Topic
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.
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TL;DR: In this paper, it is shown that the computationally fastest way to calculate mean lagged products is to begin by calculating all Fourier coefficients with a fast Fourier transform and then to fast-Fourier-retransform a sequence made up of a{k}^{2}+b_{k}−2} (where a_{k+ib_{k}) is the complex Fourier coefficient.
Abstract: The paper discusses the impact of the fast Fourier transform on the spectrum of time series analysis. It is shown that the computationally fastest way to calculate mean lagged products is to begin by calculating all Fourier coefficients with a fast Fourier transform and then to fast-Fourier-retransform a sequence made up of a_{k}^{2}+b_{k}^{2} (where a_{k}+ib_{k} are the complex Fourier coefficients). Also discussed are raw and modified Fourier periodograms, bandwidth versus stability aspects, and aims and computational approaches to complex demodulation. Appendixes include a glossary, a review of complex demodulation without fast Fourier transform, and a short explanation of the fast Fourier transform.
405 citations
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23 Dec 1994TL;DR: In this article, the authors introduce the image plane and the Two Dimensional Fourier Transform (2DFT) for image reconstruction, and present the projection-slice theorem for image restoration.
Abstract: 1. Introduction. 2. The Image Plane. 3. Two Dimensional Impulses. 4. The Two Dimensional Fourier Transform. 5. Two Dimensional Convolution. 6. The Convolution Theorem. 7. Sampling and Interpolation in Two Dimensions. 8. Digital Operations. 9. Rotational Symmetry and the Two Dimensional Fourier Transform. 10. Imaging by Convolution. 11. Diffraction Theory of Sensors. 12. Indirect Imaging and Interferometry. 13. Restoration of Images. 14. The Projection-Slice Theorem. 15. Computed Tomography. 16. Synthetic Aperture Radar. 17. Random Images and Fractals. Index.
391 citations
07 Oct 2001
TL;DR: Hypercomplex numbers, specifically quaternions, are used to define a Fourier transform applicable to color images, and the properties of the transform are developed, and it is shown that the transform may be computed using two standard complex fast Fourier transforms.
Abstract: Hypercomplex Fourier transforms based on quaternions have been proposed by several groups for use in image processing, particularly of color images. So far, however, there has not been a coherent explanation of what the spectral domain coefficients produced by a hypercomplex Fourier transform represent and this paper attempts to present such an explanation for the first time making use of the polar form of a quaternion and a separation of a quaternion spectral coefficient into components parallel and perpendicular to the hypercomplex exponentials in the transform (the basis functions).
379 citations
TL;DR: An incoherent optical data-processing method is described, which has the potential for performing discrete Fourier transforms of short length at rates far exceeding those afforded by both special-purpose digital hardware and representative coherent optical processors.
Abstract: An incoherent optical data-processing method is described, which has the potential for performing discrete Fourier transforms of short length at rates far exceeding those afforded by both special-purpose digital hardware and representative coherent optical processors.
375 citations