In this paper, we discuss the development of a sublinear sparse fourier algorithm for high-dimensional data. In 11Adaptive Sublinear Time fourier Algorithm" by Lawlor et al. (Adv. Adapt. Data Anal. 5(01):1350003,...
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In this paper, we discuss the development of a sublinear sparse fourier algorithm for high-dimensional data. In 11Adaptive Sublinear Time fourier Algorithm" by Lawlor et al. (Adv. Adapt. Data Anal. 5(01):1350003, 2013), an efficient algorithm with Theta(k log k) average-case runtime and Theta(k) average-case sampling complexity for the one-dimensional sparse FFT was developed for signals of bandwidth N, where k is the number of significant modes such that k << N. In this work we develop an efficient algorithm for sparse FFT for higher dimensional signals, extending some of the ideas in Lawlor et al. (Adv. Adapt. Data Anal. 5(01):1350003, 2013). Note a higher dimensional signal can always be unwrapped into a one-dimensional signal, but when the dimension gets large, unwrapping a higher dimensional signal into a one-dimensional array is far too expensive to be realistic. Our approach here introduces two new concepts: "partial unwrapping" and "tilting." These two ideas allow us to efficiently compute the sparse FFT of higher dimensional signals.
We extend the recent sparse fourier transform algorithm of [1] to the noisy setting, in which a signal of bandwidth N is given as a superposition of k << N frequencies and additive random noise. We present two s...
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We extend the recent sparse fourier transform algorithm of [1] to the noisy setting, in which a signal of bandwidth N is given as a superposition of k << N frequencies and additive random noise. We present two such extensions, the second of which exhibits a form of error-correction in its frequency estimation not unlike that of the beta-encoders in analog-to-digital conversion [2]. On k-sparse signals corrupted with additive complex Gaussian noise, the algorithm runs in time O(k log(k) log(N/k)) on average, provided the noise is not overwhelming. The error-correction property allows the algorithm to outperform FFTW [3], a highly optimized software package for computing the full discrete fourier transform, over a wide range of sparsity and noise values. (C) 2015 Elsevier Inc. All rights reserved.
In this paper, the concept of the two-dimensional discrete fourier transformation (2-D DFT) is defined in the general case, when the form of relation between the spatial-points (x, y) and frequency-points (omega(1), o...
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ISBN:
(纸本)9781628414899
In this paper, the concept of the two-dimensional discrete fourier transformation (2-D DFT) is defined in the general case, when the form of relation between the spatial-points (x, y) and frequency-points (omega(1), omega(2)) is defined in the exponential kernel of the transformation by a nonlinear form L(x, y;omega(1), omega(2)). The traditional concept of the 2-D DFT uses the Diaphanous form x omega(1) + y omega(2) and this 2-D DFT is the particular case of the fourier transform described by the form L(x, y;omega(1), omega(2)). Properties of the general 2-D discrete fourier transform are described and examples are given. The special case of the N x N-point 2-D fourier transforms, when N = 2(r), r > 1, is analyzed and effective representation of these transforms is proposed. The proposed concept of nonlinear forms can be also applied for other transformations such as Hartley, Hadamard, and cosine transformations.
In this work we derive two families of radix-4 factorizations for the FFT (fastfourier Transform) that have the property that both inputs and outputs are addressed in natural order. These factorizations are obtained ...
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ISBN:
(纸本)9781424432974
In this work we derive two families of radix-4 factorizations for the FFT (fastfourier Transform) that have the property that both inputs and outputs are addressed in natural order. These factorizations are obtained from another two families of radix-2 algorithms that have the same property. The radix-4 algorithms obtained have the same mathematical complexity (number of multiplications and additions) that Cooley-Tukey radix-4 algorithms but avoid de bit-reversal ordering applied to the input or at the output.
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