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Journal Article

Mathematical Analysis of Random Noise-Conclusion

01 Jan 1945-Bell System Technical Journal-Vol. 24, pp 46-156
About: This article is published in Bell System Technical Journal.The article was published on 1945-01-01 and is currently open access. It has received 807 citations till now.
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Book ChapterDOI
01 Jan 2015
TL;DR: This work detail a novel, probabilistic approach to compressive phase retrieval based on the generalized approximate message passing (GAMP) algorithm, and presents a numerical study of the proposed PR-GAMP algorithm, demonstrating its excellent phase-transition behavior, robustness to noise, and runtime.
Abstract: In phase retrieval, the goal is to recover a signal \(\boldsymbol{x} \in \mathbb{C}^{N}\) from the magnitudes of linear measurements \(\boldsymbol{Ax} \in \mathbb{C}^{M}\). While recent theory has established that M ≈ 4 N intensity measurements are necessary and sufficient to recover generic \(\boldsymbol{x}\), there is great interest in reducing the number of measurements through the exploitation of sparse \(\boldsymbol{x}\), which is known as compressive phase retrieval. In this work, we detail a novel, probabilistic approach to compressive phase retrieval based on the generalized approximate message passing (GAMP) algorithm. We then present a numerical study of the proposed PR-GAMP algorithm, demonstrating its excellent phase-transition behavior, robustness to noise, and runtime. For example, to successfully recover K-sparse signals, approximately \(M \geq 2K\log _{2}(N/K)\) intensity measurements suffice when K ≪ N and \(\boldsymbol{A}\) has i.i.d Gaussian entries. When recovering a 6k-sparse 65k-pixel grayscale image from 32k randomly masked and blurred Fourier intensity measurements, PR-GAMP achieved 99% success rate with a median runtime of only 12. 6 seconds. Compared to the recently proposed CPRL, sparse-Fienup, and GESPAR algorithms, experiments show that PR-GAMP has a superior phase transition and orders-of-magnitude faster runtimes as the problem dimensions increase.

2 citations

Journal ArticleDOI
Igor Rychlik1, Eva Sjö1
TL;DR: In this article, the joint density of (T, Z), the position and height of the global maximum of ω in a closed interval, is derived using the generalized Rice's formula.
Abstract: For a stochastic process ω with absolutely continuous sample path derivative, a formula for the joint density of (T, Z), the position and height of the global maximum of ω in a closed interval, is given. The formula is derived using the generalized Rice’s formula. The presented result can be applied both to stationary and non-stationary processes under mild assumptions on the process. The formula for the density is explicit but involves integrals that have to be computed using numerical integration. The computation of the density is discussed and some numerical examples are given.

2 citations

Journal ArticleDOI
TL;DR: In this paper, for the case of auditory energy detection in which a noise-filled interval is compared with a sine wave in noise, the psychometric functions are shown to have polynomial form.

2 citations

Journal ArticleDOI
TL;DR: The relation between two simple approaches to estimate the extreme ship response used when no, or a limited, amount of data are available is discussed in this paper, where the long term distribution of the local maxima of ship response and the expected number of upcrossings of a level by the response are compared.
Abstract: In this note the relation between two simple approaches to estimate the extreme ship response used when no, or a limited, amount of data are available is discussed. The first one employs the long term distribution of the local maxima of ship response while the second one uses the expected number of upcrossings of a level by the response. It is mathematically demonstrated that the two approaches are equivalent. However, the upcrossing method is more straightforward and convenient for practical applications, particularly for non-Gaussian responses. The full-scale measurements of a 2800 TEU container ship during the first six months of 2008 are used in the comparisons.

2 citations

Journal ArticleDOI
TL;DR: In this paper, a modified Rice formulation is proposed to generate ergodic Gaussian random noise with specified spectral properties, which is the most efficient with regard to the computation time.

2 citations