Showing papers on "Discrete-time Fourier transform published in 1991"

Fast Fourier Transforms

Abstract: Basic aspects of fourier series definition of fourier series examples of fourier series fourier series of real functions pointwise convergence of fourier series further aspects of convergence of fourier series fourier sine series and cosine series convergence of fourier sine and cosine series the discrete fourier transform (DFT) the fast fourier transform (FFT) some applications of fourier series fourier transforms properties of fourier transforms inversion of fourier transforms convolution - an introduction the convolution theorem an application of convolution in quantum mechanics filtering, frequency detection, and removal of noise summation kernals arising from poisson summation fourier optics fresnel diffraction fraunhofer diffraction circular apertures the phase transformation induced by a thin lens imaging with a single lens user's manual for fourier analysis software some computer programmes the schwarz inequality.

A restriction theorem for the Fourier transform

Abstract: In this note we will prove a (L , LP) -restriction theorem for certain submanifolds & of codimension / > 1 in an n— dimensional Euclidean space which arise as orbits under the action of a compact group K. As is well known such a result can in general only hold for 1 < p < j^y. We will show that for the submanifolds under consideration the inequality ^\\f(x)\\dKx)

Signals, systems, and transforms

Abstract: (NOTE: Each chapter begins with an Introduction and ends with a Summary and Problems). 1. Overview of Signals and Systems. Signals. Systems. 2. Continuous-Time and Discrete-Time Signals. PART A Continuous-Time Signals. Basic Continuous -Time Signals. Modification of the Variable t. Continuous-Time Convolution. PART B Discrete-Time Signals. Basic Discrete-Time Signals. Modification of the Variable n. Discrete-Time Convolution. 3. Linear Time-Invariant Systems. PART A Continuous-Time Systems. System Attributes. Continuous-Time LTI Systems. Properties of LTI Systems. Differential Equations and Their Implementation. PART B Discrete-Time Systems. System Attributes. Discrete-Time LTI Systems. Properties of LTI Systems. Difference Equations and the Their Implementation. 4. Fourier Analysis for Continuous-Time Signals. The Eigenfunctions of Continuous-Time LTI Systems. Periodic Signals and the Fourier Series. The Continuous-Time Fourier Transform. Properties and Applications of the Fourier Transform. APPLICATION 4.1 Amplitude Modulation. APPLICATION 4.2 Sampling. 5.Frequency Response of LTI Systems. APPLICATION 4.3 Filtering. 5. The Laplace Transform. The Region of Convergence. The Inverse Laplace Transform. Properties of the Laplace Transform. The System Function for LTI Systems. Differential Equations. APPLICATION 5.1 Butterworth Filters. Structures for Continuous-Time Filters. Appendix 5A The Unilateral Laplace Transform. Appendix 5B Partial-Fraction Expansion for Multiple Poles. 6. The z Transform. The Eigenfunctions of Discrete-Time LTI Systems. The Region of Convergence. The Inverse z Transform. Properties of the z Transform. The System Function to LTI Systems. Difference Equations. APPLICATION 6.1 Second-Order IIR Filters. APPLICATION 6.2 Linear-Phase FIR Filters. Structures for Discrete-Time Filters. APPENDIX 6A The Unilateral z Transform. APPENDIX 6B Partial-Fraction Expansion for Multiple Poles. 7. Fourier Analysis for Discrete-Time Signals. The Discrete-Time Fourier Transform. 2.Properties of the DTFT. APPLICATION 7.1 Windowing. 3.Sampling. 4.Filter Design by Transformation. 5.The Discrete Fourier Transform/Series. APPLICATION 7.2 FFT Algorithm. 8. State Variables. Discrete-Time Systems. Continuous-Time Systems. Operational-Amplifier Networks. Bibliography. Index.

