scholarly journals Fast Spectral Approximation of Structured Graphs with Applications to Graph Filtering

Algorithms ◽  
2020 ◽  
Vol 13 (9) ◽  
pp. 214
Author(s):  
Mario Coutino ◽  
Sundeep Prabhakar Chepuri ◽  
Takanori Maehara ◽  
Geert Leus
Keyword(s):  
Fourier Transform ◽  
Computational Models ◽  
Divide And Conquer ◽  
Spectral Approximation ◽  
Topological Properties ◽  
Irregular Domains ◽  
Graph Spectrum ◽  
Fourier Theory ◽  
Structured Graphs ◽  
Frequency Domain Properties

To analyze and synthesize signals on networks or graphs, Fourier theory has been extended to irregular domains, leading to a so-called graph Fourier transform. Unfortunately, different from the traditional Fourier transform, each graph exhibits a different graph Fourier transform. Therefore to analyze the graph-frequency domain properties of a graph signal, the graph Fourier modes and graph frequencies must be computed for the graph under study. Although to find these graph frequencies and modes, a computationally expensive, or even prohibitive, eigendecomposition of the graph is required, there exist families of graphs that have properties that could be exploited for an approximate fast graph spectrum computation. In this work, we aim to identify these families and to provide a divide-and-conquer approach for computing an approximate spectral decomposition of the graph. Using the same decomposition, results on reducing the complexity of graph filtering are derived. These results provide an attempt to leverage the underlying topological properties of graphs in order to devise general computational models for graph signal processing.

Synthesis of quantum circuits for $d$-level systems by using cosine-sine

2009 ◽  
Vol 9 (5&6) ◽  
pp. 423-443
Author(s):  
Y. Nakajima ◽  
Y. Kawano ◽  
H. Sekigawa ◽  
M. Nakanishi ◽  
S. Yamashita ◽  
...  
Keyword(s):  
Fourier Transform ◽  
Quantum Circuit ◽  
Previous Method ◽  
Quantum Circuits ◽  
Divide And Conquer ◽  
Synthesis Methods ◽  
Quantum Fourier Transform ◽  
Asymptotically Optimal ◽  
Polynomial Size

We study the problem of designing minimal quantum circuits for any operations on $n$ qudits by means of the cosine-sine decomposition. Our method is based on a divide-and-conquer strategy. In that strategy, the size of the produced quantum circuit depends on whether the partitioning is balanced. We provide a new cosine-sine decomposition based on a balanced partitioning for $d$-level systems. The produced circuit is not asymptotically optimal except when $d$ is a power of two, but, when the number of qudits $n$ is small, our method can produce the smallest quantum circuit compared to the circuits produced by other synthesis methods. For example, when $d=3$ (three-level systems) and $n=2$ (two qudits), then the number of two-qudit operations called CINC, which is a generalized versions of CNOT, is 36 whereas the previous method needs 156 CINC gates. Moreover, we show that our method is useful for designing a polynomial-size quantum circuit for the radix-$d$ quantum Fourier transform.

scholarly journals Can Schrödinger and Heisenberg Make a Contribution to the Spectral Analysis of Signals?

2018 ◽  
Author(s):  
Mario Mastriani
Keyword(s):  
Spectral Analysis ◽  
Fourier Transform ◽  
Computational Cost ◽  
Energy Output ◽  
Quantum Time ◽  
Classical World ◽  
Fourier Theory ◽  
Short Time ◽  
The Impact ◽  
Low Computational Cost

A quantum time-dependent spectrum analysis, or simply, quantum spectral analysis (QSA) is presented in this work, and it’s based on Schrödinger’s equation. In the classical world, it is named frequency in time (FIT), which is used here as a complement of the traditional frequency-dependent spectral analysis based on Fourier theory. Besides, FIT is a metric which assesses the impact of the flanks of a signal on its frequency spectrum - not taken into account by Fourier theory and let alone in real time. Even more, and unlike all derived tools from Fourier Theory (i.e., continuous, discrete, fast, short-time, fractional and quantum Fourier Transform, as well as, Gabor) FIT has the following advantages, among others: a) compact support with excellent energy output treatment, b) low computational cost, O(N) for signals and O(N2) for images, c) it does not have phase uncertainties (i.e., indeterminate phase for a magnitude = 0) as in the case of Discrete and Fast Fourier Transform (DFT, FFT, respectively). Finally, we can apply QSA to a quantum signal, that is, to a qubit stream in order to analyze it spectrally.

