Questions tagged [fourier-analysis]
Questions on the computational aspects of Fourier analysis, including the various applications of the fast Fourier transform (FFT).
120
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Can this finite difference dispersion be eliminated somehow?
I am trying to solve the wave equation
$$ {\partial ^2u(t,x) \over \partial x^2} = {\partial ^2u(t,x) \over \partial t^2} \tag1 $$
with the following boundary and initial conditions:
$$ {\partial u \...
- 227
5
votes
1
answer
763
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Taking derivative using FFT
I would like to calculate derivative of a given function ( a 1D array) using Array. Here is the code
...
1
vote
1
answer
145
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Fourier Transform with logarithmic spacing?
I have a very long time series $f(t)$ (hours) dataset taken at a very high sample rate (250 MHz) and would like to understand its frequency structure at many different frequency scales (from milli-Hz ...
- 171
5
votes
1
answer
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Prediction of sphere (i.e. roast) core temperature heated in an oven
The real-life problem
Assume I put a spherical roast with initially constant temperature of start_temp=25 (°C) into an oven with ...
- 161
0
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0
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70
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Deviation between Analytic DFT and FFT in Python
Within my work, I am trying to compare analytically retrieved power spectra with ones calculated from fft packages in python.
The problem I have, is that the analytic form of the peaks I derived does ...
- 31
0
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0
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39
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Efficiently compute the Fourier series coefficients of a piecewise trigonometric function
I am searching for the most efficient algorithm to compute the Fourier series coefficients of a number of given functions $f(\theta)$
$$a_k = \int_0^{2\pi} f(\theta)\cos(k\theta) \\
b_k = \int_0^{2\pi}...
3
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0
answers
105
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How to take the Fourier transform of a Fibonacci chain in a Python script?
This may be an easy question to answer but I am really stuck.
In several topics (especially that of quasicrystals) the Fibonacci chain's Fourier transform and diffraction pattern is mentioned. Despite ...
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138
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Deconvolution of sinc function in spectrum calculation in FTS
In Fourier transform spectroscopy (FTS) I am calculating a broadband interferogram (e.m. frequency 190-300 GHz top-hat), then back-retrieving the spectrum by FT.
Here in the figure, you can see the ...
- 61
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0
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41
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Describing a vector field
I am currently making use of a 2D vector field that I would like to be able to describe and characterize. I thought of a few metrics that I could use, but they seem limited for both a qualitative and ...
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0
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56
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How do people deal with resized grid steps while numerically integrating using discrete Fourier Transform?
I am trying to simulate light propagation on python using FFT following the Fresnel diffraction equation given on Wikipedia:
The problem with this is that the output matrix from the DFT would be ...
- 11
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0
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74
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Fourier integral over elements
Suppose I have a triangular element with vertices ${\vec{r_1},...,\vec{r_3}}$ and a function $f(\vec{r})$. I want to calculate the fourier integral over this triangle such that:
$$F(k_x,k_y)=\int \int ...
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How to obtain the exact value of wavelength from a 2D FFT amplitude vs wavenumber plot like it is obtainable from 1D FFT amplitude vs wavenumber plot?
I have a two dimensional multi modal spatial signal generated from a MATLAB code using sinusoidal functions of different wave numbers, amplitudes and phases. What I want to know is that if I have the ...
3
votes
2
answers
2k
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Solving numerically the 1D Kuramoto-Sivashinsky Equation using spectral methods
Lately, I've been trying to solve numerically the 1D Kuramoto-Sivashinsky Equation using spectral methods.
Let $\nu$ be the viscosity and $[0,L]$ the domain. The 1D equation is,
$$
u_t + uu_x + u_{xx} ...
- 173
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1
answer
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Understanding why scipy.fft.fft (fast Fourier transform) doesn't work as expected
I write the following fast Fourier transform code into my Python notebook expecting to see a plot wherein there's a spike at $1/2\pi$ since that's the frequency of the sin function, but instead I get ...
user37540
3
votes
1
answer
106
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Cauchy Lorentzian simulation on FFT with oscillation
Recently I do simulation on Lorentzian Function with FFT
Lorentzian Function is 2a/(x**2+a**2)
...
- 31
0
votes
0
answers
68
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Adaptive quadrature methods for Fourier Optics
In Fourier Optics one often needs to compute approximations to bivariate integrals like
$$
\int_{-\frac{l}{2}}^{\frac{l}{2}}\int_{-\frac{l}{2}}^{\frac{l}{2}} {\rm e}^{i\phi(\xi,\eta)}\mathrm{exp}\left[...
