Signals. 6.4 Convolution for Linear Discrete-Time Systems. 6.5 Numerical Convolution Using MATLAB. 6.6 MATLAB Laboratory Experiments on Convolution. 6.7 Chapter Six Summary. 6.8 References. 6.9 Problems. 7. System Response in Time Domain. 7.1 Solving Linear Differential Equations. 7.2 Solving Linear Difference Equations. 7.3 Discrete-Time System Impulse Response. 7.4 Continuous-Time System Impulse Response. 7.5 Complete Continuous-Time System Response. 7.6 Complete Discrete-Time System Response. 7.7 Stability of Continuous-Time Linear Systems. 7.8 Stability of Discrete-Time Linear Systems. 7.9 MATLAB Experiment on Continuous-Time Systems. 7.10 MATLAB Experiment on Discrete-Time Systems. 7.11 Chapter Seven Summary. 7.12 References. 7.13 Problems. 8. State Space Approach. 8.1 State Space Models. 8.2 Time Response from the State Equation. 8.3 Discrete-Time Models. 8.4 System Characteristic Equation and Eigenvalues. 8.5 Cayley-Hamilton Theorem. 8.6 Linearization of Nonlinear System. 8.7 State Space MATLAB Laboratory Experiments. 8.8 Chapter Eight Summary. 8.9 References. 8.10 Problems. III. SYSTEMS IN ELECTRICAL ENGINEERING. 9. Signals in Digital Signal Processing. 9.1 Sampling Theorem. 9.2 Discrete-Time Fourier Transform (DFDT). 9.3 Double Sided z-Transform. 9.4 Discrete Fourier Transform. 9.5 Discrete-Time Fourier Series. 9.6 Correlation of Discrete-Time Signals. 9.7 FIR and IIR Filters. 9.8 Laboratory Experiment on Digital Signal Processing. 9.9 Chapter Nine Summary. 9.10 References. 9.11 Problems. 10. Signals in Communication Systems. 10.1 Signal Transmission in Communications. 10.2 Signal Correlation, Energy and Power Spectra. 10.3 Hilbert Transform. 10.4 Ideal Filter. 10.5 Modulation and Demodulation. 10.6 Digital Communication System. 10.7 Communication Systems Laboratory Experiment. 10.8 Chapter Ten Summary. 10.9 References. 10.10 Problems. 11. Linear Electric Circuits. 11.1 Basic Relations. 11.2 First-Order Linear Electrical Circuits. 11.3 Second-Order Linear Electrical Circuits. 11.4 Higher-Order Linear Electrical Circuits. 11.5 Chapter Eleven Summary. 11.6 References. 11.7 MATLAB Laboratory Experiment. 11.8 Problems. 12. Linear Controls Systems. 12.1 The Essence of Feedback. 12.2 Transient Response of Second-Order Systems. 12.3 Feedback System Steady State Errors. 12.4 Feedback System Frequency Characteristics. 12.5 Bode Diagrams. 12.6 Common Dynamic Controllers: PD, PI, PID. 12.7 Laboratory Experiment on Control Systems. 12.8 Chapter Twelve Summary. 12.9 References. 12.10 Problems. Appendices. A. Linear Algebra. B. Some Results from Calculus. C. Introduction to MATLAB. D. Introduction to SIMULINK. Index.
TL;DR: In this paper, the static response of beam-and plate-like repetitive lattice structures is obtained by discrete Fourier transform, and the governing equation is set up as a single operator form with the physical stiffness operator acting as a convolution sum and containing a matrix kernel.
TL;DR: Raw amplitude and time-of-flight patterns acquired from a real sonar system are processed, demonstrating reduced error in both recognition and position estimation of objects.
TL;DR: In this paper, the Fourier amplitude spectrum is modelled as a scaled, lognormal probability density function, which is defined by modeling the probability distribution of Fourier phase differences conditional on the amplitude.
Abstract: Acceleration time histories of horizontal earthquake ground motion are obtained by inverting the discrete Fourier transform, which is defined by modelling the probability distribution of the Fourier phase differences conditional on the Fourier amplitude.