scholarly journals Designing Unimodular Waveform with Low Range Sidelobes and Stopband for Cognitive Radar via Relaxed Alternating Projection

2016 ◽  
Vol 2016 ◽  
pp. 1-9 ◽  
Author(s):  
Xiang Feng ◽  
Yang-chun Song ◽  
Zhi-quan Zhou ◽  
Yi-nan Zhao
Keyword(s):  
Fourier Transform ◽  
Phase Retrieval ◽  
Spectral Power ◽  
Local Area ◽  
Narrowband Interference ◽  
Spectral Approximation ◽  
Cognitive Radar ◽  
Excellent Performance ◽  
Alternating Projection ◽  
Novel Method

Cognitive radar could adapt the spectrum of waveforms in response to information regarding the changing environment, so as to avoid narrowband interference or electronic jamming. Besides stopband constraints, low range sidelobes and unimodular property are also desired. In this paper, we propose a Spectral Approximation Relaxed Alternating Projection (SARAP) method, to synthesize unimodular waveform with low range sidelobes and spectral power suppressed. This novel method, based on phase retrieval and relaxed alternating projection, could convert the correlation optimization into the spectrum approximation via the Fast Fourier Transform (FFT). Moreover, by virtue of the relaxation factor and accelerated factor, SARAP can exploit local area and more likely converge to the global solution. Numerical trials have demonstrated that SARAP could achieve excellent performance and computational efficiency which will facilitate the real-time design.

scholarly journals A New Surrogating Algorithm by the Complex Graph Fourier Transform (CGFT)

Entropy ◽  
2019 ◽  
Vol 21 (8) ◽  
pp. 759
Author(s):  
Belda ◽  
Vergara ◽  
Safont ◽  
Salazar ◽  
Parcheta
Keyword(s):  
Fourier Transform ◽  
Laplacian Matrix ◽  
Complex Case ◽  
Graph Spectrum ◽  
Original Graph ◽  
Negative Eigenvalues ◽  
Complex Graph ◽  
The Fourier Transform ◽  
Phase Randomization ◽  
Spectrum Amplitude

The essential step of surrogating algorithms is phase randomizing the Fourier transform while preserving the original spectrum amplitude before computing the inverse Fourier transform. In this paper, we propose a new method which considers the graph Fourier transform. In this manner, much more flexibility is gained to define properties of the original graph signal which are to be preserved in the surrogates. The complex case is considered to allow unconstrained phase randomization in the transformed domain, hence we define a Hermitian Laplacian matrix that models the graph topology, whose eigenvectors form the basis of a complex graph Fourier transform. We have shown that the Hermitian Laplacian matrix may have negative eigenvalues. We also show in the paper that preserving the graph spectrum amplitude implies several invariances that can be controlled by the selected Hermitian Laplacian matrix. The interest of surrogating graph signals has been illustrated in the context of scarcity of instances in classifier training.

scholarly journals Formation of Standing Waves Within Crystal Unitcell

2021 ◽  
Author(s):  
Bhanumoorthy Pullooru
Keyword(s):  
Fourier Transform ◽  
Structure Determination ◽  
Standing Waves ◽  
Polar Coordinates ◽  
Polar Coordinate ◽  
Fourier Theory ◽  
Nickel Crystal

Abstract We show that the historic Davisson-Germer experiment demonstrates formation of standing waves within nickel crystal unitcell. Cartesian Fourier transform cannot offer description in terms of standing waves because Cartesian Fourier theory cannot accommodate π in place of 2π. Thus, formation of standing waves within unitcell in Davisson-Germer experiment necessarily requires spherical polar coordinate description of crystal diffraction. Description in spherical polar coordinates permits to incorporate precision angles from the experiment for better convergence in structure determination calculations.

Electron Microscopy of Nucleic Acids

1974 ◽  
Vol 32 ◽  
pp. 302-303
Author(s):  
Norman Davidson
Keyword(s):  
Electron Microscopy ◽  
Nucleic Acids ◽  
Basic Protein ◽  
Molecular Biologist ◽  
Topological Properties ◽  
Protein Film ◽  
Polynucleotide Chain

The basic protein film technique for mounting nucleic acids for electron microscopy has proven to be a general and powerful tool for the working molecular biologist in characterizing different nucleic acids. It i s possible to measure molecular lengths of duplex and single-stranded DNAs and RNAs. In particular, it is thus possible to as certain whether or not the nucleic acids extracted from a particular source are or are not homogeneous in length. The topological properties of the polynucleotide chain (linear or circular, relaxed or supercoiled circles, interlocked circles, etc. ) can also be as certained.

