scholarly journals Interactive Tools for Teaching Fourier Transforms

2020 ◽  
Vol 1 (1) ◽  
pp. 4
Author(s):  
Carlos R. Baiz
Keyword(s):  
Fourier Transforms ◽  
Lesson Plan ◽  
Real Space ◽  
Fourier Space ◽  
Core Component ◽  
Interactive Approach ◽  
Interactive Teaching ◽  
Mathematical Skills ◽  
Interactive Tools ◽  
Upper Level

Fourier transforms (FT) are universal in chemistry, physics, and biology. Despite FTs being a core component of multiple experimental techniques, undergraduate courses typically approach FTs from a mathematical perspective, leaving students with a lack of intuition on how an FT works. Here, I introduce interactive teaching tools for upper-level undergraduate courses and describe a practical lesson plan for FTs. The materials include a computer program to capture video from a webcam and display the original images side-by-side with the corresponding plot in the Fourier domain. Several patterns are included to be printed on paper and held up to the webcam as input. During the lesson, students are asked to predict the features observed in the FT and then place the patterns in front of the webcam to test their predictions. This interactive approach enables students with limited mathematical skills to achieve a certain level of intuition for how FTs translate patterns from real space into the corresponding Fourier space.

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space (thus the argument that CCD cameras with a higher number of pixels to increase the resolution in Fourier space is no longer valid).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.

scholarly journals Fast and spectrally accurate evaluation of gyroaverages in non-periodic gyrokinetic-Poisson simulations

2017 ◽  
Vol 83 (4) ◽  
Author(s):  
J. Guadagni ◽  
A. J. Cerfon
Keyword(s):  
Time Complexity ◽  
Numerical Scheme ◽  
Fourier Transforms ◽  
Hankel Transform ◽  
Log P ◽  
Fourier Space ◽  
Compactly Supported ◽  
Geometric Convergence ◽  
Spatial Grid ◽  
Grid Points

We present a fast and spectrally accurate numerical scheme for the evaluation of the gyroaveraged electrostatic potential in non-periodic gyrokinetic-Poisson simulations. Our method relies on a reformulation of the gyrokinetic-Poisson system in which the gyroaverage in Poisson’s equation is computed for the compactly supported charge density instead of the non-periodic, non-compactly supported potential itself. We calculate this gyroaverage with a combination of two Fourier transforms and a Hankel transform, which has the near optimal run-time complexity$O(N_{\unicode[STIX]{x1D70C}}(P+\hat{P})\log (P+\hat{P}))$, where$P$is the number of spatial grid points,$\hat{P}$the number of grid points in Fourier space and$N_{\unicode[STIX]{x1D70C}}$the number of grid points in velocity space. We present numerical examples illustrating the performance of our code and demonstrating geometric convergence of the error.

scholarly journals Pengembangan Bahan Ajar Interaktif ‘POST’ dalam Pembelajaran Apresiasi Puisi untuk Siswa Kelas X SMA

2020 ◽  
Vol 3 (1) ◽  
pp. 14-23
Author(s):  
Nur Jamilah
Keyword(s):  
Field Trial ◽  
Lesson Plan ◽  
Trial Result ◽  
Dissemination And Implementation ◽  
Teaching Materials ◽  
Interactive Teaching ◽  
Audiovisual Media ◽  
Product Planning ◽  
Product Improvement ◽  
Product Effectiveness

Poetry appreciation, one of the literature learnings in Senior High School (SMA), is a subject that has the most important part to establish students’ intuition and introduce literature as one of Indonesia's Cultures. Poetry appreciation teaching, which is one of the literature learnings given by teachers in the 2013 Curriculum nowadays, is expected to be the bridge that connects the students’ mindset and the learning purpose. This study is a kind of Research and Development. According to the study, the researcher finds some conclusions. First, the process of developing interactive teaching materials by using ‘POST’ audiovisual media product planning takes some steps (1) data researching and collecting, and (2) product planning, (3) product developing, (4) initial trial, (5) trial result revision, (6) field trial, (7) field trial result product improvement, (8) implementation field test, (9) final product improvement, (10) dissemination and implementation to produce learning CD as teaching materials. Second, the implementation of interactive teaching materials by using ‘POST’ audiovisual media application in poetry appreciation learning which produces CD is obtained through some process: (1) learning implementation, (2) observation, (3) questionnaire. Third, the product practicality of interactive teaching materials by using ‘POST’ audiovisual media is obtained from the score of lesson plan implementation analysis result which shows 93% and the students’ response seen from observation and questionnaire. Fourth, the product effectiveness of interactive teaching materials by using ‘POST’ audiovisual media is obtained from the advance individual trial score and the students’ activity.

scholarly journals Randomized Benchmarking as Convolution: Fourier Analysis of Gate Dependent Errors

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 581
Author(s):  
Seth T. Merkel ◽  
Emily J. Pritchett ◽  
Bryan H. Fong
Keyword(s):  
Fourier Analysis ◽  
Recent Work ◽  
Fourier Transforms ◽  
Decay Rates ◽  
Alternative Proof ◽  
Mathematical Framework ◽  
Fourier Space ◽  
Clifford Group ◽  
Depolarizing Channel ◽  
The Ideal

We show that the Randomized Benchmarking (RB) protocol is a convolution amenable to Fourier space analysis. By adopting the mathematical framework of Fourier transforms of matrix-valued functions on groups established in recent work from Gowers and Hatami \cite{GH15}, we provide an alternative proof of Wallman's \cite{Wallman2018} and Proctor's \cite{Proctor17} bounds on the effect of gate-dependent noise on randomized benchmarking. We show explicitly that as long as our faulty gate-set is close to the targeted representation of the Clifford group, an RB sequence is described by the exponential decay of a process that has exactly two eigenvalues close to one and the rest close to zero. This framework also allows us to construct a gauge in which the average gate-set error is a depolarizing channel parameterized by the RB decay rates, as well as a gauge which maximizes the fidelity with respect to the ideal gate-set.

