# Tagged Questions

Questions on the computational aspects of Fourier analysis, including the various applications of the fast Fourier transform (FFT).

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### Convolution of two real functions using discrete Fourier transform (FFT): zero-padding and normalization

I want to obtain the convolution of two discretized real functions $f$ and $g$, $$c(t) = \int_{-\infty}^{+\infty} \mathrm{d}{x} \, f(x) \, g(t-x) \tag{1}$$ via discrete Fourier transform (DFT). As ...
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### Fourier in X and Chebyshev in Y direction elliptic equation

Can anybody please explain me the algorithm to solve a 2D-3D elliptic equation with Fourier in X-Z direction for periodic boundary condition and non-periodic (Chebyshev) in Y direction. I have read ...
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### How do I solve Laplace's equation in 2D using spectral methods?

I want to solve the 2D Laplace's equation: $$\frac{\partial^2 T}{\partial x^2 } + \frac{\partial^2 T}{\partial y^2 } = 0$$ with boundary conditions: T(x=0)=T(x=1)=T(y=1)=0 and T(y=0)=1 on a ...
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### Finding errors in frequency from a Fast Fourier Transform from Gaussian fitting

I took a FFT of sound in a box generated by a frequency sweep over a range of frequencies, and have an array of frequencies and their corresponding FFT amplitudes. According to models for the ...
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### How do I avoid divide-by-zero when solving the Poisson equation with Fourier transforms?

I wanted to try to implement part of the method in the following article using Fourier transforms. http://www.shodor.org/media/content/jocse/student_submissions/nocito2010/nocito2010_pdf Right now I ...
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### Numerically computing Viscous Burger

I am trying to solve the Viscous Burgers equation using the spectral method. $$u_t+uu_x = Du_{xx}$$ where $D$ is a constant (chosen to be zero) and with the initial condition $$u(x,0) = exp(-x/0.2)^2$$...
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### Numerically computing the advection equation

I am trying to write a program to compute the advection equation. $$u_t +u_x = 0$$ I use the spectral method for the spatial derivative $u_x$ and the leapfrog method for the time derivative $u_t$. ...
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### Computing modes with fourier beam propagation

For my final optics projects I have spent the last few weeks implementing the beam proportionate method with Fourier split steps. This now works really well. Now I am trying to compute the modes of ...
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### Does there exist a Fourier transform algorithm for perturbed data?

Assuming I have a length-$n$ real vector $x$ and have already computed its Fourier transform $\hat x$ (in time $O(n\log n)$), I would like to compute the Fourier transform of $y = x + \delta x$, where ...
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### Does this Algorithm (probably Fourier like) Exist for 2D Shapes? [closed]

Update: Someone changed the title to this post to a possible answer ("Fourier decomposition of parametric shapes") but I changed it to a different title as that makes it clear what I was asking. As I ...
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### Fourier techniques and periodic boundary conditions

Could somebody explain to me why periodic boundary conditions are automatically satisfied if you solve your problem assuming a Fourier series? So, if we assume a Fourier series for our solution, we ...
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### Quick and simple discrete 2D Helmholtz-Hodge Decomposition using FFTs?

For a silly screen saver I'm trying to develop, I'd like to randomly generate a divergence-free 2D array of 2D vectors, and then use it to generate a line integral convolution plot. I've heard$^1$ ...
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### Scaling factor of the inverse Fourier Transform (for convolution purposes)

I have a certain 2-D function. More properly, I have not the function itself, but the matrices $[X,Y,Z]$, where $X,Y$ are $1\times n$, and $Z$ is $n \times n$. Now, I want to calculate a a new matrix,...
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### 3D Poisson equation, Fourier and Chebyshev

I am currently trying to solve the 3D Poisson equation with a Chebyshev discretisation in the $z$ direction (from -1 to 1) and Fourier in the $x$ and $y$ (from $-\pi$ to $\pi$) I have taken the code ...
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### How can DFT of a two dimensional array be found using program for one dimensional DFT in C?

I have the program four1.c from Numerical Recipes in C to calculate the Discrete Fourier Transform (DFT) of a one dimensional array. I want to calculate the DFT of ...
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### What spline functions are used in Section 13.9 of “Numerical Recipes in C”?

I asked a similar question on MathSE but with more added fluff, but didn't really get any straight answers, so I figured I'd ask here. Computing Fourier coefficients of a function using the FFT is ...