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 was shown that information of order six suffices to determine a structure uniquely, where six is the number of atoms in a set of equal atoms that can occupy positions on a circle subdivided into equally spaced markings.
Abstract: The three-dimensional configuration of crystallized structures is obtained by reading off partial information about the Fourier transform of such structures from diffraction data obtained with an X-ray source. We consider a discrete version of this problem and discuss the extent to which `intensity only' measurements as well as `higher-order invariants' can be used to settle the reconstruction problem. This discrete version is an extension of the study undertaken by Patterson in terms of `cyclotomic sets', corresponding to arrangements of equal atoms that can occupy positions on a circle subdivided into N equally spaced markings. This model comes about when the usual three-dimensional Fourier transform is replaced by a one-dimensional discrete Fourier transform. The model in this paper considers molecules made up of atoms with possibly different (integer-valued) atomic numbers. It is shown that information of order six suffices to determine a structure uniquely.
33 citations
33 citations
33 citations
Patent•
22 Oct 1969TL;DR: In making a record of the exact Fourier transform of an array of beams of electromagnetic radiation, the phase of each of a substantial fraction of the beams is shifted by a constant amount before recording the transform as discussed by the authors.
Abstract: In making a record of the exact Fourier transform of an array of beams of electromagnetic radiation, the phase of each of a substantial fraction of the beams is shifted by a constant amount before recording the transform.
33 citations
TL;DR: A memory efficient approximation to the nonuniform Fourier transform of a support limited sequence is derived based on the theory of shift-invariant representations and an exact expression for the worst-case mean square approximation error is derived.
Abstract: The main focus of this paper is to derive a memory efficient approximation to the nonuniform Fourier transform of a support limited sequence. We show that the standard nonuniform fast Fourier transform (NUFFT) scheme is a shift invariant approximation of the exact Fourier transform. Based on the theory of shift-invariant representations, we derive an exact expression for the worst-case mean square approximation error. Using this metric, we evaluate the optimal scale-factors and the interpolator that provides the least approximation error. We also derive the upper-bound for the error component due to the lookup tablebased evaluation of the interpolator; we use this metric to ensure that this component is not the dominant one. Theoretical and experimental comparisons with standard NUFFT schemes clearly demonstrate the significant improvement in accuracy over conventional schemes, especially when the size of the uniform fast Fourier transform (FFT) is small. Since the memory requirement of the algorithm is dependent on the size of the uniform FFT, the proposed developments can lead to iterative signal reconstruction algorithms with significantly lower memory demands.
33 citations