Spectrum Analysis of Signal in Wolfram Mathematica System

The purpose of this paper is a spectrum analysis of signals of various nature, construction of the signal scalogram using Morlet wavelet, modification of the scalogram to obtain a more informative graphic representation of the signal. Spectral analysis of the signal is constructed by means of the Fourier transform. A modification of the graphical representation of the result of the wavelet transform has been developed with the help of the Mathematica system. For this, a wavelet scalogram has been used as a two-dimensional representation of the original signal. A scale has been introduced on it for the value of the signal amplitude depending on the time and period of its constituent components. This graphical representation allows us to obtain additional information about the dynamic properties of the original signal. A modification of the representation of the original signal scalogram has been developed for a more complete spectrum analysis (determination of the period of the constituent components). The paper contains an example using a modified scalogram for the analysis of a signal containing two pulses, an audio signal and white noise. The basic wavelet in this case is the Morlet wavelet. A comparison of the scalogram, obtained using the built-in function, and the modified scalogram has been made in the paper. The disadvantage of the first scalogram is the impossibility of assessing the frequency of the signal; its advantage is the ability to assess the localization of the pulse. For a modified scalogram, the advantage is the estimation of the signal periodicity, and the disadvantage is the inaccuracy in determining the range of pulse localization. For spectrum analysis in Mathematica, it is recommended to use a combination of two approaches (using a standard built-in function to determine the localization of the pulse) and a modified scalogram (to determine the periods of the constituent components).

An uncertainty principle for the basic wavelet transform

The aim of this paper is to derive the uncertainty principle which has implications in signal analysis and in quantum mechanics. First, we derive the definition of the wavelet transform in [Formula: see text]-calculus by using some weight function. Certain properties like linearity, scaling, translation, etc. were discussed. Later on, to illustrate this integral transform several results were derived for the effectiveness and performance of the proposed method. Also, this paper surveys recent applications to establish generalized uncertainty principle.

Wavelet transformation on a finite interval

Integral transformations on a finite interval with a singular basis wavelet are considered. Using a sequence of such transformations, the problem of nonparametric approximation of a function is solved. Traditionally, it is assumed that the validity condition must be met for a basic wavelet (the average value of the wavelet must be zero). The paper develops the previously proposed method of singular wavelets when the tolerance condition is not met. In this case Delta-shaped functions that participate in Parzen – Rosenblatt and Nadaray – Watson estimations can be used as a basic wavelet. The set of wavelet transformations for a function defined on a numeric axis, defined locally, and on a finite interval were previously investigated. However, the study of the convergence of the decomposition on a finite interval was carried out only in one particular case. It was due to technical difficulties when trying to solve this problem directly. In the paper the idea of evaluating the periodic continuation of a function defined initially on a finite interval is implemented. It allowed to formulate sufficient convergence conditions for the expansion of the function in a series. An example of approximation of a function defined on a finite interval using the sum of discrete wavelet transformations is given.

Software tools for analysis and synthesis of stochastic systems with high availability (XI)

The article proceeds the thematic cycle dedicated to software tools for stochastic systems with high availability (StSHA) functioning at shock disturbances (ShD) and is dedicated to wavelet synthesis according to complex statistical criteria (CsC). Short survey concerning corresponding works for mean square criteria (msc) is given. In Sect.1 basic CsC definitions and approaches are given. Sect. 2 dedicated to CsC wavelet necessary and sufficient conditions of optimality for scalar non-stationary shock StSHA (StSSHA). Methodological support is based on Haar wavelets. Sect. 4 and Sect.5 are devoted to CSK optimization ShStSHA (basic wavelet equations, algorithms, software tools and example). Several advantages of wavelet algorithms and tools are described and stated for complex ShD. Some generalization of CsC algorithms based on wavelet canonical expansion of StP in ShStSHA mentioned.

Wavelet Neural Network for Modeling Chlorophyll a Concentration Affected by Artificial Upwelling

Through bringing nutrient-rich subsurface water to the surface, the artificial upwelling technology is applied to increase the primary marine productivity which could be assessed by Chlorophyll a concentration. Chlorophyll a concentration may vary with different water physical properties. Therefore, it is necessary to study the relationship between Chlorophyll a concentration and other water physical parameters. To ensure the accuracy of predicting the concentration of Chlorophyll a, we develop several models based on wavelet neural network (WNN). In this study, we build up a three-layer basic wavelet neural network followed by three improved wavelet neural networks, which are namely genetic algorithm-based wavelet neural network (GA-WNN), particle swarm optimization-based wavelet neural network (PSO-WNN), and genetic algorithm & particle swarm optimization-based wavelet neural network (GAPSO-WNN). The experimental data were collected from Qiandao Lake, China. The performances of the proposed models are compared based on four evaluation parameters, i.e., R-square, root mean square error (RMSE), mean of error (ME), and distance (D). The modeling results show that the wavelet neural network can achieve a certain extent of accuracy in modeling the relationships between Chlorophyll a concentration and the five input parameters (salinity, depth, temperature, pH, and dissolved oxygen).

Improving the reliability of the compressor unit using the wavelet transform method

Centrifugal compressors are an integral part of modern production in such industries as gas transmission, oil refining, metallurgical, machine-building, mining, as well as in electric and heat power engineering. Interruptions in the operation or failures of compressors lead to decrease in profit or large material loss. Conditions should be created for the safe (stable) operation of centrifugal compressors. Surge is global (complete) loss of stability, an unacceptable phenomenon for a centrifugal compressor. Compressor surge protection must function during operation. The algorithms used to protect centrifugal compressors against surge have some drawbacks, which makes it impossible to reliably prevent surges. There are many methods for analyzing rapidly changing processes in the flow part of a centrifugal compressor. The wavelet theory is the most accurate and modern method. The use of the wavelet transform method for signal processing allows us to solve the problems of analyzing non-stationary processes of a centrifugal compressor to expand the acceptable range of work and build reliable operation of the anti-surge diagnostic system. In the future, it is possible to use other basic wavelet functions, for comparison and selection of the most suitable one, for the analysis of unsteady signals in a centrifugal compressor.

High-resolution characterization of geologic structures using the synchrosqueezing transform

The main factors responsible for the nonstationarity of seismic signals are the nonstationarity of the geologic structural sequences and the complex pore structure. Time-frequency analysis can identify various frequency components of seismic data and reveal their time-variant features. Choosing a proper time-frequency decomposition algorithm is the key to analyze these nonstationarity signals and reveal the geologic information contained in the seismic data. According to the Heisenberg uncertainty principle, we cannot obtain the finest time location and the best frequency resolution at the same time, which results in the trade-off between the time resolution and the frequency resolution. For instance, the most commonly used approach is the short-time Fourier transform, in which the predefined window length limits the flexibility to adjust the temporal and spectral resolution at the same time. The continuous wavelet transform (CWT) produces an “adjustable” resolution of time-frequency map using dilation and translation of a basic wavelet. However, the CWT has limitations in dealing with fast varying instantaneous frequencies. The synchrosqueezing transform (SST) can improve the quality and readability of the time-frequency representation. We have developed a high-resolution and effective time-frequency analysis method to characterize geologic bodies contained in the seismic data. We named this method the SST, and the basic wavelet is the three-parameter wavelet (SST-TPW). The TPW is superior in time-frequency resolution than those of the Morlet and Ricker wavelets. Experiments on synthetic and field data determined its validity and effectiveness, which can be used in assisting in oil/gas reservoir identification.