Multidimensional Signal Analysis

2009 ◽  
Author(s):  
Robert J Marks II
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Higher Dimensions ◽  
Two Dimensions ◽  
One Dimension ◽  
One Dimensional ◽  
Multidimensional Signals ◽  
Multidimensional Signal ◽  
The Fourier Transform

N dimensional signals are characterized as values in an N dimensional space. Each point in the space is assigned a value, possibly complex. Each dimension in the space can be discrete, continuous, or on a time scale. A black and white movie can be modelled as a three dimensional signal.Acolor picture can be modelled as three signals in two dimensions, one each, for example, for red, green and blue. This chapter explores Fourier characterization of different types of multidimensional signals and corresponding applications. Some signal characterizations are straightforward extensions of their one dimensional counterparts. Others, even in two dimensions, have properties not found in one dimensional signals. We are fortunate to be able to visualize structures in two, three, and sometimes four dimensions. It assists in the intuitive generalization of properties to higher dimensions. Fourier characterization of multidimensional signals allows straightforward modelling of reconstruction of images from their tomographic projections. Doing so is the foundation of certain medical and industrial imaging, including CAT (for computed axial tomography) scans. Multidimensional Fourier series are based on models found in nature in periodically replicated crystal Bravais lattices [987, 1188]. As is one dimension, the Fourier series components can be found from sampling the Fourier transform of a single period of the periodic signal. The multidimensional cosine transform, a relative of the Fourier transform, is used in image compression such as JPG images. Multidimensional signals can be filtered. The McClellan transform is a powerful method for the design of multidimensional filters, including generalization of the large catalog of zero phase one dimensional FIR filters into higher dimensions. As in one dimension, the multidimensional sampling theorem is the Fourier dual of the Fourier series. Unlike one dimension, sampling can be performed at the Nyquist density with a resulting dependency among sample values. This property can be used to reduce the sampling density of certain images below that of Nyquist, or to restore lost samples from those remaining. Multidimensional signal and image analysis is also the topic of Chapter 9 on time frequency representations, and Chapter 11 where POCS is applied signals in higher dimensions.

PROJECTED ENTANGLED STATES: PROPERTIES AND APPLICATIONS

2006 ◽  
Vol 20 (30n31) ◽  
pp. 5142-5153 ◽  
Author(s):  
F. VERSTRAETE ◽  
M. WOLF ◽  
D. PÉREZ-GARCÍA ◽  
J. I. CIRAC
Keyword(s):  
Numerical Algorithms ◽  
Higher Dimensions ◽  
Two Dimensions ◽  
One Dimension ◽  
Entangled States ◽  
Matrix Product ◽  
Many Body ◽  
Many Body Systems ◽  
Dimension One

We present a new characterization of quantum states, what we call Projected Entangled-Pair States (PEPS). This characterization is based on constructing pairs of maximally entangled states in a Hilbert space of dimension D2, and then projecting those states in subspaces of dimension d. In one dimension, one recovers the familiar matrix product states, whereas in higher dimensions this procedure gives rise to other interesting states. We have used this new parametrization to construct numerical algorithms to simulate the ground state properties and dynamics of certain quantum-many body systems in two dimensions.

scholarly journals A Guide on Spectral Methods Applied to Discrete Data in One Dimension

2017 ◽  
Vol 2017 ◽  
pp. 1-27 ◽  
Author(s):  
Martin Seilmayer ◽  
Matthias Ratajczak
Keyword(s):  
Spectral Analysis ◽  
Fourier Transform ◽  
Spectral Methods ◽  
Discrete Data ◽  
Time Dependent ◽  
One Dimension ◽  
One Dimensional ◽  
The Common ◽  
The Fourier Transform ◽  
Fragmented Data

This paper provides an overview about the usage of the Fourier transform and its related methods and focuses on the subtleties to which the users must pay attention. Typical questions, which are often addressed to the data, will be discussed. Such a problem can be the origin of frequency or band limitation of the signal or the source of artifacts, when a Fourier transform is carried out. Another topic is the processing of fragmented data. Here, the Lomb-Scargle method will be explained with an illustrative example to deal with this special type of signal. Furthermore, the time-dependent spectral analysis, with which one can evaluate the point in time when a certain frequency appears in the signal, is of interest. The goal of this paper is to collect the important information about the common methods to give the reader a guide on how to use these for application on one-dimensional data. The introduced methods are supported by the spectral package, which has been published for the statistical environment R prior to this article.

scholarly journals Assessing a Linear Nanosystem's Limiting Reliability from its Components

2008 ◽  
Vol 45 (03) ◽  
pp. 879-887 ◽  
Author(s):  
Nader Ebrahimi
Keyword(s):  
Present State ◽  
Size Range ◽  
Two Dimensions ◽  
The Other ◽  
One Dimension ◽  
Present Moment ◽  
One Dimensional ◽  
The One ◽  
Fixed Moment ◽  
Nanosized Systems

Nanosystems are devices that are in the size range of a billionth of a meter (1 x 10-9) and therefore are built necessarily from individual atoms. The one-dimensional nanosystems or linear nanosystems cover all the nanosized systems which possess one dimension that exceeds the other two dimensions, i.e. extension over one dimension is predominant over the other two dimensions. Here only two of the dimensions have to be on the nanoscale (less than 100 nanometers). In this paper we consider the structural relationship between a linear nanosystem and its atoms acting as components of the nanosystem. Using such information, we then assess the nanosystem's limiting reliability which is, of course, probabilistic in nature. We consider the linear nanosystem at a fixed moment of time, say the present moment, and we assume that the present state of the linear nanosystem depends only on the present states of its atoms.