- 301
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1
answer
448
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Galerkin method for heat equation
I'm working out the Galerkin method for the heat equation $$\frac{\partial u}{\partial t} - \frac{\partial^2 u}{\partial x^2} = 0$$ subject to $u(0,t)=0,u_x(1,t)=v(t)$.
I want to use a Fourier basis ...
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1
vote
0
answers
194
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Poisson equation with FFT and normalization
I'm trying to understand how to solve Poisson equation with FFT. Say, if we have the simplest periodic example
$$u_{xx}=-4\pi\cos(x)$$
The solution then should be
$$u=4\pi\cos(x)$$
I really get ...
- 11
3
votes
0
answers
44
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Is the given equation for Fractional Fourier Transform wrong?
In order to compute $$ \hat{f}_k = \sum_{m = 0}^{M - 1} e^{-2\pi i k m \theta} f_m, \ \ k = 0, ..., M - 1$$
for any $\theta$, my book states that this can be done using fractional Fourier transform ...
- 31
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0
answers
78
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How to use RODFT00 and REDFT00
I have some difficulty in implementing RODFT00 and REDFT00. I want to use them for fluid simulations.
I would really appreciate ...
- 53
1
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0
answers
109
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2
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0
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35
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Fast approximate evaluation of Fourier-Legendre series
Suppose I know that a function from $[0,\pi] \to \mathbb{R}$ may be written as
$$
\sum_{k=0}^\infty A_l \frac{2l+1}{4\pi} P_l(r)
$$
where $A_l$ all are known. Is there a way in which I may very ...
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1
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0
answers
223
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How to use discrete cosine and discrete sine transforms in fftw
I work on fluid-related simulations. I have used FFT for fluid simulation. I want to use discrete cosine transform (DCT) and discrete sine transform (DST) to transform my velocity field to wavenumbers....
- 53
2
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0
answers
22
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Normalising DFTs Correctly
I have been playing around with convolutions in scipy's signal package:
...
- 21
1
vote
1
answer
97
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Obtain velocity from imposed energy spectrum using the inverse FFT
I am trying to obtain the spatial representation of $u(x)$ (e.g. velocity) from its energy spectrum $E(k)=k^4\exp(-(k/k_0)^2)$, which is given in the frequency domain, provided $|u(k)|=\sqrt{2E(k)}$. ...
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1
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242
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Calculating the Convolution Using DFT (FFT)
I have the following convolution as part of a numerical simulation.
$$T(r)=\int \mathrm{d}^3r_2\, p(r_2)f(r_2)\alpha(r-r_2)\, .$$
My problem is that the analytical expressions for $f$ and $p$ do ...
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1
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0
answers
91
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Pseudospectral method for Rayleigh-Benard with constant temperature gradient
$$
\nabla\cdot \mathbf{u} = 0 \\
\frac{\partial \mathbf{u}}{\partial t}+\left(\mathbf{u}\cdot \nabla\right)\mathbf{u} = -\nabla p+\nu\nabla^2\mathbf{u}+\alpha g\theta\mathbf{e}_z\\
\frac{\partial\...
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1
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2
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Numpy FFT gives me a pulse shorter than it should be. Not sure what I am doing wrong
I've created a code (Python, numpy) that defines an ultrashort laser pulse in the frequency domain (pulse duration should be 4 fs), but when I perform the Fourier Transform using DFT, my pulse in the ...
1
vote
1
answer
131
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Demagnetizing field using scalar potential method
I want to calculate the stray magnetic field from a ferromagnet using the scalar potential method (1). The problem consists of a ferromagnetic cuboid divided into small cuboidal cells in which the ...
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0
votes
1
answer
221
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Specifying mesh spacing for DFT in numpy
I was testing the .fft package of numpy 1.16.1 in Python 3.7.2. In particular I was trying to verify that the transform resembles the analytical one for: $$f(x) = \mathrm{exp}\left[-\left(\frac{x-5}{2}...
user30058
1
vote
1
answer
1k
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FFT solver for the Poisson problem with Dirichlet boundary conditions
I am trying to solve the Poisson problem with Dirichlet boundary condition in 1D:
\begin{equation}
\begin{array}{rcl}
- \mu \Delta u & = & f~in~[0,1], \\
u(0) & = & 0, \\
u(1) & = ...
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2
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1
answer
619
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Incorporating a potential barrier in a wave-packet simulation (Fourier Transform method)
I'm trying to simulate the scattering of a wave-packet at a potential barrier in Python. I'm using a Fourier Transform method (not sure if its the same as the Split-Step method), where I apply Fourier ...
