Questions tagged [fourier-transform]
For questions about Fourier transforms, how they are used, and implementation details.
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Numerically computing envelope of Gibbs oscillation
If I numerically compute the envelope of $\sin(\pi t)$ using a Hilbert transform, I obtain exactly what I expect:
If I do the same for $\mathrm{sinc}(t)$, still I obtain an envelope which agrees with ...
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How to plot the power spectrum
I have an array of data whose columns are solution vectors to a system of ODEs at a specific time. I want to plot the power spectrum of a solution at a specific time, but when I attempt this I get ...
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Numerical integration in Fourier space over 3D grid
I am attempting to implement a model outlined in this paper:
General magnetostatic shape–shape interactions
Background
This model allows the calculation of magnetostatic interaction energies between ...
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Fast Fourier Transform on Meshes
I have a (closed, manifold, oriented) triangular mesh for which I build a matrix $L\in\mathbb{R}^{n\times n}$ discretising the negated Laplace-Beltrami operator. The matrix $L$ is symmetric positive ...
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Complex matrix logarithm discontinuity by solving inverse Fourier integral by alternative method to FFT
NOTE: This code is a piece of code I am using for a master's thesis, so I do not expect someone to do the work for me, but I gladly accept suggestions of any kind.
However, I am trying to get the ...
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Helmholtz decomposition of a vector field in Fourier space with Python
I have a 3D vector field and I want to extract its divergence-free part (also called transverse component), using the Helmholtz decomposition.
In principle, this can be done in the Fourier space, as ...
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Spectral Intensity of complex signal
I'm simulating an electromagnetic wave that has a real and imaginary part. Something like:
$$ E(x,t) = A(x,t) e^{-i(\omega t - k x)} $$
Where $A(x,t)$ is some complex amplitude. Then taking the ...
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Numerical solution of nonlinear water wave equation with Dirchlet-Neumann operator
I've been trying now for quite some time to numerically solve the nonlinear water wave equation [Craig and Sulem, JCP (1993)] by using FFT to discretize the space. I present my code below. By testing ...
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How to accelerate a convolution (laplace kernel) with FFT
I have the following computation I'm trying to program and accelerate with the FFT.
$$
\phi(x) = \sum_{y \in Y} K(x, y) q(x), \> \> \forall x \in X
$$
Where $X$ and $Y$ are sets of Cartesian ...
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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 ...
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How to get the inverse FFt in this Fortran code?
I find this fft algorithm on the link
The code looks simple and easy to implement. But it does not have inverse fast Fourier transformation. A brief search on the internet shows that to get the ...
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Is it possible to realign sliding-window FFT-filtered output on time-line with the original signal when it is used for noise-filtering in real-time?
In real-time systems, an output data can be taken into sliding-window Fourier Transformation directly without waiting for new data. Then Fourier Transformed data can be altered (remove high ...
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Free Time Dependent Schrodinger Equation with Inhomogeneous Dirichlet boundary
There exists a FFT-based method to solve the poisson equation in inhomogeneous Dirichlet boundary condition using the sine-transform. For example, Which fourier series is needed to solve a 2D poisson ...
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Computing convolution on non-uniform sample
How to efficiently convolve the function $h(t)=H(t)e^{-t}$ with a function $x(t)$ sampled non-uniformly, i.e. $\{x(t_0), x(t_1), ..., x(t_{N-1})\}$?
$H(t)$ is the Heaviside step function, and the ...
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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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Does DCT diagonalize the FD discretisation of the Laplacian with Neumann boundary conditions?
If one has the Poisson problem (assume $\int_{\Omega} f = 0$ and $\int_{\Omega} u = 0$):
\begin{alignat}{3}
\Delta u(x) &= f(x), &\quad&x\in\Omega \\
\partial_nu(x) &= 0, &\quad&...
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Padding length and error analysis of discrete convolution by FFT
The standard algorithm for discrete convolution of two vectors $x\in \mathbb{R}^{n}$ and $y \in \mathbb{R}^{m}$ is (in essence) a FFT of the two input vectors, multiplication of the two elementwise, ...
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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 ...
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How to perform FFT from plane-wave basis function coefficients to real space?
I have a 3D grid in real space of grid spacing $L$ and say 21 grid points in each direction, containing e.g. a charge distribution. This is stored as a numpy array of shape ...
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Numerical solution of 2D wave equation using Fourier transform and finite differences
This is the $2$-dimensional wave equation
$$ u_{tt} = u_{xx} + u_{yy} $$
with initial condition $u(x,y,0)=f(x,y)$ and $u_{t}(x,y,0) = 0$.
The inverse Fourier transform used is
$$ u(x,y,t) = \iint \hat{...
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Why not use the convolution theorem for explicit timestepping?
Consider the advection equation
\begin{equation}
\frac{\partial C}{\partial t} + u\frac{\partial C}{\partial x} + v\frac{\partial C}{\partial y} = 0
\end{equation}
I want to do a forward time, center ...
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Bounds condition for IFT to obtain a $1/f$ time-series
I am coding a function to obtain a randomized time-series from a given $\frac{1}{f}$ law.
The randomization is obtained by introducing a random phase in the function.
I experience a problem in the ...
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How do people deal with resized grid steps while numerically integrating using discrete Fourier Transform?
I am trying to simulate light propagation on python using FFT following the Fresnel diffraction equation given on Wikipedia:
The problem with this is that the output matrix from the DFT would be ...