The Fourier amplitude spectrum is modelled as a scaled, lognormal probability density function. Three parameters are necessary to define the Fourier amplitude spectrum. They are the total energy of the accelerogram, the central frequency, and the spectral bandwidth.
The Fourier phase differences are simulated conditional on the Fourier amplitudes. The amplitudes are classified into three categories: small, intermediate and large. For each amplitude category, a beta distribution or a combination of a beta distribution and a uniform distribution are defined for the phase differences. Seven parameters are needed to completely define the phase difference distributions: two for each of the three beta distributions, and the weight of the uniform distribution for phase differences corresponding to small Fourier amplitudes.
Approximately 300 uniformly processed horizontal ground motion records from recent California earthquakes are used to develop prediction formulas for the model parameters, as well as to validate the simulation model. The moment magnitude of the earthquakes ranges from 5.8 to 7.3. The source to site distance for all the records is less than 100 km. Copyright © 2002 John Wiley & Sons, Ltd.
Patent•
18 Oct 2002TL;DR: In this article, a method for coding in frequency, module and phase a digital representation, in the space field, of a ring-shaped element, including the steps of: applying to any point of the element a polar conversion at constant angle, whereby the element is unfolded in rectangular form; transferring, to the frequency field, any points of the converted rectangular shape by means of a Fourier transform; filtering the discrete data resulting from the transfer by at least one real, bidimensional, band-pass filter, oriented along the phase axis; applying a Hilbert transform to the filtering results
Abstract: A method for coding in frequency, module and phase a digital representation, in the space field, of a ring-shaped element, including the steps of: applying to any point of the element a polar conversion at constant angle, whereby the element is unfolded in rectangular form; transferring, to the frequency field, any point of the converted rectangular shape by means of a Fourier transform; filtering the discrete data resulting from the transfer by means of at least one real, bidimensional, band-pass filter, oriented along the phase axis; applying a Hilbert transform to the filtering results; applying an inverse Fourier transform to the results of the Hilbert transform; and extracting phase and module information in the space field.
TL;DR: In this paper, the Fourier transform is considered as a Henstock/Kurzweil integral and sufficient conditions for its existence are given for continuous Fourier transformation. But the Riemann-Lebesgue lemma fails.
Abstract: The Fourier transform is considered as a Henstock/Kurzweil integral. Sufficient conditions are given for the existence of the Fourier transform and necessary and sufficient conditions are given for it to be continuous. The Riemann-Lebesgue lemma fails: Henstock/Kurzweil Fourier transforms can have arbitrarily large point-wise growth. Convolution and inversion theorems are established. An appendix gives sufficient conditions for interchanging repeated Henstock/Kurzweil integrals and gives an estimate on the integral of a product. AMS (MOS) subject classification: 42A38, 26A39
01 Apr 2002
TL;DR: In this paper, the authors introduce the notion of fractional Fourier transform, which may be considered as a fractional power of the classical Fourier trans-form, and give an introduction to the denition, the properties and computational aspects of both the continuous and discrete Fourier transforms.
Abstract: In this survey paper we introduce the reader to the notion of the fractional Fourier transform, which may be considered as a fractional power of the classical Fourier trans- form. It has been intensely studied during the last decade, an attention it may have partially gained because of the vivid interest in time-frequency analysis methods of signal processing, like wavelets. Like the complex exponentials are the basic functions in Fourier analysis, the chirps (signals sweeping through all frequencies in a certain interval) are the building blocks in the fractional Fourier analysis. Part of its roots can be found in optics where the fractional Fourier transform can be physically realized. We give an introduction to the denition, the properties and computational aspects of both the continuous and discrete fractional Fourier transforms. We include some examples of applications and some possible generalizations.
TL;DR: It is shown that the fractional Fourier transform is a suitable mechanism with which to analyze the diffraction patterns produced by a one-dimensional object because its intensity distribution is partially described by a linear chirp function.
Abstract: We show that the fractional Fourier transform is a suitable mechanism with which to analyze the diffraction patterns produced by a one-dimensional object because its intensity distribution is partially described by a linear chirp function. The three-dimensional location and the diameter of a fiber can be determined, provided that the optimal fractional order is selected. The effect of compaction of the intensity distribution in the fractional Fourier domain is discussed. A few experimental results are presented.
