Sparse phase retrieval of structured signals by Prony's method

The phase retrieval problem consists in the recovery of a complex-valued signal from the magnitudes of its Fourier transform. Restricting ourselves to the case of sparse structured signals f, which can be represented as a linear combination of N arbitrary translations of a given generator function, we show that almost all f can be recovered from 𝒪 (N²) intensity measurements |ℱ[f](ω)| up to trivial ambiguities. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On signal reconstruction from FROG measurements

Phase retrieval refers to recovering a signal from its Fourier magnitude. This problem arises naturally in many scientific applications, such as ultra-short laser pulse characterization and diffraction imaging. Unfortunately, phase retrieval is ill-posed for almost all one-dimensional signals. In order to characterize a laser pulse and overcome the ill-posedness, it is common to use a technique called Frequency-Resolved Optical Gating (FROG). In FROG, the measured data, referred to as FROG trace, is the Fourier magnitude of the product of the underlying signal with several translated versions of itself. The FROG trace results in a system of phaseless quartic Fourier measurements. In this paper, we prove that it suffices to consider only three translations of the signal to determine almost all bandlimited signals, up to trivial ambiguities. In practice, one usually also has access to the signal's Fourier magnitude. We show that in this case only two translations suffice. Our results significantly improve upon earlier work.     

Compressive Sampling via Random Convolution

Several recent results in compressive sampling show that a sparse signal (i.e. a signal which can be compressed in a known orthobasis) can be efficiently acquired by taking linear measurements against random test functions. In this paper, we show that these results can be extended to measurements taken by convolving with a random pulse and then subsampling. The measurement scheme is universal in that it complements (with high probability) any fixed orthobasis we use to represent the signal. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Similarity analysis of DNA sequences based on the EMD method

DNA sequences can be translated into 2D graphs and into numerical sequences; we call the numerical sequences nonlinear signal sequences. We can use the empirical mode decomposition (EMD) method to divide nonlinear signal sequences into a group of well-behaved intrinsic mode functions (IMFs) and a residue, so that we can compare the similarities among DNA sequences conveniently and intuitively. This work tests the method’s suitability by using the mitochondria of four different species.     

Phase retrieval techniques for radar ambiguity problems

The radar ambiguity function plays a central role in the theory of radar signals. Its absolute value (¦A(u)¦) measures the correlation between the signal u emitted by the radar transmitter and its echo after reaching a moving target. It is important to know signals that give rise to ambiguity functions of given shapes. Therefore, it is also important to know to what extent ¦A(u)¦ determines the signal. This problem is called the radar ambiguity problem by Bueckner [5]. Using methods developed for phase retrieval problems, we give here a partial answer for some classes of time limited (compactly supported) signals. In doing so, we also obtain results for Pauli's problem; in particular, we build functions that have infinitely many Pauli partners.     

Frame Analysis of Irregular Periodic Sampling of Signals and Their Derivatives

Given a bandlimited signal, we consider the sampling of the signal and some of its derivatives in a periodic manner. The mathematical concept of frames is utilized in the analysis of the properties of the sequence of sampling functions. The frame operator of this sequence is expressed as a matrix-valued function multiplying a vector-valued function. An important property of this matrix is that the maximum and minimum eigenvalues are equal (in some sense) to the upper and lower frame bounds. We present a method for finding the dual frame and, thereby, a method for reconstructing the signal from its samples. Using the matrix approach we prove that the sequence of sampling functions is always complete in the cases of critical sampling and oversampling. A sufficient condition for the sequence of sampling functions to constitute a frame is derived. We show that if no sampling of the signal itself is involved, the sampling is not stable and cannot be stabilized by oversampling. Examples are considered, and the frame bounds in the case of sampling of the signal and its first derivative are calculated explicitly. Finally, the matrix approach can be similarly applied to other problems of signal representation.     