Weighting effect on the discrete time Fourier transform of noisy signals

Abstract: The effect of weighting on the uncertainty of the discrete time Fourier transform (DTFT) samples of a signal corrupted by additive noise is investigated. Making very weak assumptions, it is shown how the adopted window sequence and the autocovariance function of the noise affect the second-order stochastic moments of the frequency-domain data. The relationship obtained extends the results reported in the literature and is useful in many frequency-domain estimation problems. It is shown how the knowledge of the second-order moments of the transform has allowed the application of the least squares technique for the estimation of the parameters of a multifrequency signal in the frequency-domain. The estimator obtained is very useful when high-accuracy results are required under real-time constraints. The procedure exhibits a better accuracy than similar frequency-domain methods proposed in the literature. >

Signal processing apparatus and method for iteratively determining Arithmetic Fourier Transform

Abstract: A signal processing apparatus and method for iteratively determining the inverse Arithmetic Fourier Transform (AFT) of an input signal by converting the input signal, which represents Fourier coefficients of a function that varies in relation to time, space, or other independent variable, into a set of output signals representing the values of a Fourier series associated with the input signal. The signal processing apparatus and method utilize a process in which a data set of samples is used to iteratively compute a set of frequency samples, wherein each computational iteration utilizes error information which is calculated between the initial data and data synthesized using the AFT. The iterative computations converge and provide AFT values at the Farey-fraction arguments which are consistent with values given by a zero-padded Discrete Fourier Transform (DFT), thus obtaining dense frequency domain samples without interpolation or zero-padding.

A stochastic roundoff error analysis for the fast Fourier transform

Abstract: We study the accuracy of the output of the Fast Fourier Transform by estimating the expected value and the variance of the accompanying linear forms in terms of the expected value and variance of the relative roundoff errors for the elementary operations of addition and multiplication. We compare the results with the corresponding ones for the direct algorithm for the Discrete Fourier Transform, and we give indications of the relative performances when different rounding schemes are used. We also present the results of numerical experiments run to test the theoretical bounds and discuss their significance.

Design of 2-D FIR filters by nonuniform frequency sampling

Abstract: A method for the frequency-sampling design of two-dimensional FIR filters with nonuniformly spaced samples is presented. By imposing some mild constraints on sample location in the 2-D frequency plane, the method always provides a unique design solution. Important characteristics of the method are design flexibility through the use of nonuniform samples and computational efficiency. This method is compared with the uniform sampling, inverse discrete Fourier transform (DFT) approach and also with a general method for filter design called arbitrary sampling. The method presented is shown to require much less computation than the arbitrary sampling approach, which may lead to possible degenerate cases where there is no unique solution for the filter. The method proposed does not lead to such degeneracies and possesses more flexibility than the uniform sampling method. Examples are given in order to compare the new method with the uniform sampling method. >

Fourier transform of a linear distribution with triangular support and its applications in electromagnetics

Abstract: A three-dimensional Fourier transform (FT) of a linear function with triangular support is derived in its coordinate-free representation. The Fourier transform of this distribution is derived in three steps. First, the 2-D FT of a constant (top hat) function is obtained. Next, the distribution is generalized to a linearly varying function. Finally, the formulation is extended to a coordinate-free representation which is the 3-D FT of the 2-D function defined over a surface. This formulation is applied to the near-field computation, yielding accurate numerical solutions. >

Estimation of the acoustic field directionality by use of planar and volumetric arrays via the Fourier series method and the Fourier integral method

Abstract: The possibility of estimating the directionality of a stationary homogeneous noise field, directly from the element outputs of a line array, is investigated and found to be feasible for large arrays without encountering ill conditioning This technique, called inverse beamforming for historical reasons, is applicable to line and planar as well as volumetric arrays, and requires no more than two‐dimensional fast Fourier transforms (FFTs) for its realization Derivations and results are presented both for a Fourier series method and a Fourier integral method

Aliasing effects in Fourier transforms of monotonically decaying functions and the calculation of viscoelastic moduli by combining transforms over different time periods

