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: It is shown that R supports a length m discrete Fourier transform if and only if m is a divisor of O(R) and this result is applied to the homomorphic images of rings-of algebraic integers.
Abstract: Necessary and sufficient conditions for a direct sum of local rings to support a generalized discrete Fourier transform are derived. In particular, these conditions can be applied to any finite ring. The function O(N) defined by Agarwal and Burrus for transforms over ZN is extended to any finite ring R as O(R) and it is shown that R supports a length m discrete Fourier transform if and only if m is a divisor of O(R) This result is applied to the homomorphic images of rings-of algebraic integers.
46 citations
TL;DR: In this article, the authors proved the uncertainty relation in the general case of a complex Fourier transform and with somewhat extended definitions of durations, and showed that the uncertainty principle holds true by appropriate definitions of the durations even if discrete signals are considered.
Abstract: It has recently been shown that the uncertainty principle holds true by appropriate definitions of the durations even if discrete signals are considered. A basic inequality was derived in the particular case where the Fourier transform is real. As an extension to this work, the authors prove the uncertainty relation in the general case of a complex Fourier transform and with somewhat extended definitions of durations. >
45 citations
TL;DR: A modified Fourier-Hankel method that is accurate, computationally efficient, and has the ability to filter noise in the inversion process is proposed for applying to experimental data.
Abstract: Errors in discrete Abel inversion methods using Fourier transform techniques have been analyzed. The Fourier expansion method is very accurate but sensitive to noise. The Fourier-Hankel method has a significant systematic negative deviation, which increases with the radius; inversion error of the method can be reduced by adjusting the value of a factor. With a decrease of the factor both methods show a noise filtering property. Based on the analysis, a modified Fourier-Hankel method that is accurate, computationally efficient, and has the ability to filter noise in the inversion process is proposed for applying to experimental data.
45 citations
TL;DR: In this paper, the Gerchberg-Saxton (GS) algorithm and a fractional Fourier transform (FFT) were combined to deal with the problem of phase retrieval from two intensity measurements.
Abstract: Recently the combination of the Gerchberg–Saxton (GS) algorithm and a fractional Fourier transform was proposed to implement beam shaping in the fractional Fourier domain [ Zalevsky , Opt. Lett.21, 842 (1996)]. We generalize this idea to deal with the problem of phase retrieval from two intensity measurements in a fractional Fourier transform system. The relevant equations for determining the unknown phases are derived, based on the general theory of amplitude–phase retrieval in an optical system. The unitarity condition of the fractional Fourier transform in a practical optical system with finite aperture is discussed. For different fractional orders P, the phase retrieval of several typical model images is studied in detail. A comparison of the GS and our algorithms is given, based on numerical simulations. It follows that our algorithm can offer the desired phase in all cases considered. However, the GS algorithm may fail when the transform system is nonunitary.
45 citations
TL;DR: The HTR algorithm is outlined, and it is shown that its performance compares favorably to the popular convolution-backprojection algorithm.
Abstract: A relatively unexplored algorithm is developed for reconstructing a two-dimensional image from a finite set of its sampled projections. The algorithm, referred to as the Hankel-transform-reconstruction (HTR) algorithm, is polar-coordinate based. The algorithm expands the polar-form Fourier transform F(r, theta ) of an image into a Fourier series in theta ; calculates the appropriately ordered Hankel transform of the coefficients of this series, giving the coefficients for the Fourier series of the polar-form image f(p, phi ); resolves this series, giving a polar-form reconstruction; and interpolates this reconstruction to a rectilinear grid. The HTR algorithm is outlined, and it is shown that its performance compares favorably to the popular convolution-backprojection algorithm. >
45 citations