Showing papers on "Discrete Fourier transform published in 1998"

Interpolation and extrapolation using a high-resolution discrete Fourier transform

Abstract: We present an iterative nonparametric approach to spectral estimation that is particularly suitable for estimation of line spectra. This approach minimizes a cost function derived from Bayes' theorem. The method is suitable for line spectra since a "long tailed" distribution is used to model the prior distribution of spectral amplitudes. Since the data themselves are used as constraints, phase information can also be recovered and used to extend the data outside the original window. The objective function is formulated in terms of hyperparameters that control the degree of fit and spectral resolution. Noise rejection can also be achieved by truncating the number of iterations. Spectral resolution and extrapolation length are controlled by a single parameter. When this parameter is large compared with the spectral powers, the algorithm leads to zero extrapolation of the data, and the estimated Fourier transform yields the periodogram. When the data are sampled at a constant rate, the algorithm uses one Levinson recursion per iteration. For irregular sampling, the algorithm uses one Cholesky decomposition per iteration. The performance of the algorithm is illustrated with three different problems that arise in geophysical data: (1) harmonic retrieval from a time series contaminated with noise; (2) linear event detection from a finite aperture array of receivers, (3) interpolation/extrapolation of gapped data. The performance of the algorithm as a spectral estimator is tested with the Kay and Marple (1981) data set.

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

An accurate algorithm for nonuniform fast Fourier transforms (NUFFT's)

Abstract: Based on the (m, N, q)-regular Fourier matrix, a new algorithm is proposed for fast Fourier transform (FFT) of nonuniform (unequally spaced) data. Numerical results show that the accuracy of this algorithm is much better than previously reported results with the same computation complexity of O(N log/sub 2/ N). Numerical examples are shown for the applications in computational electromagnetics.

Free-space beam propagation between arbitrarily oriented planes based on full diffraction theory: a fast Fourier transform approach

Abstract: A fast and accurate numerical method for free-space beam propagation between arbitrarily oriented planes is developed. The only approximation made in the development of the method was that the vector nature of light was ignored. The method is based on evaluating the Rayleigh–Sommerfeld diffraction integral by use of the fast Fourier transform with a special transformation to handle tilts and offsets of planes. The fundamental aspects of a software package based on the developed method are presented. A numerical example realized with the software package is presented to establish the validity of the method.

A new scheme for generating and measuring active, reactive, and apparent power at power frequencies with uncertainties of 2.5/spl times/10/sup -6/

Abstract: A new method for generating and measuring active, reactive, and apparent power at power frequencies has been devised. It makes use of digital signal synthesis and discrete Fourier transform (DFT) evaluation based on a single master clock. This results in a significant reduction of synchronizing errors and thus in an uncertainty of only 2.5/spl times/10/sup -6/ (k=1).

High-resolution image reconstruction with multisensors

Abstract: This article considers the problem of reconstructing a high-resolution image from multiple undersampled, shifted, degraded frames with subpixel displacement errors. This leads to a formulation involving a periodically shift-variant system model. The maximum a posteriori (MAP) estimation scheme is used subject to the assumption that the original high-resolution image is modeled by a stationary Markov-Gaussian random field. The resulting MAP formulation is expressed as a complex linear matrix equation, where the characterizing matrix involves the periodic block Toeplitz with Toeplitz block (BTTB) blur matrix and banded-BTTB inverse covariance matrix associated with the original image. By approximating the periodic-BTTB and the banded-BTTB matrices with, respectively, the periodic block circulant with circulant block (BCCB) and the banded-BCCB matrices, it is shown that the computation-intensive MAP formulation can be decomposed into a set of smaller matrix equations by using the two-dimensional discrete Fourier transform. Exact solutions are also considered through the use of the preconditioned conjugate gradient algorithm. Computer simulations are given to illustrate the procedure. © 1998 John Wiley & Sons, Inc. Int J Imaging Syst Technol, 9, 294–304, 1998

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.

Efficient moving-window DFT algorithms

Abstract: The authors deal with the real-valued, moving-window discrete Fourier transform. After reviewing the basic recursive versions appearing in the literature, additional recursive equations are presented. Then, these equations are combined so that nonrecursive expressions involving only consecutive discrete Fourier transform (DFT) sine components are obtained for both the DFT cosine component and squared harmonic amplitude. The computational complexity of this new scheme is finally studied and compared to that of existing methods, showing that, in most practical situations, a reduction in the operation count is achieved.

