Showing papers on "Discrete-time Fourier transform published in 1998"

Fourier analysis and applications

Abstract: Signals and Systems.- Periodic Signals.- The Discrete Fourier Transform and Numerical Computations.- The Lebesgue Integral.- Spaces.- Convolution and the Fourier Transform of Functions.- Analog Filters.- Distributions.- Convolution and the Fourier Transform of Distributions.- Filters and Distributions.- Sampling and Discrete Filters.- Current Trends: Time-Frequency Analysis.- References.

Signal processing and linear systems

Abstract: BACKGROUND B1 Complex Numbers B2 Sinusoids B3 Sketching Signals B4 Cramer's Rule B5 Partial Fraction Expansion B6 Vectors and Matrices B7 Miscellaneous CHAPTER 1 INTRODUCTION TO SIGNALS AND SYSTEMS 11 Size of a Signal 12 Classification of Signals 13 Some Useful Signal Operations 14 Some Useful Signal Models 15 Even and Odd Functions 16 Systems 17 Classification of Systems 18 System Model: Input-Output Description CHAPTER 2 TIME-DOMAIN ANALYSIS OF CONTINUOUS-TIME SYSTEMS 21 Introduction 22 System Response to Internal Conditions: Zero-Input Response 23 The Unit Impulse Response h(t) 24 System Response to External Input: Zero-State Response 25 Classical Solution of Differential Equations 26 System Stability 27 Intuitive Insights into System Behavior 28 Appendix 21: Determining the Impulse Response CHAPTER 3 SIGNAL REPRESENTATION BY FOURIER SERIES 31 Signals and Vectors 32 Signal Comparison: Correlation 33 Signal Representation by Orthogonal Signal Set 34 Trigonometric Fourier Series 35 Exponential Fourier Series 36 Numerical Computation of D[n 37 LTIC System response to Periodic Inputs 38 Appendix CHAPTER 4 CONTINUOUS-TIME SIGNAL ANALYSIS: THE FOURIER TRANSFORM 41 Aperiodic Signal Representation by Fourier Integral 42 Transform of Some Useful Functions 43 Some Properties of the Fourier Transform 44 Signal Transmission through LTIC Systems 45 Ideal and Practical Filters 46 Signal Energy 47 Application to Communications: Amplitude Modulation 48 Angle Modulation 49 Data Truncation: Window Functions CHAPTER 5 SAMPLING 51 The Sampling Theorem 52 Numerical Computation of Fourier Transform: The Discrete Fourier Transform (DFT) 53 The Fast Fourier Transform (FFT) 54 Appendix 51 CHAPTER 6 CONTINUOUS-TIME SYSTEM ANALYSIS USING THE LAPLACE TRANSFORM 61 The Laplace Transform 62 Some Properties of the Laplace Transform 63 Solution of Differential and Integro-Differential Equations 64 Analysis of Electrical Networks: The Transformed Network 65 Block Diagrams 66 System Realization 67 Application to Feedback and Controls 68 The Bilateral Laplace Transform 69 Appendix 61: Second Canonical Realization CHAPTER 7 FREQUENCY RESPONSE AND ANALOG FILTERS 71 Frequency Response of an LTIC System 72 Bode Plots 73 Control System Design Using Frequency Response 74 Filter Design by Placement of Poles and Zeros of H(s) 75 Butterworth Filters 76 Chebyshev Filters 77 Frequency Transformations 78 Filters to Satisfy Distortionless Transmission Conditions CHAPTER 8 DISCRETE-TIME SIGNALS AND SYSTEMS 81 Introduction 82 Some Useful Discrete-Time Signal Models 83 Sampling Continuous-Time Sinusoids and Aliasing 84 Useful Signal Operations 85 Examples of Discrete-Time Systems CHAPTER 9 TIME-DOMAIN ANALYSIS OF DISCRETE-TIME SYSTEMS 91 Discrete-Time System Equations 92 System Response to Internal Conditions: Zero-Input Response 93 Unit Impulse Response h[k] 94 System Response to External Input: Zero-State Response 95 Classical Solution of Linear Difference Equations 96 System Stability 97 Appendix 91: Determining Impulse Response CHAPTER 10 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS 101 Periodic Signal Representation by Discrete-Time Fourier Series 102 Aperiodic Signal Representation by Fourier Integral 103 Properties of DTFT 104 DTFT Connection with the Continuous-Time Fourier Transform 105 Discrete-Time Linear System Analysis by DTFT 106 Signal Processing Using DFT and FFT 107 Generalization of DTFT to the Z-Transform CHAPTER 11 DISCRETE-TIME SYSTEM ANALYSIS USING THE Z-TRANSFORM 111 The Z-Transform 112 Some Properties of the Z-Transform 113 Z-Transform Solution of Linear Difference Equations 114 System Realization 115 Connection Between the Laplace and the Z-Transform 116 Sampled-Data (Hybrid) Systems 117 The Bilateral Z-Transform CHAPTER 12 FREQUENCY RESPONSE AND DIGITAL FILTERS 121 Frequency Response of Discrete-Time Systems 122 Frequency Response From Pole-Zero Location 123 Digital Filters 124 Filter Design Criteria 125 Recursive Filter Design: The Impulse Invariance Method 126 Recursive Filter Design: The Bilinear Transformation Method 127 Nonrecursive Filters 128 Nonrecursive Filter Design CHAPTER 13 STATE-SPACE ANALYSIS 131 Introduction 132 Systematic Procedure for Determining State Equations 133 Solution of State Equations 134 Linear Transformation of State Vector 135 Controllability and Observability 136 State-Space Analysis of Discrete-Time Systems ANSWERS TO SELECTED PROBLEMS SUPPLEMENTARY READING INDEX Each chapter ends with a Summary