在一类非平稳干扰下的极大信噪比线性滤波问题

<正> 随机过程的滤波问题无论在理论上还是在工程上都是十分重要的,极大信噪比滤波是最佳滤波中重要的一类.比如雷达中的信号检测,图象信号的处理,以及一般的数字通信等。但已有的结果都比较简单,随着工程技术的发展,日愈迫切地提出在时变信道下极大信噪比的滤波问题。关于具有二阶矩的非平稳过程的理论已有不少一般性的讨     

Attractive Periodic Orbits in Nonlinear Discrete-time Neural Networks with Delayed Feedback

We consider the discrete-time system x ( n )= g x ( n m 1)+ f ( y ( n m k )), y ( n )= g y ( n m 1)+ f ( x ( n m k )), n ] N describing the dynamic interaction of two identical neurons, where g ] (0,1) is the internal decay rate, f is the signal transmission function and k is the signal transmission delay. We construct explicitly an attractive 2 k -periodic orbit in the case where f is a step function (McCulloch-Pitts Model). For the general nonlinear signal transmission functions, we use a perturbation argument and sharp estimates and apply the contractive map principle to obtain the existence and attractivity of a 2 k -periodic orbit. This is contrast to the continuous case (a delay differential system) where no stable periodic orbit can occur due to the monotonicity of the associated semiflow.     

Sampling Theorem for Entire Functions of Exponential Growth

Applying the theory of generalized functions we obtain the Shannon sampling theorem for entire functions F(z) of exponential growth and give its error estimate which shows how much the error depends on the sampling size and bandwidth for given domain of the signal F(z). As an application we obtain a uniqueness theorem for entire functions and temperature functions.     

Windowed Fourier transform of two-dimensional quaternionic signals

In this paper, we generalize the classical windowed Fourier transform (WFT) to quaternion-valued signals, called the quaternionic windowed Fourier transform (QWFT). Using the spectral representation of the quaternionic Fourier transform (QFT), we derive several important properties such as reconstruction formula, reproducing kernel, isometry, and orthogonality relation. Taking the Gaussian function as window function we obtain quaternionic Gabor filters which play the role of coefficient functions when decomposing the signal in the quaternionic Gabor basis. We apply the QWFT properties and the (right-sided) QFT to establish a Heisenberg type uncertainty principle for the QWFT. Finally, we briefly introduce an application of the QWFT to a linear time-varying system.     

Construction of data-adaptive orthogonal wavelet bases with an extension of principal component analysis

We show that orthonormal bases of functions with multiscale compact supports can be obtained from a generalization of principal component analysis. These functions, called multiscale principal components (MPCs), are eigenvectors of the correlation operator expressed in different vector subspaces. MPCs are data-adaptive functions that minimize their correlation with the reference signal. Using MPCs, we construct orthogonal bases which are similar to dyadic wavelet bases. We observe that MPCs are natural wavelets, i.e. their average is zero or nearly zero if the signal has a dominantly low-pass spectrum. We show that MPCs perform well in simple data compression experiments, in the presence or absence of singularities. We also introduce concentric MPCs, which are orthogonal basis functions having multiscale concentric supports. Use as kernels in convolution products with a signal, these functions allow to define a wavelet transform that has a striking capacity to emphasize atypical patterns.     

The Mellin–Parseval formula and its interconnections with the exponential sampling theorem of optical physics

In this paper, we establish a Mellin version of the classical Parseval formula of Fourier analysis in the case of Mellin bandlimited functions, and its equivalence with the exponential sampling formula (ESF) of signal analysis, in which the samples are not equally spaced apart as in the classical Shannon theorem, but exponentially spaced. Two quite different examples are given illustrating the truncation error in the ESF. We employ Mellin transform methods for square-integrable functions.     

Consecutive minimum phase expansion of physically realizable signals with applications

In digital signal processing, it is a well know fact that a causal signal of finite energy is front loaded if and only if the corresponding analytic signal, or the physically realizable signal, is a minimum phase signal, or an outer function in the complex analysis terminology. Based on this fact, a series expansion method, called unwinding adaptive Fourier decomposition (AFD), to give rise to positive frequency representations with rapid convergence was proposed several years ago. It appears to be a promising positive frequency representation with great potential of applications. The corresponding algorithm, however, is complicated due to consecutive extractions of outer functions involving computation of Hilbert transforms. This paper is to propose a practical algorithm for unwinding AFD that does not depend on computation of Hilbert transform, but, instead, factorizes out the Blaschke product type of inner functions. The proposed method significantly improves applicability of unwinding AFD. As an application, we give the associated Dirac‐type time‐frequency distribution of physically realizable signals. Copyright © 2015 John Wiley & Sons, Ltd.     