Abstract: An analysis is presented of the discrete Fourier transform of a monotonically decaying function, represented by a sum of exponentials with negative exponents. The results are compared with those of a previous analysis based on the analytical Fourier transform, which proposed a method for extending the frequency range of the Fourier transform of experimental data by combining transforms performed over different time periods. The principle of the method is confirmed but comparison shows that results derived for the analytical transform cannot always be applied directly to the discrete transform. Modifications are therefore proposed which improve the accuracy and mitigate the aliasing effects evident in short-time transforms while keeping the computing time to a minimum. These results are particularly applicable to stress relaxation and creep in viscoelastic materials and an example from articular cartilage shows the compliance modulus over a frequency range from 10-3 Hz to 230 Hz from one creep experiment of duration 18.7 min.

Use of Fourier domain real-plane zeros to overcome a phase retrieval stagnation

Abstract: The iterative Fourier transform algorithm, although it has been demonstrated to be a practical phase retrieval algorithm, suffers from certain stagnation problems. Specifically, there exists a stripe stagnation problem, in which stagnated reconstructed images exhibit stripelike features throughout the image, which is particularly difficult to overcome. Previous solutions to this problem used multiple reconstructions and did not address the cause. In this paper a new procedure that uses only a single image is developed that estimates the locations of real-plane zeros from either the measured Fourier modulus data or a stagnated reconstruction and uses this information in the iterative Fourier transform algorithm to force the complex-valued Fourier data to have real-plane zeros at the correct locations. It is shown that this procedure overcomes the stripe stagnation.

Application of the Hartley transform for the analysis of the propagation of nonsinusoidal waveforms in power systems

Abstract: Because the Fourier transform causes the convolution operation to become a simple complex product, it has been used to solve power system problems. A similar convolution property of the Hartley transform is used to calculate transients and nonsinusoidal waveshape propagation in electric power systems. The importance of this type of calculation relates to the impact of loads, particularly electronic loads, whose demand currents are nonsinusoidal. An example is given in which the Hartley transform is used to assess the impact of an electronic load with a demand which contains rapidly changing current. The authors also present a general introduction to the use of Hartley transforms for electric circuit analysis. A brief discussion of the error characteristics of discrete Fourier and Hartley solutions is presented. Because the Hartley transform is a real transformation, it is more computationally efficient then the Fourier or Laplace transforms. >

The techniques of the generalized fast Fourier transform algorithm

Abstract: A general method of deriving DFT (discrete Fourier transform) algorithms, generalised fast Fourier transform algorithms, is presented. It is shown that a special case of the method is equivalent to nesting of FFTs. The application of the method to the case where N has mutually prime factors results in a new interpretation of the permutations characteristic of this class of algorithms. It is shown that the equalization of FFTs leads to results which are different from the widely used intuitive ones. The high efficiency of split-radix FFTs is explained. It is shown that the formulae of the method can be easily adapted for deriving algorithms for the cosine/sine DFT. A set of FFTs that has smaller arithmetical and/or memory complexities than any algorithm known is presented. In particular, a method of deriving split-radix-2/sup s/ FFTs requiring N log/sub 2/ N-3N+4 real multiplications and 3N log/sub 2/ N-3N+4 additions for any s>1 is presented. >

Generalized discrete Hartley transforms

Abstract: The discrete Hartley transform (DHT) is generalized into 4 classes in the same way as the generalized discrete Fourier transform. The fast algorithms for the resulting transforms are derived. The generalized transforms are expected to be useful in applications such as digital filter banks, computing convolution and signal representation. The fast computation of skew-circular convolution with the generalized transforms is discussed in detail. >

Determination of transfer function of two dimensional generalised systems using the discrete Fourier transform

Abstract: A systematic and conceptually simple algorithm is presented for the determination of the transfer function for two-dimensional generalised or singular systems. The method uses the discrete Fourier transform (DFT) and can easily be applied. The simplicity and efficiency of the algorithm are illustrated by two examples.