A convolution and product theorem for the fractional Fourier transform

Abstract: The fractional Fourier transform (FRFT), which is a generalization of the Fourier transform, has many applications in several areas, including signal processing and optics. Almeida (see ibid., vol.4, p.15-17, 1997) and Mendlovic et al. (see Appl. Opt., vol.34, p.303-9, 1995) derived fractional Fourier transforms of a product and of a convolution of two functions. Unfortunately, their convolution formulas do not generalize very well the classical result for the Fourier transform, which states that the Fourier transform of the convolution of two functions is the product of their Fourier transforms. This paper introduces a new convolution structure for the FRFT that preserves the convolution theorem for the Fourier transform and is also easy to implement in the designing of filters.

Discrete fractional Hartley and Fourier transforms

Abstract: This paper is concerned with the definitions of the discrete fractional Hartley transform (DFRHT) and the discrete fractional Fourier transform (DFRFT). First, the eigenvalues and eigenvectors of the discrete Fourier and Hartley transform matrices are investigated. Then, the results of the eigendecompositions of the transform matrices are used to define DFRHT and DFRFT. Also, an important relationship between DFRHT and DFRFT is described, and numerical examples are illustrated to demonstrate that the proposed DFRFT is a better approximation to the continuous fractional Fourier transform than the conventional defined DFRFT. Finally, a filtering technique in the fractional Fourier transform domain is applied to remove chirp interference.

Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters

Abstract: We provide a general treatment of optical two-dimensional fractional Fourier transforming systems. We not only allow the fractional Fourier transform orders to be specified independently for the two dimensions but also allow the input and output scale parameters and the residual spherical phase factors to be controlled. We further discuss systems that do not allow all these parameters to be controlled at the same time but are simpler and employ a fewer number of lenses. The variety of systems discussed and the design equations provided should be useful in practical applications for which an optical fractional Fourier transforming stage is to be employed.

The Nonuniform Discrete Fourier Transform and Its Applications in Signal Processing

Abstract: 1. Introduction. 2. The Nonuniform Discrete Fourier Transform. 3. 1-D Fir Filter Design Using the NDFT. 4. 2-D Fir Filter Design Using the NDFT. 5. Antenna Pattern Synthesis with Prescribed Nulls. 6. Dual-Tone Multi-Frequency Signal Decoding. 7. Conclusions. References. Index.

