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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IBM1
TL;DR: The problem of establishing the correspondence between the discrete transforms and the continuous functions with which one is usually dealing is described and formulas and empirical results displaying the effect of optimal parameters on computational efficiency and accuracy are given.
251 citations
251 citations
19 May 2012
TL;DR: If one assumes that the Fast Fourier Transform is optimal, the algorithm for the exactly k-sparse case is optimal for any k = nΩ(1), and the first known algorithms that satisfy this property are shown.
Abstract: We consider the problem of computing the k-sparse approximation to the discrete Fourier transform of an n-dimensional signal. We show: An O(k log n)-time randomized algorithm for the case where the input signal has at most k non-zero Fourier coefficients, and An O(k log n log(n/k))-time randomized algorithm for general input signals. Both algorithms achieve o(n log n) time, and thus improve over the Fast Fourier Transform, for any k=o(n). They are the first known algorithms that satisfy this property. Also, if one assumes that the Fast Fourier Transform is optimal, the algorithm for the exactly k-sparse case is optimal for any k = nΩ(1). We complement our algorithmic results by showing that any algorithm for computing the sparse Fourier transform of a general signal must use at least Ω(k log (n/k) / log log n) signal samples, even if it is allowed to perform adaptive sampling.
250 citations
Book•
17 Jul 2003
TL;DR: In this article, the ubiquitous convolution was used for multidimensional Fourier analysis and the Discrete Fourier Transform (DFT) transform was used to transform the Fourier series into a discrete Fourier transform.
Abstract: Introduction.- Preparations.- Laplace and Z Transforms.- Fourier Series.- L^2 Theory.- Separation of Variables.- Fourier Transforms.- Distributions.- Multi-Dimensional Fourier Analysis.- Appendix A: The ubiquitous convolution.- Appendix B: The Discrete Fourier Transform.- Appendix C: Formulae.- Appendix D: Answers to exercises.- Appendix E: Literature.
248 citations