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# Tag Info

Accepted

### Taking derivative using FFT

I believe this stems from the fact that your function $f(x) = x^2$ does not have continuous derivatives once it is extended periodically like $$\tilde{f}(x) = f(x \ \mathrm{mod} \ 12 -6),$$ which ...
• 2,776

### 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 ...
• 1,031

### 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 (...
• 256

### 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 ...
• 56.1k
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 ...
• 2,269
Accepted

### Can this finite difference dispersion be eliminated somehow?

It isn't the spike that's causing the dispersion. The scheme you use has a dispersion relationship whereby waves of different frequency travel at different speeds. Every numerical scheme has such a ...
• 56.1k
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. ...
• 220

### Error propagation through an FFT

Assembled from comments of @AlexE: The Fourier transform is linear, so the error in the Fourier domain is the Fourier transform of the error in the spatial (original) domain. So, if $\sigma$ is ...
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 ...
• 211

### 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: <...

### Why do problems arise in FFT for smaller value of df in Python?

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 ...
• 6,129

### 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 ...
• 11.5k