Dynamic Origin-Destination Demand Estimation with Multiday Link Traffic Counts for Planning Applications

2003 ◽  
Vol 1831 (1) ◽  
pp. 30-38 ◽  
Author(s):  
Xuesong Zhou ◽  
Xiao Qin ◽  
Hani S. Mahmassani
Keyword(s):  
Real Time ◽  
Demand Estimation ◽  
Test Bed ◽  
Estimation Model ◽  
Interactive Approach ◽  
Traffic Demand ◽  
Level Problem ◽  
Upper Level ◽  
Demand Patterns

A dynamic origin–destination demand estimation model for planning applications with real-time link counts from multiple days is presented. Based on an iterative bilevel estimation framework, the upper-level problem is to minimize both the deviation between estimated link flows and real-time link counts and the deviation between estimated time-dependent demand and given historical static demand. These two types of deviations are combined into a weighted objective function, where the weighting value is determined by an interactive approach to obtain the best compromise solution. The single-day formulation is further extended to use link counts from multiple days to estimate the variation in traffic demand over multiple days. A case study based on the Irvine test bed network is conducted to illustrate the methodology and estimate day-to-day demand patterns. The application illustrates considerable benefits in analyzing the demand dynamics with multiday data.

scholarly journals Reconnection studies under different types of turbulence driving

2012 ◽  
Vol 19 (2) ◽  
pp. 297-314 ◽  
Author(s):  
G. Kowal ◽  
A. Lazarian ◽  
E. T. Vishniac ◽  
K. Otmianowska-Mazur
Keyword(s):  
Magnetic Reconnection ◽  
Three Dimensional ◽  
Real Space ◽  
Weak Turbulence ◽  
Fourier Space ◽  
Velocity Fluctuations ◽  
Different Types ◽  
Injection Power ◽  
Exact Relations ◽  
Varying Viscosity

Abstract. We study a model of fast magnetic reconnection in the presence of weak turbulence proposed by Lazarian and Vishniac (1999) using three-dimensional direct numerical simulations. The model has been already successfully tested in Kowal et al. (2009) confirming the dependencies of the reconnection speed Vrec on the turbulence injection power Pinj and the injection scale linj expressed by a constraint Vrec ~ Pinj1/2linj3/4and no observed dependency on Ohmic resistivity. In Kowal et al. (2009), in order to drive turbulence, we injected velocity fluctuations in Fourier space with frequencies concentrated around kinj = 1/linj, as described in Alvelius (1999). In this paper, we extend our previous studies by comparing fast magnetic reconnection under different mechanisms of turbulence injection by introducing a new way of turbulence driving. The new method injects velocity or magnetic eddies with a specified amplitude and scale in random locations directly in real space. We provide exact relations between the eddy parameters and turbulent power and injection scale. We performed simulations with new forcing in order to study turbulent power and injection scale dependencies. The results show no discrepancy between models with two different methods of turbulence driving exposing the same scalings in both cases. This is in agreement with the Lazarian and Vishniac (1999) predictions. In addition, we performed a series of models with varying viscosity ν. Although Lazarian and Vishniac (1999) do not provide any prediction for this dependence, we report a weak relation between the reconnection speed with viscosity, Vrec ~ ν−1/4.

Dynamic Origin-Destination Demand Estimation Using Turning Movement Counts

2008 ◽  
Vol 2085 (1) ◽  
pp. 39-48 ◽  
Author(s):  
Hamed Alibabai ◽  
Hani S. Mahmassani
Keyword(s):  
A Priori ◽  
Demand Estimation ◽  
Estimation Model ◽  
Interactive Approach ◽  
The Us ◽  
Level Problem ◽  
Upper Level ◽  
The One ◽  
Dependent Demand

A dynamic origin-destination (O-D) demand estimation model is presented that uses turning movement counts as observations. Based on an iterative bilevel estimation framework, the upper-level problem is to minimize a weighted objective function of the deviation between simulated link flows and real-time link counts and the deviation between estimated time-dependent demand and an a priori historical O-D table, where the weighting value is determined by an interactive approach to obtain the best compromise solution. A case study was performed on the US-29 network in Maryland to compare the estimated tables of this approach with the one obtained from the traditional method, which uses only approach link volume counts. The application illustrates considerable benefits of using turning movements instead of approach volumes in matching observed counts.

ON GENERATING RANDOM POTENTIALS

2012 ◽  
Vol 11 (04) ◽  
pp. 1250026 ◽  
Author(s):  
M. SUÑÉ SIMON ◽  
J. M. SANCHO ◽  
A. M. LACASTA
Keyword(s):  
Real Space ◽  
Fourier Space ◽  
Random Potentials ◽  
Practical Applications ◽  
Gaussian Potential ◽  
Brownian Particles ◽  
Straightforward Method ◽  
Transport And Diffusion ◽  
Potential Landscape ◽  
And Diffusion

The study of transport and diffusion of Brownian particles in disorder media needs the generation of random potentials with well prescribed statistical properties. Here we present a straightforward method to build a Gaussian potential landscape with an arbitrary spatial correlation with the only requirement of isotropy. The method has the particularity that, although it uses the Fourier space, all its constraints and information are in real space. As practical applications we construct three types of Gaussian disordered correlations: Normal, exponential and power-law. These three cases cover a variety of physical situations.

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