Electron-density maps

2002 ◽  
Author(s):  
David Blow
Keyword(s):  
Fourier Transform ◽  
Electron Density ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Unit Cell ◽  
Density Maps ◽  
Wave Cycle ◽  
Density Map ◽  
The Fourier Transform ◽  
Electron Density Map

When everything has been done to make the phases as good as possible, the time has come to examine the image of the structure in the form of an electron-density map. The electron-density map is the Fourier transform of the structure factors (with their phases). If the resolution and phases are good enough, the electron-density map may be interpreted in terms of atomic positions. In practice, it may be necessary to alternate between study of the electron-density map and the procedures mentioned in Chapter 10, which may allow improvements to be made to it. Electron-density maps contain a great deal of information, which is not easy to grasp. Considerable technical effort has gone into methods of presenting the electron density to the observer in the clearest possible way. The Fourier transform is calculated as a set of electron-density values at every point of a three-dimensional grid labelled with fractional coordinates x, y, z. These coordinates each go from 0 to 1 in order to cover the whole unit cell. To present the electron density as a smoothly varying function, values have to be calculated at intervals that are much smaller than the nominal resolution of the map. Say, for example, there is a protein unit cell 50 Å on a side, at a routine resolution of 2Å. This means that some of the waves included in the calculation of the electron density go through a complete wave cycle in 2 Å. As a rule of thumb, to represent this properly, the spacing of the points on the grid for calculation must be less than one-third of the resolution. In our example, this spacing might be 0.6 Å. To cover the whole of the 50 Å unit cell, about 80 values of x are needed; and the same number of values of y and z. The electron density therefore needs to be calculated on an array of 80×80×80 points, which is over half a million values. Although our world is three-dimensional, our retinas are two-dimensional, and we are good at looking at pictures and diagrams in two dimensions.

Diffraction by crystals

2002 ◽  
Author(s):  
David Blow
Keyword(s):  
Fourier Transform ◽  
Three Dimensional ◽  
Incident Beam ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Dimensional Structure ◽  
Angle Of Incidence ◽  
X Ray ◽  
Max Von Laue ◽  
The Fourier Transform

In Chapter 4 many two-dimensional examples were shown, in which a diffraction pattern represents the Fourier transform of the scattering object. When a diffracting object is three-dimensional, a new effect arises. In diffraction by a repetitive object, rays are scattered in many directions. Each unit of the lattice scatters, but a diffracted beam arises only if the scattered rays from each unit are all in phase. Otherwise the scattering from one unit is cancelled out by another. In two dimensions, there is always a direction where the scattered rays are in phase for any order of diffraction (just as shown for a one-dimensional scatterer in Fig. 4.1). In three dimensions, it is only possible for all the points of a lattice to scatter in phase if the crystal is correctly oriented in the incident beam. The amplitudes and phases of all the scattered beams from a three-dimensional crystal still provide the Fourier transform of the three-dimensional structure. But when a crystal is at a particular angular orientation to the X-ray beam, the scattering of a monochromatic beam provides only a tiny sample of the total Fourier transform of its structure. In the next section, we are going to find what is needed to allow a diffracted beam to be generated. We shall follow a treatment invented by Lawrence Bragg in 1913. Max von Laue, who discovered X-ray diffraction in 1912, used a different scheme of analysis; and Paul Ewald introduced a new way of looking at it in 1921. These three methods are referred to as the Laue equations, Bragg’s law and the Ewald construction, and they give identical results. All three are described in many crystallographic text books. Bragg’s method is straightforward, understandable, and suffices for present needs. I had heard J.J. Thomson lecture about…X-rays as very short pulses of radiation. I worked out that such pulses…should be reflected at any angle of incidence by the sheets of atoms in the crystal as if these sheets were mirrors.…It remained to explain why certain of the atomic mirrors in the zinc blende [ZnS] crystal reflected more powerfully than others.

Pego everywhere

2016 ◽  
Vol 15 (04) ◽  
pp. 1650074 ◽  
Author(s):  
Przemysław Górka ◽  
Tomasz Kostrzewa
Keyword(s):  
Fourier Transform ◽  
Abelian Groups ◽  
General Version ◽  
Pontryagin Duality ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
The Fourier Transform ◽  
Compact Subsets

In this note we show the general version of Pego’s theorem on locally compact abelian groups. The proof relies on the Pontryagin duality as well as on the Arzela–Ascoli theorem. As a byproduct, we get the characterization of relatively compact subsets of [Formula: see text] in terms of the Fourier transform.

The use of the Hartley transform in geophysical applications

Geophysics ◽  
1990 ◽  
Vol 55 (11) ◽  
pp. 1488-1495 ◽  
Author(s):  
R. Saatcilar ◽  
S. Ergintav ◽  
N. Canitez
Keyword(s):  
Fourier Transform ◽  
Seismic Data ◽  
Integral Transform ◽  
Complex Multiplication ◽  
One Dimensional ◽  
Hartley Transform ◽  
And Migration ◽  
The Fourier Transform ◽  
Processing Techniques ◽  
Phase Spectra

The Hartley transform (HT) is an integral transform similar to the Fourier transform (FT). It has most of the characteristics of the FT. Several authors have shown that fast algorithms can be constructed for the fast Hartley transform (FHT) using the same structures as for the fast Fourier transform. However, the HT is a real transform and for this reason, since one complex multiplication requires four real multiplications, the discrete HT (DHT) is computationally faster than the discrete FT (DFT). Consequently, any process requiring the DFT (such as amplitude and phase spectra) can be performed faster by using the DHT. The general properties of the DHT are reviewed first, and then an attempt is made to use the FHT in some seismic data processing techniques such as one‐dimensional filtering, forward seismic modeling, and migration. The experiments show that the Hartley transform is two times faster than the Fourier transform.

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