Discrete-time Fourier transform

About: Discrete-time Fourier transform is a research topic. Over the lifetime, 5072 publications have been published within this topic receiving 144643 citations. The topic is also known as: DTFT.

A Framework for Discrete Integral Transformations I—The Pseudopolar Fourier Transform

Abstract: The Fourier transform of a continuous function, evaluated at frequencies expressed in polar coordinates, is an important conceptual tool for understanding physical continuum phenomena. An analogous tool, suitable for computations on discrete grids, could be very useful; however, no exact analogue exists in the discrete case. In this paper we present the notion of pseudopolar grid (pp grid) and the pseudopolar Fourier transform (ppFT), which evaluates the discrete Fourier transform at points of the pp grid. The pp grid is a type of concentric-squares grid in which the radial density of squares is twice as high as usual. The pp grid consists of equally spaced samples along rays, where different rays are equally spaced in slope rather than angle. We develop a fast algorithm for the ppFT, with the same complexity order as the Cartesian fast Fourier transform; the algorithm is stable, invertible, requires only one-dimensional operations, and uses no approximate interpolations. We prove that the ppFT is invertible and develop two algorithms for its inversion: iterative and direct, both with complexity $O(n^{2}\log{n})$, where $n \times n$ is the size of the reconstructed image. The iterative algorithm applies conjugate gradients to the Gram operator of the ppFT. Since the transform is ill-conditioned, we introduce a preconditioner, which significantly accelerates the convergence. The direct inversion algorithm utilizes the special frequency domain structure of the transform in two steps. First, it resamples the pp grid to a Cartesian frequency grid and then recovers the image from the Cartesian frequency grid.

Handbook of Fourier Analysis & Its Applications

Abstract: 1. Introduction 2. Fundamentals of Fourier Analysis 3. Fourier Analysis in Systems Theory 4. Fourier Transforms in Probability, Random Variables and Stochastic Processes 5. The Sampling Theory 6. Generalizations of the Sampling Theorem 7. Noise and Error Effects 8. Multidimensional Signal Analysis 9. Time-Frequency Representations 10. Signal Recovery 11. Signal and Image Synthesis: Alternating Projections Onto Convex Sets 12. Mathematical Morphology and Fourier Analysis on Time Sales 13. Applications 14. Appendices 15. Reference

Fast Fourier transform method of computing difference equations and simulating filters

Abstract: Two methods for using the fast Fourier transform to reduce the number of arithmetic operations and, therefore the time required for computing discrete, preformulated, and finite convolutions are listed and justified. Under the idealistic assumption that the impulse response of a preformulated difference equation terminates, a theorem is proved that these two methods can be modified to compute such difference equations. This theorem makes plausible the application of these methods when the impulse response does not terminate, provided that the impulse response decays to a small value. In such cases, the fast Fourier transform can be used to compute approximations to the solutions, although usually this use of the fast Fourier transform offers no reduction in the amount of time required for computing the definition of the difference equation. However, if a filtering operation is specified as a frequency response, the fast Fourier transform can be used to compute the filtering operation directly without need of formulating a difference equation, although this simplification is achieved at the cost of a moderate increase (e.g., twice) in the amount of computation time.

On the Use of a Lower Sampling Rate for Broken Rotor Bar Detection With DTFT and AR-Based Spectrum Methods

Abstract: Broken rotor bars in an induction motor create asymmetries and result in abnormal amplitude of the sidebands around the fundamental supply frequency and its harmonics. Motor current signature analysis (MCSA) techniques are applied to inspect the spectrum amplitudes at the broken rotor bar specific frequencies for abnormality and to decide about broken rotor bar fault detection and diagnosis. In this paper, we have demonstrated with experimental results that the use of a lower sampling rate with a digital notch filter is feasible for MCSA in broken rotor bar detection with discrete-time Fourier transform and autoregressive-based spectrum methods. The use of the lower sampling rate does not affect the performance of the fault detection, while requiring much less computation and low cost in implementation, which would make it easier to implement in embedded systems for motor condition monitoring.

A matrix theory proof of the discrete convolution theorem

Abstract: In this paper we prove the discrete convolution theorem by means of matrix theory. The proof makes use of the diagonalization of a circulant matrix to show that a circular convolution is diagonalized by the discrete Fourier transform. The diagonalization of the circular convolution shows that the eigenvalues of a circular convolution operator are identical with the discrete Fourier frequency spectrum.