# Tagged Questions

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### Why would the discrete fourier transform “see” signals like this? What is the origin of spectral leakage?

The discrete fourier transform of $x = (x_{0},\dots,x_{N-1})$ is defined as $\displaystyle X_{k} = \sum_{n=0}^{N-1} x_{n}\omega^{kn}_{N}$ where $\omega^{kn}_{N} = e^{-2\pi ik/N}$ and ...
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### Discrete Fourier Transform Interpretation

Using Mathematica I took the Discrete Fourier Transform (DFT) of a vector whose entries are volumes of a particular stock. The power spectrum is plotted below: There are two questions that I have ...
0answers
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### Using the Modulation property of the Fourier Transform

I'm working on a problem: Let $X(w)$ be the Fourier transform of $x(t)$. Find the transform of $y(t)=x(5t+3)\sin(2t)$ in terms of X(w). I am table to take the Fourier transform of $x(5t+3)$ and ...
1answer
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### abs(x)cos(x) in Fourier space

I am working on some problems concerning Fourier Transform and I am facing something I don't understand. I am trying to understand what is the representation of the function f(x)=abs(x)cos(x) in the ...
1answer
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### Fourier Transform of rational function

So I have this function: $$f(t)=\frac{1}{(1-it)^{n+1}}$$ And I have the Fourier Transform defined as $$\hat{f}(\lambda)=\frac{1}{\sqrt{2\pi}}\int_\mathbb{R}f(t)e^{-\lambda.it}dt$$ Now my ...
0answers
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### Invariant functions under integral transforms

We all know Fourier transform has invariants such as $e^{-x^2}$, and another MSE post has shown the non-existence of invariant function under Hilbert transform using Fourier transform. I am wondering ...
1answer
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### Fourier transformation: Determining the axis

I need some help with the Fourier transformation of my data. My original data is a Distance VS Time: upon doing a Fourier Transform, I get the following: I understand that normally after a ...
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### Is a Fourier transform a change of basis, or is it a linear transformation?

I've frequently heard that a Fourier transform is "just a change of basis". However, I'm not sure whether that's correct, in terms of the terminology of "change of basis" versus "transformation" in ...