Estimation of Signal Parameters Using SSA and Linear Transformation of Covariance Matrix or Data Matrix

Polarization filters are widely used for denoising seismic data. These filters are applied in the field of seismology, microseismic monitoring, vertical seismic profiling and subsurface imaging. They are primarily useful to suppress ground-roll in seismic reflection data and enhance P and S wave arrivals. Traditional implementations of the polarization filters involved the analysis of the covariance matrix or the SVD decomposition of a three-component seismogram matrix. The linear polarization level, known as rectilinearity, is expressed as a function of the covariance matrix eigenvalues or by the data matrix singular values. Wavefield records that are linearly polarized are amplified while others are attenuated. Besides the described implementations, we introduced a new time domain polarization filter based on the analysis of the seismic data correlation matrix. The principal idea is to extend the notion of the correlation coefficient in 3D space. This new filter avoids the need for diagonalization of the covariance matrix or SVD decomposition of data matrix, which are often time consuming. The new implementation facilitates the choice of the rectilinearity threshold: we demonstrate that linear polarization in 3D can be represented as three classic 2D correlations. A good linear polarization is detected when a high linear correlation between the three seismogram components is mutually observed. The tuning parameters of the new filter are the length of the time window, the filter order, and the rectilinearity threshold. Tests using synthetic seismograms show that optimal results are reached with a filter order that spans between 2 and 4, a rectilinearity threshold between 0.3 and 0.7, and a window length that is equivalent to one to three times the period of the signal wavelet. Compared to covariance-based filters, the new filter can enhance the signal-to-noise ratio by 6 to 20 dB and reduces computational costs by 25%.

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TRANSFORMATIONS, GENERATION AND MEASUREMENT OF PHOTONIC STATES THAT POSSESS A GAUSSIAN DENSITY MATRIX

International Journal of Modern Physics B ◽

10.1142/s0217979292001456 ◽

1992 ◽

Vol 06 (20) ◽

pp. 3309-3325

Author(s):

ROBERT R. TUCCI

Keyword(s):

Density Matrix ◽

Covariance Matrix ◽

Linear Transformation ◽

Covariance Matrices ◽

Squeezed States ◽

Linear Transformations ◽

Gaussian Density ◽

Mean Fields ◽

Photonic States ◽

Multi Mode

In this paper, we consider issues of importance in dealing with 1-mode and 2-mode photonic states that possess a Gaussian density matrix. By a Gaussian density matrix or a Gaussian “rho”, we mean an exponential of a quadratic polynomial of the creation and annihilation operators of the modes. Gaussian-rho states, which include the socalled squeezed states of light, are uniquely specified by the covariance matrix of their modes (provided we assume zero mean fields). We discuss the set of all possible linear transformations of N modes into an equal number of modes. This set is shown to be a group that is isomorphic to one of the symplectic groups. We propose a device, which we call a covmuter, that can perform any linear transformation in this group. Given a fixed covariance matrix Vo of a Gaussian-rho state, we find all covariance matrices V for which there exists a multi-mode linear transformation that takes Vo to V. We study the conservation laws obeyed by such transformations. Finally, we show that a covmuter can be used both to generate and to measure 1-mode and 2-mode Gaussian-rho states.

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On the Reduction of Transmission Complexity in MIMO-WCDMA Frequency-Selective Fading Orientations via Eigenvalue Analysis

Electronics ◽

10.3390/electronics7100239 ◽

2018 ◽

Vol 7 (10) ◽

pp. 239 ◽

Cited By ~ 1

Author(s):

P. Gkonis ◽

D. Kaklamani ◽

I. Venieris ◽

C. Dervos ◽

M. Chrysomallis ◽

...

Keyword(s):

Covariance Matrix ◽

Principal Component ◽

Complexity Reduction ◽

Data Matrix ◽

Transmission Matrix ◽

Selective Fading ◽

Matrix Formulation ◽

Frequency Selective Fading ◽

Frequency Selective ◽

Code Division Multiple

In this paper, a novel transmission strategy for Mutliple Input Multiple Output Wideband Code Division Multiple Access (MIMO-WCDMA) orientations operating in frequency-selective fading environments is investigated, in terms of overall algorithmic complexity reduction. To this end, Principal Component Analysis (PCA) is employed on the received data matrix, in order to define the significant terms that are taken into account during transmission matrix formulation. According to the presented results, feedback information of only the primary eigenvector of the corresponding covariance matrix of the received data matrix is required, in order to maintain the mean Bit Error Rate (BER) at acceptable levels. In particular, a complexity reduction of up to 10% can be achieved, when comparing BER values derived by the selection of all components of the received covariance matrix during transmission matrix formulation, and the corresponding BER when selecting half of the components. This reduction is maintained to 10%, when considering a realistic four-element antenna design; however, in this case mean BER inaccuracy is further reduced to 1%.

