Structured random measurements in signal processing

Krahmer, Felix; Rauhut, Holger

doi:10.1002/gamm.201410010

Cited by 26 publications

(28 citation statements)

References 120 publications

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“…Randomness plays a key role in many theoretical results, and often leads to good empirical results [12]. Near optimal conditions for Gaussian, sub-Gaussian, Bernoulli and random Fourier matrices have been derived [16], [17] (and references therein). However, the measurement matrix in FAR differs from these random matrices, so that previous theoretical results are not directly applicable.…”

Section: Introductionmentioning

confidence: 99%

Analysis of Frequency Agile Radar via Compressed Sensing

Huang

Liu

et al. 2018

IEEE Trans. Signal Process.

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Frequency agile radar (FAR) is known to have excellent electronic counter-countermeasures (ECCM) performance and the potential to realize spectrum sharing in dense electromagnetic environments. Many compressed sensing (CS) based algorithms have been developed for joint range and Doppler estimation in FAR. This paper considers theoretical analysis of FAR via CS algorithms. In particular, we analyze the properties of the sensing matrix, which is a highly structured random matrix. We then derive bounds on the number of recoverable targets. Numerical simulations and field experiments validate the theoretical findings and demonstrate the effectiveness of CS approaches to FAR.

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Section: Introductionmentioning

confidence: 99%

Analysis of Frequency Agile Radar via Compressed Sensing

Huang

Liu

et al. 2018

IEEE Trans. Signal Process.

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“…The utility of this new, decoupled form Q L dec (a, a , w) of L(a, w) is that it introduces extra randomness -Q L dec (a, a , w) is now a chaos process of a conditioned on a . This makes analyzing sup w∈S Q L dec (a, a , w) amenable to existing analysis of suprema of chaos processes for random circulant matrices [44]. However, achieving the decoupling requires additional work; the most general existing results on decoupling pertain to tetrahedral polynomials, which are polynomials with no monomials involving any power larger than one of any random variable.…”

Section: Why Decoupling?mentioning

confidence: 99%

“…In comparison, the contraction region we show for the random convolutional model is larger O(1/polylog(n)), which is achievable in the initialization stage via the spectral method. For a more detailed review of this subject, we refer the readers to Section 4 of [44].The convolutional measurement can also be reviewed as a single masked coded diffraction patterns [34,50], since a x = F −1 ( a x), where a is the Fourier transform of a and x is the oversampled Fourier transform of x. The sample complexity for coded diffraction patterns m ≥ Ω(n log 4 n) in [34] suggests that the dependence of our sample complexity on C x for convolutional phase retrieval might not be necessary and can be improved in the future.…”

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confidence: 99%

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Convolutional Phase Retrieval via Gradient Descent

Zhang

Eldar

et al. 2020

IEEE Trans. Inform. Theory

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We study the convolutional phase retrieval problem, of recovering an unknown signal x ∈ C n from m measurements consisting of the magnitude of its cyclic convolution with a given kernel a ∈ C m . This model is motivated by applications such as channel estimation, optics, and underwater acoustic communication, where the signal of interest is acted on by a given channel/filter, and phase information is difficult or impossible to acquire. We show that when a is random and the number of observations m is sufficiently large, with high probability x can be efficiently recovered up to a global phase shift using a combination of spectral initialization and generalized gradient descent. The main challenge is coping with dependencies in the measurement operator. We overcome this challenge by using ideas from decoupling theory, suprema of chaos processes and the restricted isometry property of random circulant matrices, and recent analysis of alternating minimization methods. ). Most of the work has been conducted when QQ and YZ were with Department of Electrical Engineering and Data Science Institute at Columbia University. An extended abstract of the current work has appeared in NeurIPS'17 [1]. 1 The convolution can be written in matrix-vector form as a x = Caιn→mx, where Ca ∈ R m×m denotes the circulant matrix generated by a and ιn→m is a zero padding operator. In other words, a x = Ax with A ∈ C m×n being a matrix formed by the first n columns of Ca. arXiv:1712.00716v3 [stat.CO] 6 Oct 2019Structured random measurements. The study of structured random measurements in signal processing has quite a long history [44]. For compressed sensing [45], the work [46-48] studied random Fourier measurements, and later [13,14] proved similar results for partial random convolution measurements. However, the study of structured random measurements for phase retrieval is still quite limited. In particular, [49] and [50] studied t-designs and coded diffraction patterns (i.e., random masked Fourier measurements) using semidefinite programming. Recent work studied nonconvex optimization using coded diffraction patterns [34] and STFT measurements [51], both of which minimize a nonconvex objective similar to (1.5). These measurement models are motivated by different applications. For instance, coded diffraction is designed for imaging applications such as X-ray diffraction imaging, STFT can be applied to frequency resolved optical gating [52] and some speech processing tasks [53]. Both of the results show iterative contraction in a region that is at most O(1/ √ n)-close to the optimum. Unfortunately, for both results either the radius of the contraction region is not large enough for initialization to reach, or they require extra artificial technique such as resampling the data. In comparison, the contraction region we show for the random convolutional model is larger O(1/polylog(n)), which is achievable in the initialization stage via the spectral method. For a more detailed review of this subject, we refer the readers to Section 4 of [44]....

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“…by exploiting prior knowledge about the signal of interest) for sampling the signals. Second, those approaches which attempt to improve the structure of an initially random measurement matrix using optimization techniques [5][6][7][8][9][10]. In this paper, the first family of approaches is addressed.…”

Section: Introductionmentioning

confidence: 99%

A block-wise random sampling approach: Compressed sensing problem

Abolghasemi

Ferdowsi

Sanei

2015

JAIDM

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The focus of this paper is to consider the compressed sensing problem. It is stated that the compressed sensing theory, under certain conditions, helps relax the Nyquist sampling theory and takes smaller samples. One of the important tasks in this theory is to carefully design measurement matrix (sampling operator). Most existing methods in the literature attempt to optimize a randomly initialized matrix with the aim of decreasing the amount of required measurements. However, these approaches mainly lead to sophisticated structure of measurement matrix which makes it very difficult to implement. In this paper we propose an intermediate structure for the measurement matrix based on random sampling. The main advantage of blockbased proposed technique is simplicity and yet achieving acceptable performance obtained through using conventional techniques. The experimental results clearly confirm that in spite of simplicity of the proposed approach it can be competitive to the existing methods in terms of reconstruction quality. It also outperforms existing methods in terms of computation time.

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Structured random measurements in signal processing

Cited by 26 publications

References 120 publications

Analysis of Frequency Agile Radar via Compressed Sensing

Analysis of Frequency Agile Radar via Compressed Sensing

Convolutional Phase Retrieval via Gradient Descent

A block-wise random sampling approach: Compressed sensing problem

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