An application of the fourier transform to sections of star bodies

Abstract: We express the volume of central hyperplane sections of star bodies inR n in terms of the Fourier transform of a power of the radial function, and apply this result to confirm the conjecture of Meyer and Pajor on the minimal volume of central sections of the unit balls of the spacesl with 0

The simple genetic algorithm and the walsh transform: Part i, theory

Abstract: This paper is the first part of a two-part series. It proves a number of direct relationships between the Fourier transform and the simple genetic algorithm. (For a binary representation, the Walsh transform is the Fourier transform.) The results are of a theoretical nature and are based on the analysis of mutation and crossover. The Fourier transform of the mixing matrix is shown to be sparse. An explicit formula is given for the spectrum of the differential of the mixing transformation. By using the Fourier representation and the fast Fourier transform, one generation of the infinite population simple genetic algorithm can be computed in time O(cl log2 3), where c is arity of the alphabet and l is the string length. This is in contrast to the time of O(c3l) for the algorithm as represented in the standard basis. There are two orthogonal decompositions of population space that are invariant under mixing. The sequel to this paper will apply the basic theoretical results obtained here to inverse problems and asymptotic behavior.

Fractional fourier transform of generalized functions

Abstract: In recent years the fractional Fourier transform (FRFT), which is a generalization of the Fourier transform, has been the focus of many research papers because of its application in several areas, including signal processing and optics. In this paper, we extend the fractional Fourier transform to different spaces of generalized functions using two different techniques, one analytic and the other algebraic. The algebraic approach requires the introduction of a new convolution operation for the fractional Fourier transform that makes the transform of a convolution of two functions almost equal to the product of their transform.

An efficient 3-D near-field ISAR algorithm

Abstract: This paper presents a new three-dimensional (3-D) near-field inverse synthetic aperture radar (ISAR) imaging technique. A 3-D ISAR image can be obtained by processing coherently the backscattered fields as a function of the frequency and two rotation angles about axes which are mutually orthogonal. Most of the existing ISAR algorithms are based on the Fourier transform and as such can tolerate only small amounts of wavefront curvature. Wavefront curvature must be taken into account when imaging an object in the near-field. Near-field ISAR imaging of large objects using a direct Fourier inversion may result in images which are increasingly unfocused at points which are more distant from the center of rotation. An algorithm based on an azimuth convolution between a near-field focusing function and the frequency domain backscattered fields is discussed. This convolution is efficiently implemented by using fast Fourier transform (FFT) techniques. Furthermore, in order to further alleviate the computational load of the algorithm, the discrete Fourier transform (DFT) of the focusing function is evaluated by means of the stationary phase method. Experimental results show that this technique is precise and virtually impulse invariant.

Fourier transform, wavelets, Prony analysis: tools for harmonics and quality of power

Abstract: The Fourier transform is a very useful tool for signal studies. Nevertheless there are many problems in using it; but these problems are very well known and correctly explained in literature. Wavelets are not usual in power network analysis. However, they are easy to use and give good results; the edge effects are transient and the computation time may be reasonable. Prony analysis is only found in a few papers about power networks. There are few high-performance decomposition programs. The best ones remain sensitive to noise. They require a long observation time with many samples. But, when the analysis succeed, this method is the most powerful to explain what happens in a power network transient. This paper explains as simply as possible the wavelet and the Prony analyses and shows, qualitatively, their performances and their limits.

Basic analog of Fourier series on a $q$-quadratic grid

Abstract: We prove orthogonality relations for some analogs of trigonometric functions on a g-quadratic grid and introduce the corresponding g-Fourier series. We also discuss several other properties of this basic trigonometric system and the g-Fourier series.

Numerical calculation of fractional Fourier transforms with a single fast-Fourier-transform algorithm

Abstract: A numerical algorithm based on a single fast Fourier transform is proposed. Its precision and calculation efficiency show better performance than those of previously published algorithms. It is also shown that if specific conditions are met, the numerical calculations of two successive fractional Fourier transforms produce results that are similar to the analytical solution.

Recursive algorithm for phase retrieval in the fractional fourier transform domain.