Algorithms for processing fourier transform phase of signals

Abstract: The studies presented in this thesis represent an attempt to process the Fourier transform (FT) phase of signals for feature extraction. Although the FT magnitude and phase spectra are independent functions of frequency features of a signal, most techniques for feature extraction from a signal are bked upon manipulating the the FT magnitude only. The phase spectrum of the signal corresponds to time delay corresponding to each of the sinusoidal components of the signal. In the context of additive noise, the time delay may not be significantly corrupted and the phase spectrum might be considered to be a more reliable source for estimating the features in a noisy signal. Although the importance of phase in signals is realised by researchers, very few attempts have been made to process the FT phase of signals for the extraction of features. Features of a signal, for example, resonance information, is completely masked by the inevitable wrapping of the phase spectrum. An alternative to processing the phase spectrum is processing the group delay function. The group delay function is the negative derivative of the (unwrapped) FT phase spectrum. The group delay function can be computed directly from the time domain signal.The group delay function possesses additive and high resolution properties, in that it shows a squared magnitude behaviour in the vicinity of a resonance. But the group delay function in general is not well behaved for all classes of signals. Zeros in the z-transform of a signal that are close to the unit circle cause large amplitude spikes to appear in the group delay function. The polarity of a spike depends on the location of the zero with respect to the unit circle. These large amplitude spikes mask the information about resonances. The research effort in this thesis focusses on the development of algorithms for manipulating the group delay function to suppress the information corresponding to the zeros of th signal that are close to unit circle in the z-domain and emphasise the features of of a signal. To demonstrate the usefulness of the algorithms developed, these algorithms are used to estimate (a) formant and pitch data from speech signals and ( b ) estimate spectra of auto-regressive processes and sinusoids in noise. The research effort in this thesis shows that the phase spectrum (or rather the group delay function) of a signal can be usefully processed to reliably extract features of a signal. ACKNOWLEDCEMENT I express my appreciation to Prof.B.Yegnanarayana for his constant help, excellent guidance and constructive criticisms throughout the course of this work. I thank Prof. R. ~a~arajan, Head, Department of Computer Science and Engineering, for making the various facilities in the department available to me: I owe my special thanks to Madhu Murthy and C.P.Mariadassou for some fruitful discussions. I thank G. V. Ramana Rao and R. Ramaseshan for reading my thesis and making useful suggestions. I would like to thank all my colleagues of the Speech and Vision Lab who have helped me in one way or the other. I thank Vatsala for providing me a shoulder whenever I was depressed. Finally, I thank my husband M. V. N. Murthy for his support and perseverence throughout the course of this work.

Learning the Fourier spectrum of probabilistic lists and trees

Abstract: We observe that the Linial, Mansour, and Nissan method of learning boolean concepts (under uniform sampling distribution) by reconstructing their Fourier represent ation [LMN89] extends when the concepts are probabilistic in the sense of Kearns and Shapire [KS90]. We show that probabilistic decision lists, and more generally probabilistic decision trees with at most one occurrence of each literal, can be approximate ed by polynomially small Fourier represent ations, and that the non-negligible Fourier coefficients can be efficiently identified and estimated. Hence, all such concepts are learnable in polynomial time under uniform sampling distribution. This is the first instance where Fourier methods result in polynomial learning algorithms: the polynomiality of our results should be contrasted to the np”lylogn complexities in the analogous cases of [LMN89] and [M90]. The new ingredient of our work that allows us to achieve this polynomiality is that via refined Fourier analysis we are able to isolate the polynomially small set of non-negligible Fourier coefficients that reside in a super-polynomially large area of the spectrum. We further observe that several more general concept classes have slightly super-polynomial (npolyk)gn ) learning algorithms. These classes include all polynomial-size probabilistic decision trees, their convex combinations, etc. A concrete special case which results in polynomial learnabil“Bdl ColIl]lltl[\icalioI]s Research, Morristown NJ 07960. aidlo((!fl ash .Ixdlcorc.con]. flkll (bmmnnicat.ions Research, hlorrist.own NJ 07!w0, ]I~illail(@)fl&sll .l}cllcorc. col]). ity is the weighted arithmetization of k-DNF.