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3
votes
2
answers
832
views
Generate high n quantum harmonic oscillator states numerically
How can I generate the higher $n$ quantum harmonic oscillator wavefunction (in position space) numerically? Here, higher means around $n=500$, or say $n=2000$, where $n$ is the $n$th oscillator ...
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3
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79
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Rudin lecture -- if f(x) is not integrable on some interval, does it not have a Fourier Series expansion on that interval?
I found an old lecture on YouTube given by Walter Rudin (1990, in Wisconsin), and towards the beginning he mentions that if $f(x)$ were not integrable, on some interval, it would be obvious that it ...
1
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0
answers
447
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What does the Jackson Kernel measure?
A certain filter I'm writing uses two different kernels. The Fejer kernel (which is common) and the Jackson kernel:
$$ \Delta_T(x) = T \,\left( \frac{\sin \pi T x}{\pi T x}\right)^2 \quad\text{and}...
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1
answer
117
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FFTW on subarray with MPI
With the guru interfaces of FFTW, I can apply transforms only to parts of a multidimensional array by modifying the fftw_iodim ...
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1
vote
0
answers
38
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How to numerically transform a 2D Fourier spectrum with arbitrary frequency shift to center frequency?
Suppose $F(u,v)$ is the center frequency Fourier representation of some $f(x,y)$ in 2D.
$$
f(x,y)=\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}F(u,v)e^{2\pi i (xu+yv)}dudv
$$
In ...
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0
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Discrete sine and cosine transform for mixed derivatives
Using sine and cosine transforms to solve Poisson's equation with Dirichlet boundary conditions seem quite standard nowadays (see, e.g., here or Table 2 in this paper). In the case of Poisson's ...
- 71
6
votes
3
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4k
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Error propagation through an FFT
If I take the Fourier transform of data $x \pm \sigma$, is there a standard approach to what the error in the outputs will be? Would the best way be a direct evaluation of the upper and lower bounds?
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0
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295
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Fourier transform spherical system
I need to take the Fourier transform of a 3D function $h(r)=h(|r|)$ so that I can invert a convolution problem.
What is the best way to do this with Python?
I know the the FT is equivalent to a sine ...
- 171
1
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0
answers
122
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Code for solving the heat equation on the semi-infinite rod
Cross posted in mathematica.SE.
Question : I want to test the solution which is given below is right by Matlab/Maple/Mathematica.
Please look the post in mathstackexhange
or
Please look below.
...
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votes
1
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232
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Fourier characteristics of repeated numerical derivative
Background
I am trying to analyse fourier characteristics of a derivative. For example if I have a first order derivative approximated as following:
$$\frac{\partial \Psi(x)}{\partial x} = \frac{\...
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2
votes
0
answers
152
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Precision not improving by decreasing step-size in nonlinear Schrödinger
I tried to simulate soliton propagation by solving the nonlinear Schrödinger equation using the split-step Fourier method. The following is an example of the Matlab code copied from a textbook.
...
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votes
2
answers
281
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von Neumann stability analysis for spatial variable flux
Can we use the von Neumann stability analysis to investigate the stability of the discrete form of the following problem?
$$u_t+\frac{\partial(x^2u)}{\partial x}=S(u,x)\ .$$
Please give some hint ...
- 33
1
vote
1
answer
116
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fft with non uneven spacing between the value of the signal
I am trying to implement in C or C++ a solution for a fft and Ifft when the signal values are not obtained at a constant rate, making it having a desviation between the values and the periodic ones. I ...
2
votes
1
answer
1k
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1-D turbulent energy spectra in homogeneous direction (non-isotropic)
I am trying to compute the one-dimensional energy spectra for my channel-flow simulation. I have already written a post-processing script to achieve this; however, I need to validate my code before ...
- 103
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0
answers
76
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Perfect filtering of high frequencies in 2D FFT (Multidimensional 2/3 Rule)
Let $u_n$ be an array containing discrete values of the function $u(x,y)$. Performing a 2D FFT to this array we obtain $\hat{u_n}$ representing the values of $\hat{u}(k_x,k_y)$. I would like to ...
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1
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Von Neumann stability analysis with a constant term
I have a question concerning the von Neumann stability analysis of finite difference approximations of PDEs. There seem to be a wealth of online source explaining the application of this stability ...
- 143
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0
answers
69
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How to do numerical computation of $L^p$ norm of a $p$ dimensional trigonometric polynomial
Id like to know methods for numerical computation of $L^2$ norm of a two dimensional trigonometric polynomial.
I have the coefficients. If I want to compute the L^1 norm, I can do so by sampling in ...
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