Abstract: We first discuss the discrete fractional Fourier transform and present some essential properties. We then propose a recursive algorithm to implement phase retrieval from two intensities in the fractional Fourier transform domain. This approach can significantly simplify computational manipulations and does not need an initial phase estimate compared with conventional iterative algorithms. Simulation results show that this approach can successfully recover the phase from two intensities.

Error concealment or correction of speech, image and video signals

Abstract: A method of error concealment or correction, suitable for signals such as speech, image and video signals, and particularly for such signals as transmitted over wireless and ATM channels. The method has improved stability for large block sizes and bunched errors. It is based on a modification of the discrete Fourier transform which is equivalent to a permutation of the Fourier coefficients. Thus, as with the conventional Fourier transform, setting a contiguous set of coefficients equal to zero in the transmitted signal enables error concealment techniques to be used at the receiver. However, with the modified transform, the zeroes are not bunched together in the spectrum, so the instability problems that arise when the conventional Fourier transform is used are mollified, and much larger block sizes can be used.

An efficient Fourier method for 3-D radon inversion in exact cone-beam CT reconstruction

Abstract: The radial derivative of the three-dimensional (3-D) radon transform of an object is an important intermediate result in many analytically exact cone-beam reconstruction algorithms. The authors briefly review Grangeat's (1991) approach for calculating radon derivative data from cone-beam projections and then present a new, efficient method for 3-D radon inversion, i.e., reconstruction of the image from the radial derivative of the 3-D radon transform, called direct Fourier inversion (DFI). The method is based directly on the 3-D Fourier slice theorem. From the 3-D radon derivative data, which is assumed to be sampled on a spherical grid, the 3-D Fourier transform of the object is calculated by performing fast Fourier transforms (FFTs) along radial lines in the radon space. Then, an interpolation is performed from the spherical to a Cartesian grid using a 3-D gridding step in the frequency domain. Finally, this 3-D Fourier transform is transformed back to the spatial domain via 3-D inverse FFT. The algorithm is computationally efficient with complexity in the order of N/sup 3/ log N. The authors have done reconstructions of simulated 3-D radon derivative data assuming sampling conditions and image quality requirements similar to those in medical computed tomography (CT).

Fast DFT algorithms for length N=q*2/sup m/

Abstract: This paper presents a general split-radix algorithm which can flexibly compute the discrete Fourier transforms (DFT) of length q*2/sup m/ where q is an odd integer Comparisons with previously reported algorithms show that substantial savings on arithmetic operations can be made Furthermore, a wider range of choices on different sequence lengths is naturally provided

Iterative algorithm for nonuniform inverse fast Fourier transform (NU-IFFT)

Abstract: A nonuniform inverse fast Fourier transform (NU-IFFT) for nonuniformly sampled data is realised by combining the conjugate-gradient fast Fourier transform (CG-FFT) method with the newly developed nonuniform fast Fourier transform (NUFFT) algorithms. An example application of the algorithm in computational electromagnetics is presented.

Efficient computation of Fourier transforms on compact groups

Abstract: This article genralizes the fast Fourier transform algorithm to the computation of Fourier transforms on compact Lie groups. The basic technique uses factorization of group elements and Gel'fand-Tsetlin bases to simplify the computations, and may be extended to treat the computation of Fourier transforms of finitely supported distributions on the group. Similar transforms may be defined on homogeneous spaces; in that case we show how special function properties of spherical functions lead to more efficient algorithms. These results may all be viewed as generalizations of the fast Fourier transform algorithms on the circle, and of recent results about Fourier transforms on finite groups.

A fast filter-bank for adaptive Fourier analysis

Abstract: Adaptive Fourier analyzers have been developed for measuring periodic signals with unknown or changing fundamental frequency. Typical applications are vibration measurements and active noise control related to rotating machinery and calibration equipment that can avoid the changes of the line frequency by adaptation. Higher frequency applications have limitations since the computational complexity of these analyzers are relatively high as the number of the harmonic components to be measured (or suppressed) is usually above 50. In this paper, based on the concept of transformed domain signal processing, a fast filter-bank structure is proposed to reduce the above computational complexity. The first step of the suggested solution is the application of the filter-bank version of the fast Fourier transform or any other fast transformations that convert input data into the transformed domain. These fast transform structures operate as single-input multiple-output filter-banks, however, they can not be adapted since their efficiency is due to their special symmetry. As a second step, the adaptation of the filter-bank is performed at the transform structure's output by adapting a simple linear combiner to the fundamental frequency of the periodic signal to be processed.