TL;DR: In this article, the Fourier transform of the Vlasov equation analytically in velocity space was studied and the resulting equation was solved numerically by using outgoing wave boundary conditions.
TL;DR: In this paper, a numerical method for computing a function, given its Laplace transform function on the real axis, was proposed, which is based on the Fourier series expansion of the unknown function and approximated using a Tikhonov regularization method.
Abstract: We propose a numerical method for computing a function, given its Laplace transform function on the real axis. The inversion algorithm is based on the Fourier series expansion of the unknown function and the Fourier coefficients are approximated using a Tikhonov regularization method. The key point of this approach is the use of the regularization scheme in order to improve the conditioning of the discrete problem: the value of the regularization parameter is that giving a tradeoff between the discretization error, including the regularization error, and the conditioning of the discrete problem.
TL;DR: An alternative method for evaluating the Fourier transform, based on the chirp z transform, is discussed, by which a considerable improvement in efficiency can be obtained without loss of accuracy.
Abstract: Computation of the readout signal of an optical disk involves Fourier transforms from the objective lens pupil to the disk and, after interaction with the disk, from the disk to the objective pupil. Traditionally, the complex two-dimensional Fourier transform is numerically evaluated as a two-dimensional fast Fourier transform. To obtain sufficient resolution in the involved planes, one must choose sampling grid sizes of typically 1024 × 1024 or higher, resulting in a substantial computation time if the calculation is to be repeated many times. Discussed is an alternative method for evaluating the Fourier transform, based on the chirp z transform, by which a considerable improvement in efficiency can be obtained without loss of accuracy.
11 Aug 2002
TL;DR: The method presented in this contribution provides accurate approximations of the continuous Fourier transform with similar time complexity and allows to compute numerical Fourier transforms in a broader domain of frequency than the usual half-period of the DFT.
Abstract: The classical method of numerically computing the Fourier transform of digitized functions in one or in d-dimensions is the so-called discrete Fourier transform (DFT), efficiently implemented as Fast Fourier Transform (FFT) algorithms. In many cases the DFT is not an adequate approximation of the continuous Fourier transform. The method presented in this contribution provides accurate approximations of the continuous Fourier transform with similar time complexity. The assumption of signal periodicity is no longer posed and allows to compute numerical Fourier transforms in a broader domain of frequency than the usual half-period of the DFT. In image processing this behavior is highly welcomed since it allows to obtain the Fourier transform of an image without the usual interferences of the periodicity of the classical DFT. The mathematical method is developed and numerical examples are presented.
TL;DR: In this article, the Fourier transform of the extended Lorentzian energy distribution becomes the exponential, but only for times $tg~0,$ a time asymmetry which is in conflict with the unitary group time evolution of standard quantum mechanics.
Abstract: The Fourier transform is often used to connect the Lorentzian energy distribution for resonance scattering to the exponential time dependence for decaying states. However, to apply the Fourier transform, one has to bend the rules of standard quantum mechanics; the Lorentzian energy distribution must be extended to the full real axis $\ensuremath{-}\ensuremath{\infty}lEl\ensuremath{\infty}$ instead of being bounded from below $0l~El\ensuremath{\infty}$ (Fermi's approximation). Then the Fourier transform of the extended Lorentzian becomes the exponential, but only for times $tg~0,$ a time asymmetry which is in conflict with the unitary group time evolution of standard quantum mechanics. Extending the Fourier transform from distributions to generalized vectors, we are led to Gamow kets, which possess a Lorentzian energy distribution with $\ensuremath{-}\ensuremath{\infty}lEl\ensuremath{\infty}$ and have exponential time evolution for $tg~{t}_{0}=0$ only. This leads to probability predictions that do not violate causality.
TL;DR: In this paper, a Fourier transform based algorithm for the reconstruction of functions from their nonstandard sampled Radon transform is proposed, which incorporates recently developed fast Fourier transforms for nonequispaced data.
Abstract: In this paper, we suggest a new Fourier transform based algorithm forthe reconstruction of functions from their nonstandard sampled Radon transform. The algorithm incorporates recently developed fast Fourier transforms for nonequispaced data. We estimate the-corresponding aliasing error in dependence on the sampling geometry of the Radon transform and confirm our theoretical results by numerical examples.
