L2 Boundedness of Discrete Double Hilbert Transform Along polynomials

We will show $L^{2}$ boundedness of Discrete Double Hilbert Transform along polynomials satisfying some conditions. Double Hilbert exponential sum along polynomials:$\mu(\xi)$ is Fourier multiplier of discrete double Hilbert transform along polynomials. In chapter 1, we define the reverse Newton diagram. In chapter 2, We make approximation formula for the multiplier of one valuable discrete Hilbert transform by study circle method. In chapter 3, We obtain result that $\mu(\xi)$ is bounded by constants if $|D|\geq2$ or all $(m,n)$ are not on one line passing through the origin. We study property of $1/(qt^{n})$ and use circle method (Propsotion 2.1) to calculate sums. We also envision combinatoric thinking about $\mathbb{N}^{2}$ lattice points in j-k plane for some estimates. Finally, we use geometric property of some inequalities about $(m,n)\in\Lambda$ to prove Theorem 3.3. In chapter 4, We obtain the fact that $\mu(\xi)$ is bounded by sums which are related to $\log_{2}({\xi_{1}-a_{1}\slash {q}})$ and $\log_{2}({\xi_{2}-a_{2}\slash {q}})$ and the boundedness of double Hilbert exponential sum for even polynomials with torsion without conditions in Theorem 3.3. We also use $\mathbb{N}^{2}$ lattice points in j-k plane and Proposition 2.1 which are shown in chapter 2 and some estimates to show that Fourier multiplier of discrete double Hilbert transform is bounded by terms about $\log$ and integral this with torsion is bounded by constants.

Prediction of Protein–Protein Interactions in Arabidopsis, Maize, and Rice by Combining Deep Neural Network With Discrete Hilbert Transform

Protein–protein interactions (PPIs) in plants play an essential role in the regulation of biological processes. However, traditional experimental methods are expensive, time-consuming, and need sophisticated technical equipment. These drawbacks motivated the development of novel computational approaches to predict PPIs in plants. In this article, a new deep learning framework, which combined the discrete Hilbert transform (DHT) with deep neural networks (DNN), was presented to predict PPIs in plants. To be more specific, plant protein sequences were first transformed as a position-specific scoring matrix (PSSM). Then, DHT was employed to capture features from the PSSM. To improve the prediction accuracy, we used the singular value decomposition algorithm to decrease noise and reduce the dimensions of the feature descriptors. Finally, these feature vectors were fed into DNN for training and predicting. When performing our method on three plant PPI datasets Arabidopsis thaliana, maize, and rice, we achieved good predictive performance with average area under receiver operating characteristic curve values of 0.8369, 0.9466, and 0.9440, respectively. To fully verify the predictive ability of our method, we compared it with different feature descriptors and machine learning classifiers. Moreover, to further demonstrate the generality of our approach, we also test it on the yeast and human PPI dataset. Experimental results anticipated that our method is an efficient and promising computational model for predicting potential plant–protein interacted pairs.

Hilbert transform revisited – Proper orthogonal decomposition applied to analytical signals of flow fields

The modes delivered by proper orthogonal decomposition (POD) are uncorrelated as per definition; but interestingly, they are not necessarily independent in terms of spatio-temporal flow-pattern dynamics. For instance, periodic structures that travel as waves through a series of snapshots often consist of pairs of modes with harmonic functions shifted 90 degree in phase and/or a spatial offset by a quarter of the spatial wave length of the convective flow pattern. Identification of such pairs, however, largely builds upon experience, visual inspection and/or the analysis of the reconstructed coefficients in cyclograms (Lissajous figures). This effort becomes even more challenging if measurement noise or other spurious information contaminates the raw data under consideration. One possibility to automatically pair corresponding patterns with common POD algorithms is the immediate application of the POD method to complex data (see Pfeffer et al., 1990). As outlined by Horel (1984), the Hilbert transform is a well-known and straight forward means to obtain the required extension of the original signal with an appropriate 90 degrees phase shift, which is independent of the fundamental frequencies. The complex extension of the original (real) signal Xi and its (discrete) Hilbert transform HT{Xi} as the imaginary part Xi +iHT{Xi} with the imaginary unit i is commonly known as the so-called analytical signal.

Advanced Signal Processing Methods for Inspection of Aircraft Structural Materials

Aircraft, their assemblies, and units must provide high durability and reliability, and maintain mechanical and technological characteristics throughout the life span of the aircraft. Different elements of aircraft structures work under mechanical loads, over a wide temperature range, with varying degrees of exposure to corrosive environments. Aircraft structural materials have a variation in the characteristics values and require the various testing methods for their inspection.In many NDT methods applied in aviation materials testing, signals that could be represented by a narrowband processes model are used. Known methods of their processing are focused on determining and analyzing the signals amplitude characteristics, but the information resource contained in phase characteristics is not used.In the article, the methodology for signal processing and determining phase characteristics in the time domain are discussed. It is based on the combination of the discrete Hilbert transform and the deterministic and statistical methods of the phase measurement. There are given examples of the application of the methodology for pulsed eddy current testing of electrically conductive materials and products, ultrasonic thickness measurement of products made of materials have significant ultrasonic attenuation, the realization impulse variant of acoustic impedance flaw detection of products made of composite materials. The examples have shown that the proposed signal processing methodology enables to determine new information parameters and signal characteristics for the industry, and extend the scope of known NDT methods.

A causal imaging condition for reverse time migration using the Discrete Hilbert transform and its efficient implementation on GPU

Reverse time migration (RTM) has shown a significant advantage over other imaging algorithms for imaging complex subsurface structures. However, low-wavenumber noise severely contaminates the image, which is one of the main issues in the RTM algorithm. To attenuate the undesired low-wavenumber noise, the causal imaging condition based on wavefield decomposition has been proposed. First, wavefield decompositions are performed to separate the wavefields as up-going and down-going wave components, respectively. Then, to preserve causality, it constructs images by correlating wave components that propagate in different directions. We build a causal imaging condition in this paper. Not only does it consider the up/down wavefield decomposition, but it also applies the decomposition on the horizontal direction to enhance the image quality especially for steeply dipping structures. The wavefield decomposition is conventionally achieved by the frequency-wavenumber (F-K) transform that is very computationally intensive compared with the wave propagation process of the RTM algorithm. To improve the efficiency of the algorithm, we propose a fast implementation to perform wavefield separation using the discrete Hilbert transform via the Graphics Processing Unit. Numerical tests on both the synthetic models and a real data example demonstrate the effectiveness of the proposed method and the efficiency of the optimized implementation scheme. This new imaging condition shows its ability to produce high image quality when applied to both the RTM stack image and also the angle domain common image gathers. The comparison of the total elapsed time for different methods verifies the efficiency of the optimized algorithm.