Analysis of DFT-based frequency excision algorithms for direct-sequence spread-spectrum communications

Abstract: The capacity of direct-sequence spread-spectrum modulation to reject narrow-band interference can be significantly improved by eliminating narrow-band energy at the receiver in a process called frequency excision. This paper considers several algorithms that operate on the real-time discrete Fourier transform (DFT) of the received signal to perform frequency excision. The case in which only the signal and additive white Gaussian noise (AWGN) are present at the receiver is considered as a means of comparing the relative performance of different algorithms that operate without knowledge of the power spectral density of the interference. An approach for analysis, using the postcorrelation signal-to-noise ratio (SNR) as the figure of merit, is presented that is valid for a broad class of spreading modulations. First, the algorithm that sets a fixed fraction of the frequency domain record to zero is examined using rank-order statistics as an analytical tool. This result is then generalized to confirm previous estimates of SNR degradation for the algorithm that sets all values that exceed a threshold to zero. These results are again generalized to apply to the algorithm that sets a fixed fraction of the band to a fixed amplitude while retaining phase information in an algorithm called fraction clip. The relative performances of several clip algorithm options are derived as special cases. Finally, a performance measure of the algorithms in the presence of multiple narrow-band interference is provided and illustrated with an example.

Efficient dual-tone multifrequency detection using the nonuniform discrete Fourier transform

Abstract: The International Telecommunication Union (ITU) recommendations for dual-tone multifrequency (DTMF) signaling are not met by conventional DTMF detectors. We present an efficient DTMF detection algorithm based on the nonuniform discrete Fourier transform that meets all of the ITU recommendations. The key innovations are the use of two sliding windows and development of sophisticated timing tests. Our algorithm requires no buffering of input samples. To perform DTMF detection on n telephone channels, our algorithm requires approximately n MIPS on a digital signal processor (DSP), 75+30n words of data memory, and 1000 words of program memory. Using the new algorithm, a single fixed-point DSP can perform ITU-compliant DTMF detection on the 24 telephone channels of a T1 time-division multiple multiplexed telecommunications line.

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.

The Fractional Wave Packet Transform

Abstract: We introduce the concept of the Fractional Wave Packet Transform(FRWPT), based on the idea of the Fractional Fourier Transform(FRFT) and Wave Packet Transform(WPT). We show a version of the resolution of the identity and some properties of FRWPT connected with those of FRFT and WPT.

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.

Power and area efficient fast fourier transform processor

Abstract: A fast Fourier transform (FFT) processor is constructed using discrete Fourier transform (DFT) butterfly modules having, in preferred example embodiments, sizes greater than 4. In a first example embodiment, the FFT processor employs size-8 butterflies. In a second example embodiment, the FFT processor employs size-16 butterflies. In addition, low power, fixed coefficient multipliers are employed to perform nontrivial twiddle factor multiplications in each butterfly module. The number of different, nontrivial twiddle factor multipliers is reduced by separating trivial and nontrivial twiddle factors and by taking advantage of twiddle factor symmetries in the complex plane and/or twiddle factor decomposition. In accordance with these and other factors, the present invention permits construction of an FFT processor with minimal power and IC chip surface area consumption.

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.

Accurate algorithms for nonuniform fast forward and inverse Fourier transforms and their applications

Abstract: Regular fast Fourier transform (FFT) algorithms require uniformly sampled data. In many practical situations, however, the input data is nonuniform, and hence the regular FFT does not apply. To overcome this difficulty the authors have proposed an accurate algorithm for the nonuniform forward FFT (NUFFT) based on a new class of matrices, the regular Fourier matrices. For the nonuniform inverse FFT (NU-IFFT) algorithm, the conjugate-gradient method and the regular FFT algorithm are combined to speed up a matrix inversion. Numerical results show that these algorithms are more than one order of magnitude more accurate than existing algorithms.

Simultaneous time- and frequency-domain extrapolation

Abstract: In this paper, given the early-time response and the low-frequency response of a causal system, we simultaneously extrapolate them in the time and frequency domains. The approach is iterative and is based on a simple discrete Fourier transform. Simultaneous extrapolation in time and frequency domains is further enhanced by using the matrix pencil technique in the time domain and the Cauchy method in the frequency domain. The results are further enhanced through the Hilbert transform, hence enforcing the physical constraints of the system and thereby guaranteeing a causal extrapolation in time. It is, therefore, possible to generate information over a larger domain from limited data. It is important to note that through this extrapolation, no new information is created. The early-time and low-frequency data are complementary and contain all the desired information. The key is to extract this information in an efficient and accurate manner. The electric current on a scatterer is used as an example for the method.