Comparison of Calculated and Recorded Electron Energy-Loss Spectra of Aluminium and Carbon

1990 ◽  
Vol 48 (2) ◽  
pp. 64-65
Author(s):  
L. Reimer ◽  
R. Oelgeklaus
Keyword(s):  
Fourier Transform ◽  
Energy Loss ◽  
Electron Energy ◽  
Rotational Symmetry ◽  
Mean Free Path ◽  
Single Scattering ◽  
Specimen Thickness ◽  
Two Dimensional ◽  
Image Position ◽  
Electron Energy Loss

Quantitative electron energy-loss spectroscopy (EELS) needs a correction for the limited collection aperture α and a deconvolution of recorded spectra for eliminating the influence of multiple inelastic scattering. Reversely, it is of interest to calculate the influence of multiple scattering on EELS. The distribution f(w,θ,z) of scattered electrons as a function of energy loss w, scattering angle θ and reduced specimen thickness z=t/Λ (Λ=total mean-free-path) can either be recorded by angular-resolved EELS or calculated by a convolution of a normalized single-scattering function ϕ(w,θ). For rotational symmetry in angle (amorphous or polycrystalline specimens) this can be realised by the following sequence of operations :(1)where the two-dimensional distribution in angle is reduced to a one-dimensional function by a projection P, T is a two-dimensional Fourier transform in angle θ and energy loss w and the exponent -1 indicates a deprojection and inverse Fourier transform, respectively.

FR-IR Molecular Microanalysis System

1990 ◽  
Vol 48 (2) ◽  
pp. 270-271
Author(s):  
John A. Reffner ◽  
William T. Wihlborg
Keyword(s):  
Fourier Transform ◽  
Fourier Transform Infrared ◽  
Integrated System ◽  
Morphological Examination ◽  
Fully Integrated ◽  
Ir Microscopy ◽  
Normal Functions ◽  
Infrared Microspectroscopy ◽  
Infrared Spectral ◽  
Ft Ir

The IRμs™ is the first fully integrated system for Fourier transform infrared (FT-IR) microscopy. FT-IR microscopy combines light microscopy for morphological examination with infrared spectroscopy for chemical identification of microscopic samples or domains. Because the IRμs system is a new tool for molecular microanalysis, its optical, mechanical and system design are described to illustrate the state of development of molecular microanalysis. Applications of infrared microspectroscopy are reviewed by Messerschmidt and Harthcock.Infrared spectral analysis of microscopic samples is not a new idea, it dates back to 1949, with the first commercial instrument being offered by Perkin-Elmer Co. Inc. in 1953. These early efforts showed promise but failed the test of practically. It was not until the advances in computer science were applied did infrared microspectroscopy emerge as a useful technique. Microscopes designed as accessories for Fourier transform infrared spectrometers have been commercially available since 1983. These accessory microscopes provide the best means for analytical spectroscopists to analyze microscopic samples, while not interfering with the FT-IR spectrometer’s normal functions.

The extended Fourier algorithm: Application in discrete optics and electron holography

1994 ◽  
Vol 52 ◽  
pp. 918-919
Author(s):  
E. Voelkl ◽  
L. F. Allard
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
Sampling Rate ◽  
Electron Holography ◽  
Optical Bench ◽  
Fourier Space ◽  
Ccd Cameras ◽  
Simulated Image ◽  
Initial Sampling ◽  
Fourier Algorithm

The conventional discrete Fourier transform can be extended to a discrete Extended Fourier transform (EFT). The EFT allows to work with discrete data in close analogy to the optical bench, where continuous data are processed. The EFT includes a capability to increase or decrease the resolution in Fourier space (thus the argument that CCD cameras with a higher number of pixels to increase the resolution in Fourier space is no longer valid). Fourier transforms may also be shifted with arbitrary increments, which is important in electron holography. Still, the analogy between the optical bench and discrete optics on a computer is limited by the Nyquist limit. In this abstract we discuss the capability with the EFT to change the initial sampling rate si of a recorded or simulated image to any other(final) sampling rate sf.

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