Uncertainty principles and optimally sparse wavelet transforms

In this paper we introduce a new localization framework for wavelet transforms, such as the 1D wavelet transform and the Shearlet transform. Our goal is to design nonadaptive window functions that promote sparsity in some sense. For that, we introduce a framework for analyzing localization aspects of window functions. Our localization theory diverges from the conventional theory in two ways. First, we distinguish between the group generators, and the operators that measure localization (called observables). Second, we define the uncertainty of a signal transform as a whole, instead of defining the uncertainty of an individual window. We show that the uncertainty of a window function, in the signal space, is closely related to the localization of the reproducing kernel of the wavelet transform, in phase space. As a result, we show that using uncertainty minimizing window functions, results in representations which are optimally sparse in some sense.     

The Complexity of Gene Placement

We focus on algorithmic problems related to deriving gene locations on DNA sequences of closely related species by using comparative mapping data. Conventional genetic mapping generates intervals on the DNA sequence of given species for potential gene positions. The simultaneous analysis of gene intervals in related species, e.g., human and mouse, may eliminate some of the ambiguities and lead to better estimates of gene locations. We address the problem of eliminating the ambiguities in gene orders by means of minimizing the number of conserved regions among the species. This is equivalent to the problem of choosing gene coordinates (gene placement) that satisfy the genetic mapping constraints and minimize the breakpoint distance between genomes. We first show that the gene ordering problem is hard: there is no polynomial-time approximation scheme unless P = NP, even under the restrictions that: (1) the order of genes in one of species is known, or (2) at most two intervals overlap at any location on the map of any of the species. Then we provide two polynomial-time algorithms under restriction (1) above; the first approximates the problem within a factor of 3, and the second exactly solves the problem under the additional restriction that (3) no more than O((log n)/(log log n)) intervals overlap at a location on the map of any of the species. We also prove the tractability of the general problem when there is a single conserved region (i.e., when there exists a gene placement resulting in identical gene orders).     

Sign or root of unity ambiguities of certain Gauss sums

Gauss sums play an important role in number theory and arithmetic geometry. The main objects of study in this paper are Gauss sums over the finite field with q elements. Recently, the problem of explicit evaluation of Gauss sums in the small index case has been studied in several papers. In the process of the evaluation, it is realized that a sign (or a root of unity) ambiguity unavoidably occurs. These papers determined the ambiguities by the congruences modulo L, where L is certain divisor of the order of Gauss sum. However, such method is unavailable in some situations. This paper presents a new method to determine the sign (root of unity) ambiguities of Gauss sums in the index 2 case and index 4 case, which is not only suitable for all the situations with q being odd, but also comparatively more efficient and uniform than the previous method.     

Thoughts on John of Saxony's method for finding times of true syzygy

This article examines John of Saxony's iterative method for finding the times from mean to true syzygy (i.e., conjunction or opposition of the Moon and Sun). It argues that the method, composed c. 1330, contains several ambiguities, but is so robust that only one of these ambiguities affects the time correction. Furthermore, the method yields times of true syzygy that correspond, to the nearest minute, to the time when the true elongation, as computed by the planetary equations of the 1483 Alfonsine Tables, makes its closest approach to 0° or 180°. Hence John's method yields “exact” Alfonsine times, unlike all other known medieval methods or tables that only approximate those results. It will also be shown that John Somer (1380s) and Regiomontanus (1440–1450s) wielded John's method with considerable computational skill.     

Nonstationary Gabor frames for

Recently, continuous‐time nonstationary Gabor (NSG) frames were introduced in adaptive signal analysis. They allow for efficient reconstruction with flexible sampling and varying window functions. In this paper, we focus on the existence and construction of NSG frames in the discrete‐time setting. We provide existence results for painless NSG frames and for NSG frames with fast decaying window functions. We also construct NSG frames with noncompactly supported window functions from a known painless NSG frame. Some examples are provided to illustrate the general theory.     

Pattern analysis of the genetic code

The genetic code is examined in a new and systematic fashion: we consider the code as mapping of one finite set (the 64 codons) to another (the 20 amino acids). Given a class of mappings simpler than the actual code, we ask which mappings best approximate it. This leads to an analysis of the effects of ambiguities (codon degeneracy) in one or two positions. With the 0–1 metric (counting the amino acids as equal or not equal), the codon third base degeneracy is apparent, but the first and second positions are indistinguishable; with the integrated amino acid “distance” metric compiled by Sneath (J. Theoret. Biol. 12 (1966), 157–195), the analysis ranks the information content of the three codon positions as follows: 2nd > 1st > 3rd. We discuss possible further applications of this approach to patterns in the genetic code and other codes.