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Analyzing Raman Maps of Pharmaceutical Products by Sample—Sample Two-Dimensional Correlation

Applied Spectroscopy ◽

10.1366/0003702053946047 ◽

2005 ◽

Vol 59 (5) ◽

pp. 630-638 ◽

Cited By ~ 21

Author(s):

Slobodan Šašić ◽

Donald A. Clark ◽

John C. Mitchell ◽

Martin J. Snowden

Keyword(s):

Covariance Matrix ◽

Principal Component ◽

Pharmaceutical Products ◽

Correlation Spectroscopy ◽

Data Matrix ◽

Two Dimensional ◽

Data Set ◽

Spectral Matrix ◽

Projection Approach ◽

Few Data

Sample–sample (SS) two-dimensional (2D) correlation spectroscopy is applied in this study as a spectral selection tool to produce chemical images of real-world pharmaceutical samples consisting of two, three, and four components. The most unique spectra in a Raman mapping spectral matrix are found after analysis of the covariance matrix. (This is obtained by multiplying the original mapping data matrix by itself.) These spectra are identified by analyzing the slices of the covariance matrix at the positions where covariance values are at maxima. Chemical images are subsequently produced in a univariate fashion by visually selecting the wavenumbers in the extracted spectra that are least overlapped. The performance of SS 2D correlation is compared with principal component analysis in terms of highlighting the most prominent spectral differences across the whole data set (which typically comprises several thousand spectra) and determining the total number of species present. In addition, the selection of the unique spectra by SS 2D correlation is compared with the selection obtained by the orthogonal projection approach (OPA). Both comparisons are found to be satisfactory and demonstrate that a quite simple SS 2D correlation routine can be used for producing reliable images of unknown samples. The main benefit of using SS 2D correlation is that it is based on a few data processing commands that can be executed separately and produce results that are closely related to the chemical features of the system.

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The Multidimensional Algorithm of Cumulative Sums for Detecting Changes in Time Series Characteristics

Vestnik MEI ◽

10.24160/1993-6982-2021-1-86-94 ◽

2021 ◽

Vol 1 (1) ◽

pp. 86-94

Author(s):

Gennadiy F. Filaretov ◽

◽

Pavel S. Simonenkov ◽

Keyword(s):

Time Series ◽

Covariance Matrix ◽

Linear Transformation ◽

Reference Data ◽

Distribution Functions ◽

Diagonal Form ◽

False Alarms ◽

Average Delay ◽

Simultaneous Change ◽

Monitoring Algorithm

The article presents a cumulative sum algorithm intended to detect a sudden step-like change in the probabilistic characteristics of a monitored time series when such a change (“disorder”) is associated with a simultaneous change in both the location characteristics and the dispersion characteristics of the corresponding distribution functions. In the general case of a multidimensional time series, the disorder is associated with a jump in the values of the mathematical expectation vector (the vector of means) and covariance matrix entries. To solve this problem, it is proposed to use a preliminary linear transformation of the time series values, as a result of which the covariance matrix is transformed to the unity form before disordering and to the diagonal form after disordering. The change in the vector of means is analyzed, and the main relations describing the considered detection algorithm are derived. It is noted that by using the above-mentioned linear transformation it is possible to simplify the obtaining of the reference data necessary for synthesizing the monitoring algorithm with the predetermined properties. As an example, a particular case of a one-dimensional time series and a disorder in the form of a simultaneous change in the mean and variance is considered. For this case, reference data obtained by applying the simulation method are given, using which it is possible to find the monitoring algorithm triggering threshold and estimate the average delay time of detecting the specified disorder from the given interval between false alarms. This study is a logical continuation and further development of the approach to construction of multidimensional algorithms for detecting disorders [1].

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