Fourier transforms of Schwartz functions on Chébli-Trimèche hypergroups

Abstract: The Fourier transform for Schwartz spaces on Chebli-Trimeche hypergroups is studied, leading to results on approximation to the identity for functions and distributions on the half-line. In particular it is shown that the heat and Poisson kernels on the half-line form approximate units in various function spaces. A characterization of the convolution of a tempered distribution and a Schwartz function is also given.

The Discrete Fourier Transform and the Fast Fourier Transform

Abstract: The preceding chapters have made extensive mention of the Fourier transform (FT), the discrete Fourier transform (DFT), and the fast Fourier transform (FFT). This chapter examines the relationship between the FT and the DFT, discusses the FFT algorithm as a means of computing the DFT much more rapidly than can be achieved with the DFT algorithm directly, and presents some practical guidelines for using the FFT.

The Fourier transform in biomedical engineering

Abstract: 1 Introduction to the Fourier Transform.- 1.1 Introduction.- 1.2 Basic Functions.- 1.3 Sines, Cosines and Composite waves.- 1.4 Orthogonality.- 1.5 Waves in time and space.- 1.6 Complex numbers. A Mathematical Tool.- 1.7 The Fourier transform.- 1.8 Fourier transforms in the physical world: The Lens as an FT computer.- 1.9 Blurring and convolution.- 1.9.1 Blurring.- 1.9.2 Convolution.- 1.10 The "Point" or "Impulse" response function..- 1.11 Band-limited functions.- 1.12 Summary.- 1.13 Bibliography.- 2 The 1-D Fourier Transform.- 2.1 Introduction.- 2.2 Re-visiting the Fourier transform.- 2.3 The Sampling Theorem.- 2.4 Aliasing.- 2.5 Convolution.- 2.6 Digital Filtering.- 2.7 The Power Spectrum.- 2.8 Deconvolution.- 2.9 System Identification.- 2.10 Summary.- 2.11 Bibliography.- 3 The 2-D Fourier Transform.- 3.1 Introduction.- 3.2 Linear space-invariant systems in two dimensions.- 3.3 Ideal systems.- 3.4 A simple X-ray imaging system.- 3.5 Modulation Transfer Function (MTF).- 3.6 Image processing.- 3.7 Tomography.- 3.8 Computed Tomography.- 3.9 Summary.- 3.10 Bibliography.- 4 The Fourier Transform in Magnetic Resonance Imaging.- 4.1 Introduction.- 4.2 The 2-D Fourier transform.- 4.3 Magnetic Resonance Imaging.- 4.3.1 Nuclear Magnetic Resonance.- 4.3.2 Excitation, Evolution, and Detection.- 4.3.3 The Received Signal: FIDs and Echos.- 4.4 MRI.- 4.4.1 Localization: Magnetic Field Gradients.- 4.4.2 The MRI Signal Equation.- 4.4.3 2-D Spin-Warp Imaging.- 4.4.4 Fourier Sampling: Resolution, Field-of-View, and Aliasing.- 4.4.5 2-D Multi-slice and 3-D Spin Warp Imaging.- 4.4.6 Alternate k -space Sampling Strategies.- 4.5 Magnetic Resonance Spectroscopic Imaging.- 4.5.1 Nuclear Magnetic Resonance Spectroscopy: 1-D.- 4.5.2 Magnetic Resonance Spectroscopic Imaging: 2-D, 3-D, and 4-D.- 4.6 Motion in MRI.- 4.6.1 Phase Contrast Velocity Imaging.- 4.6.2 Phase Contrast Angiography.- 4.7 Conclusion.- 4.8 Bibliography.- 5 The Wavelet Transform.- 5.1 Introduction.- 5.1.1 Frequency analysis: Fourier transform.- 5.2 Time-Frequency analysis.- 5.2.1 Generalities.- 5.2.2. How does time-frequency analysis work?.- 5.2.3 Windowed Fourier transform.- 5.2.4 Wavelet transform.- 5.3 Multiresolution Analysis.- 5.3.1 Scaling Functions.- 5.3.2 Definition.- 5.3.3 Scaling Relation.- 5.3.4 Relationship of multiresolution analysis to wavelets.- 5.3.5 Multiresolution signal decomposition.- 5.3.6 Digital filter interpretation.- 5.3.7 Fast Wavelet Transform Algorithm.- 5.3.8 Multidimensional Wavelet Transforms.- 5.3.9 Fourier vs. Wavelet Digital Signal Processing.- 5.4 Applications.- 5.4.1 Image Compression.- 5.4.2 Irregular heart beat detection from EKG signals.- 5.5 Summary.- 5.6 Bibliography.- 6 The Discrete Fourier Transform and Fast Fourier Transform.- 6.1 Introduction.- 6.2 From Continuous to Discrete.- 6.2.1 The comb function.- 6.2.2 Sampling.- 6.2.3 Interpreting DFT data in a cyclic buffer.- 6.3 The Discrete Fourier Transform.- 6.4 The Fast Fourier Transform.- 6.4.1 The DFT as a matrix equation.- 6.4.2 Simplifying the transition matrix.- 6.4.3 Signal-flow-graph notation.- 6.4.4 The DFT expressed as a signal flow graph.- 6.4.5 Speed advantages of the FFT.- 6.5 Caveats to using the DFT/FFT.- 6.6 Conclusion.- 6.7 Bibliography.

