# Tag Info

### Approximate $h$ in $F(\theta)=\sin \theta \int_{-L}^{+L}h(z)e^{-ikz\cos \theta} \,dz$

This is a linear integral equation (because the right hand side is linear in $h$) and presumably the problem is ill-posed (for one, because of the multiplication by $\sin(\theta)$ but also because the ...
Accepted

### Numerically computing the advection equation

I see several issues: The DFT computed with fft puts the zero mode at the beginning of the array, and if you want to compute the derivative, it is necessary to ...

### Fourier transform by FFT : by using cubic splines to interpolate between data points, do we change the frequency content of the Fourier transform?

The frequency content of the interpolated signal is significantly influenced by the interpolation basis. If you have a band-limited function that you have adequately sampled (i.e. satisfying Nyquist ...

### Numpy FFT gives me a pulse shorter than it should be. Not sure what I am doing wrong

Running your code, it seems like your pulse looks kinda like this: (sorry for not adding units to the plots, I used the same as you, i.e. t is in fs and w in rad/fs) So, the FWHM is not correct (...
Accepted

### Complete (an incomplete) explanation of the phenomenon of "aliasing", when using Fourier series to approximate functions?

This turned out much longer than I planned so I hope it is useful and can be a resource for others with the same questions. I believe you mean to ask primarily about the discrete Fourier transform (...
Accepted

### How do I avoid divide-by-zero when solving the Poisson equation with Fourier transforms?

You are solving Poisson's equation: $$\nabla^2 \Phi = 4\pi G \rho.$$ Notice that if $\Phi$ is a solution, then so is $\Phi+C$ for any constant $C$. Furthermore, $C$ will have no effect on your ...

### Fourier transform by FFT : by using cubic splines to interpolate between data points, do we change the frequency content of the Fourier transform?

The interpolation indeed affects the Fourier transform. @Steve already gives the correct answer in general, but I want to give you an example that helps the intuition more. Think for example that you ...
Accepted

### Von Neumann stability analysis with a constant term

From your link, consider the definition of the round off error and the statement "Since the exact solution must satisfy the discretized equation exactly, the error must also satisfy the discretized ...
Accepted

### Does this Algorithm (probably Fourier like) Exist for 2D Shapes?

You can write a curve as a parametric equation ($x(t)$, $y(t)$) for a range say $0 < t < 1$. Then you can have a Fourier decomposition of both $x(t)$ and $y(t)$. A rectangle can be written as ...
Accepted

### Quick and simple discrete 2D Helmholtz-Hodge Decomposition using FFTs?

The easiest way to generate a divergence-free 2D function is to use a streamfunction, $\psi$. Where $$u=\frac{\partial\psi}{\partial y}\quad \rm{and}\quad v=-\frac{\partial \psi}{\partial x}$$ Once ...
Accepted

### MATLAB FFT Differentiation

1) You have misinterpreted the order the fft2 output elements are stored. This is a very common mistake, since it is very easy to get confused. Thankfully, there is an easy way to avoid such bugs. ...
Accepted

### Fourier characteristics of repeated numerical derivative

Regarding point 1: generally, applying a discretized derivative (stencil) twice is not equivalent to applying an equivalent (i.e. same number of points) stencil for the second derivative. Example in ...

### Rudin lecture -- if f(x) is not integrable on some interval, does it not have a Fourier Series expansion on that interval?

The point is to guarantee that the Fourier coefficients exist, not to make really fine distinctions about what might happen if the guarantee is violated. Furthermore, you would be hard pressed to find ...

### Solving numerically the 1D Kuramoto-Sivashinsky Equation using spectral methods

Indeed the problem was that you were trying to calculate your nonlinear term incorrectly and you forgot that: $$\mathcal{F}[(f(x))^{2}] \neq (\mathcal{F}[f(x)])^{2}$$ I changed your code to this: <...
The discrete Fourier transform for a signal of period $T$ with $N$ samples reads in its inverse or reconstruction form as $$y(t)=\frac1{N}\sum_{k=-N/2}^{N/2}c_k e^{i2\pi k\frac{t}{T}}$$ with ...