TL;DR: An improved format for Shah convolution Fourier transform (SCOFT) detection that utilizes the spatial resolution of a charge-coupled device rather than a fixed optical mask to perform a Shah or sine convolution over a fluorescence signal is described.
Abstract: This paper describes an improved format for Shah convolution Fourier transform (SCOFT) detection that utilizes the spatial resolution of a charge-coupled device (CCD) rather than a fixed optical mask to perform a Shah or sine convolution over a fluorescence signal. The laser-induced fluorescence from a 9-mm section of microfabricated channel is collected with a CCD at 28 Hz. Each image frame is multiplied by a convolution function to modulate the collected signal through space. Each frame is then summed to generate an intensity-versus-time data set for Fourier analysis. The fluorescence signal oscillates at a frequency dependent upon both the convolution function multiplied across each data frame and the velocity of fluorescent microspheres or a plug of fluorescent dye flowing through the channel. This SCOFT technique affords more flexibility over formats that employ a physical mask and provides data that can be optimized for signal-to-noise (S/N) or resolution information. A 1000-fold improvement in S/N ...
TL;DR: In this article, the behavior of rectangular partial sums of the Fourier series of functions of several variables having bounded λ-variation is considered and it is proved that if a continuous function is also continuous in harmonic variation, then its Fourier-series uniformly converges in the sense of Pringsheim.
Abstract: The behaviour of rectangular partial sums of the Fourier series of functions of several variables having bounded {lambda}-variation is considered. It is proved that if a continuous function is also continuous in harmonic variation, then its Fourier series uniformly converges in the sense of Pringsheim. On the other hand, it is demonstrated that in dimensions greater than 2 there always exists a continuous function of bounded harmonic variation with Fourier series divergent over cubes at the origin.
TL;DR: Using the Windowed Fourier Transform, a number of fundamental problems from diffraction theory are studied using a representation of continuous wavefields by superpositions of beams that are continuously parameterised in phasespace and which propagate along ray trajectories.
Abstract: For high-frequency fields, which can be separated into superpositions of a few distinct components with rapidly varying phases and slowly varying amplitudes, phase-space representations exhibit a strong localisation in which the coefficients are negligible over most of the phase space. This leads, potentially, to a very large reduction in the computational cost of computing propagators. Using the Windowed Fourier Transform, a number of fundamental problems from diffraction theory are studied using a representation of continuous wavefields by superpositions of beams that are continuously parameterised in phase-space and which propagate along ray trajectories. The existence of noncanonical WFT coefficients is observed, due to the nonuniqeness of the WFT. Numerical evaluations require discrete finite bases. The discrete Wilson basis is generated by a discrete sampling of the windowed Fourier Transform in the phase-space. The sampling is optimal, in the sense that the smallest number of coefficients is generated in an orthogonal basis.
TL;DR: By using this Fourier-based method, the use of large filters or infinite impulse response filters in multiresolution analysis becomes manageable in terms of computation costs.
Abstract: Wavelet transforms are often calculated by using the Mallat algorithm. In this algorithm, a signal is decomposed by a cascade of filtering and downsampling operations. Computing time can be important but the filtering operations can be speeded up by using fast Fourier transform (FFT)-based convolutions. Since it is necessary to work in the Fourier domain when large filters are used, we present some results of Fourier-based optimization of the sampling operations. Acceleration can be obtained by expressing the samplings in the Fourier domain. The general equations of the down- and upsampling of digital multidimensional signals are given. It is shown that for special cases such as the separable scheme and Feauveau’s quincunx scheme, the samplings can be implemented in the Fourier domain. The performance of the implementations is determined by the number of multiplications involved in both FFT-convolution-based and Fourier-based algorithms. This comparison shows that the computational costs are reduced when the proposed implementation is used. The complexity of the algorithm is O(N log N). By using this Fourier-based method, the use of large filters or infinite impulse response filters in multiresolution analysis becomes manageable in terms of computation costs. Mesh simplification based on multiresolution “detail relevance” images illustrates an application of the implemenentation. © 2002 SPIE and IS&T.