System analysis and signal processing: with emphasis on the use of MATLAB

Abstract: 1. Getting Started in Matlab and an Introduction to Systems and Signal Processing. 2. Impulse Functions, Impulse Responses, and Convolution. 3. The Steady State Response of Analogue Networks to Sinusoids and to the complex exponential EJWT. 4. Phasors. 5. Line Spectra and the Fourier Series. 6. Spectral Density Functions and the Fourier Transform. 7 The Sampling and Digitization of Signals. 8. The Discrete Fourier Transform. 9. The Fast Fourier Transform and Some Applications. 10. The Steady State Response of Analogue Systems By Consideration of the 11. Natural Responses, Transients and Stability. 12. The Laplace Transform. 13. Synthesis of Analogue Filters. 14. An Introduction to Digital Networks and the Z-Transform. 15. Synthesis of Digital Filters. 16. Correlation. 17. Processing Techniques for Bandpass Signals. Index.

Convergence and divergence of decreasing rearranged Fourier series

Abstract: In a number of useful applications, e.g., data compression, the appropriate partial sums of the Fourier series are formed by taking into consideration the size of the coefficients rather than the size of the frequencies involved. The purpose of this paper is to show the limitations of that method of summation. We use several results from the number theory to construct counterexamples to Lp-convergence for p < 2. We also show how to obtain positive results if we combine the two points of view, i.e., cutting on frequencies and the size of coefficients at the same time. This can be considered as a kind of uncertainty principle for Fourier sums.

Apparatus for and method of processing image and information recording medium

Abstract: An image processing apparatus capable of extracting widely viewed features of an entire image and speedily performing image processing, and a method and an information recording medium for such processing. Image data of an original image is obtained by imaging the entire original image with an image pickup unit at a time. An amplitude distribution of a signal is obtained from the image data by fast Fourier transform performed by a fast Fourier transform section of a signal processing unit. An amplitude replacement section replaces the amplitude distribution with a predetermined function using the distance from a center of a frequency plane as a parameter. An inverse fast Fourier transform section forms an image corresponding to the original image by inverse fast Fourier transform from a phase distribution of points obtained by the fast Fourier transform and from an amplitude distribution obtained by the above-described replacement.

High-precision digital automated measurement of degree of coherence in the Thompson and Wolf experiment

Abstract: We propose a high-precision digital automated quantitative determination of the modulus of the complex degree of coherence. The Thompson and Wolf experiment is repeated, using a CCD and a measurement method based on the fast Fourier transform. The experimental results agree very well with the predictions of the theory.