Convolution relation within the three-dimensional diffraction image.

Abstract: The amplitude in the image of a point source is (a constant times) the three-dimensional Fourier transform of the lens pupil, i.e., of the amplitude distribution of the converging waves in angle space. If the three-dimensional pupil has axial symmetry, it can be factored into a spherical shell and a one-dimensional function of distance along the axis. The amplitude in the image is given by the convolution of the Fourier transforms of the factors. On the axis, the amplitude is the Fourier transform of the one-dimensional function. Therefore the amplitude anywhere in the image is the convolution of the amplitude on the axis and the Fourier transform of the spherical shell.

Fast algorithms for the continuous wavelet transform

Abstract: It is shown that filter banks arise naturally when implementing the continuous wavelet transform (CWT). The conditions under which the CWT can be computed exactly using discrete filter banks are determined, and fast CWT algorithms are derived. The complexity of the resulting algorithms increases linearly with the number of octaves. They are easily implemented by repetitive application of identical cells, to which various methods are applied for reducing the number of operations: FFT (fast Fourier transform) algorithms are most efficient for large filter lengths; for small lengths, fast running FIR (finite impulse response) algorithms are preferred. >

Signals and systems : an introduction

Abstract: Signal description system description system response to a sinusoidal input the Fourier series and Fourier transform the Laplace and transforms feedback systems.

Advanced Signal Processing Techniques

Abstract: : The research efforts under this program have been in the area of parallel and connectionist implementations of signal processing functions such as filtering, transforms, convolution and pattern classification and clustering. The topics which have been found to be the most promising, in as far as their novelty, and applicability to the classification of pulses are covered in the following four sections: (1) Algebraic Transforms and Classifiers, (2) An Interaction Model of Neural-Net based Associative Memories, (3) Models of Adaptive Neural-Net based Pattern Classifiers, and (4) Discrete Fourier Transforms on Hypercubes.

New formulation of discrete Wigner-Ville distribution

Abstract: A new formulation of the discrete Wigner-Ville distribution is presented which can be implemented directly using standard fast Fourier transform techniques. For a non-negative frequency resolution of N points, only an N point FFT is needed.

Distortion reduction in projection imaging by manipulation of fourier transform of projection sample

Abstract: Distortion is reduced in images derived by Fourier transform methods from projection samples by manipulating the Fourier transformed values by a predetermined procedure of a kind that subjects the image values to unwanted distortion, and then applying to the projection samples or image values a correction function that corresponds to the predefined procedure.

L 2 Estimates in Nonlinear Fourier Analysis

Abstract: One of the goals of classical Fourier analysis is to obtain L p estimates for linear operators that commute with translations. The case of p = 2 plays a special role, because of Placherel’s theorem, and because of the availability of methods from the Calderon- Zygmund school for deriving L p estimates from the L 2 case. However, the relevance of Plancherel’s theorem fades swiftly to oblivion when we shift our attention to nonlinear objects. (The Calderon-Zygmund technology does not.) In this paper we present an approach to dealing with nonlinear functionals that commute with translations and which satisfy nonlinear versions of the Calderon-Zygmund conditions.

On computing the discrete Fourier transform of a linear phase sequence

Abstract: Efficient fast Fourier transform (FFT) algorithms to compute the forward and inverse discrete Fourier transforms (DFT) of a sequence with linear-phase characteristic are examined. These reduce the computational requirements as regards a complex FFT by large factors and should be used whenever applicable. The case when the DFT coefficients are real-valued leads to further reductions in computational requirements. Though the redundancy in the linear-phase situation is exactly 50%, the computational requirements and implementation are quite different from the real-valued FFT which uses a similar symmetry relation. The code for such implementations can be easily written by simple restructuring of a complex FFT algorithm. >