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How do you correctly implement Scipy's FFT procedures to produce a low-pass filter - image processing
I'm following this low-pass filter example in the text "Image Operators: Image Processing in Python 1st Edition" by Jason M. Kinser, but can't seem to duplicate their results.
The text's ...
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3D Cooley-Tukey FFT
To compute the $N$-point DFT
$$ X[k] = \sum_{n=0}^{N-1} x[n] W_N^{kn} $$
where $N = N_1 N_2$, we can write the indices as $n = N_2 n_1 + n_2$ and $k = k_1 + N_1 k_2$, (effectively packing the data ...
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2D DFT for lower frequencies only; is there something significantly faster than numpy.fft.fft2 (throwing away high frequencies)?
I do a lot of 2D discrete FFT in python using np.fft.fftshift(np.fft.fft2(y)), then throw away 90% or more of the array, keeping only the central low-frequency area....
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Fast evaluation of trigonometric polynomials
Suppose you have a trigonometric polynomial of the form
\begin{equation*}
x(t) = \sum_{k = 0}^N a_k \cos(2 \pi k f_0 t).
\end{equation*}
Using Clenshaw algorithm, one can evaluate this polynomial in $...
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Solving Poisson-like PDE with FFT
Problem
I have an $n\times n$ grid, and each point on the grid is assigned two values: a score, and an (inverse) speed factor. There is a "turtle" moving along the grid, and it's goal is to ...
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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 ...
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Improving efficiency of FFT for large time window and single frequency pulses
Spectral methods for pulse propagation usually require at least one FFT and one iFFT for each step. In my case I have a two-dimensional radially symmetric electric field (one dimension in space, one ...
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Reason behind different outputs for Fast Fourier Transform in Numpy and Matlab
Here is the output of Numpy
np.fft.ifft([0, 4, 0, 0])
array([ 1.+0.j, 0.+1.j, -1.+0.j, 0.-1.j]) # may vary
Here is the output of Matlab
...
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Fourier transform in finite element
I have a finite element solver where I am using tetrahedral elements. I am solving for electric potentials and then calculate the current densities in each element, which are constant in each element. ...
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The derivative of a gauss function via FFT and IFFT in Python
I have a problem with computing a derivative of a Gauss function using FFT and IFFT from NumPy library. I use the fact that
$$
\begin{equation}
\frac{d}{dx}f(x) = \frac{1}{\sqrt{2\pi}}\int{ike^{ikx}\...
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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 ...
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How to take convolution of two arrays in Python by using NumPy?
Generally, we know that if we have this relation between Fourier transforms of three functions in frequency domain as:
$$\mathfrak{F}\{\mathsf{P}(t)\} = \mathfrak{F}\{\mathsf{Z}(t)\}\mathfrak{F}\{\...
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Problems with simulation of a spatial filter 4f setup (Python)
I have a question about my code which computes numerically the output field of a 4f setup with a pinhole in the middle which works as a spatial filter.
My setup consists of two lenses with 50mm focal ...
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Computing numeric derivative via FFT - SciPy
I wrote the following code to compute the approximate derivative of a function using FFT:
...
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How to define a dimensionless Objective function for determining how peaked a curve is?
I have attached 2 plots for FFT spectra. One is considered good and one is bad.
The good one is classified on the basis of how closely spaced the frequencies and the bad is based on how multiple ...
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Solving a spectral system by reducing it to a single frequency - Feasability of approach?
I'm trying to solve the linear non-paraxial pulse propagation equation
$$\partial_z\hat{E}=ik_z\hat{E}$$
for a field defined as $$E=E(r, t, z)$$
The equation given above uses $$\hat{E}=\hat{E}(k_\perp,...
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Problem implementing convolutions exactly with the FFT
I'm trying to perform convolutions as defined mathematically $f \star g (\tau)= \int_{\mathcal{R}}f(t-\tau)g(t) dt$ in a numerical simulation. Hence, my signal is a sampling of points $f(x_i)$.
I ...
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Fourier spectral method for coordinate transformed heat equation
As the title said, I want to solve a coordinate transformed heat equation using fourier spectral method. In particular, I am interested in transforming an uniform grid into an adaptive non-uniform ...
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Advantage of fractional Fourier transform over multiscale wavelet?
What could be the arguments of using fractional Fourier transform instead of multiscale wavelet for data analysis ?
Optimization of the good time-frequency domain parameter? good in the sens of best ...
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Computation of triple nested loops as a convolution product?
I'm trying to compute efficiently the following
\begin{equation}
A_j = \sum_{l'=1}^{\infty}\sum_{k= 0}^{K-1} L_{l'}T_ke^{2\pi i \frac{k}{K}j}\epsilon_{l',k}
\end{equation}
for $j = 0,1, \ldots, K-2,K-...
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FFT convolution works only with certain domain length
in my quest to understand how I can use FFT to compute integrals (see my other question click, still no answer there), I came across the fact that a convolution of two functions can be calculated by ...
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Normalising DFTs Correctly
I have been playing around with convolutions in scipy's signal package:
...
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242
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Smoothing FFT result
I am trying to calculate the spectrum of Bremmstrahlung, which involves calculating the Fourier transformed acceleration. I am solving a non-linear ODE to numerically calculate the acceleration in the ...
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Differences between Discrete Fourier Transform and Continuous Fourier Transform?
I am trying to visualize the time dependence of a free particle given an initial wave-function using Python and I just wanted to know if I could use the in built FFT implementation from NumPy to find ...
