All Questions
Tagged with fourier-transform signal-analysis
20 questions
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69
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Is there an generalisation of convolution theorem to integral transforms
Basic convolutions can be computed efficiently by taking fourier transforms and applying the convolution theorem. Is there something analogous for a more general transform, where we have a varying ...
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1
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106
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Spectral theory: a key to unlocking efficient insights in network datasets
In the context of directed or undirected graphs, matrices such as adjacency and Laplacian matrices are commonly used. The eigenbasis of these matrices addresses some practical implications, such as ...
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1
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81
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The Discrete Fourier Transform (DFT) decomposes any signal into four orthogonal signal components [closed]
Let $F=(w^{kl})_{k,l=0}^{n-1}$ be the discrete Fourier matrix of size $n$ where $w=\exp\left(-\frac{2\pi i}{n}\right)$.
It is a well-known that $F_n^4 = I_n$ where $I_n$ represents the identity ...
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2
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235
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Theoretical/Practical Implications of DFT Eigenvectors
Discrete Fourier transform (DFT) has only four distinct eigenvalues: $±1$ and $±i$. For large matrices , each eigenvalue $λ$ yields a multidimensional eigenspace, allowing linear combinations of ...
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113
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Is this formula for 2D Fourier integral of diffraction kernel correct?
Well I have a function parametrized by $z$
$$g_z(x,y) = \frac{z}{i \lambda r^2} e^{i k r}, \quad r = \sqrt{x^2+y^2+z^2},$$
where $\lambda > 0$ is real constant and $k = \frac{2\pi}{\lambda}$. This ...
- 151
3
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106
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A new arranging of discrete sine transform
Let $n$ be even and consider the discrete sine transform of type 5 which is the matrix
$$S=\left(\sin(k+1)(l+1)\frac{\pi}{n+\frac12}\right)_{k,l=0}^{n-1}$$
Let us denote by $s_{-,l}$ the $l^{\text{...
- 12.9k
6
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1
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491
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Harmonic analysis for a beginner
I am currently dealing with discrete Fourier transform and correlation technique to construct the spectrum of a broad band signal. It's already known that if I have enough observations of the signal, ...
- 188k
3
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1
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2k
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Deconvolution using the discrete Fourier transform
Summary: From discrete convolution theorem, it is understandable that we need 2N-1 point DFT of both sequences in order to avoid circular convolution. If we need to do deconvolution of a given ...
- 2,821
4
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1
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520
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The main topics (issues, problems) of the Fourier transform
To explain what we are looking for, let's have a quick review on some points in Fourier transform on periodic functions in both continuous and discrete cases. We emphasize that our attention is ...
- 2,605
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57
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The meaning of the frequency in continuous signals
Suppose that for a given signal $x:\mathbb{R}\to \mathbb{C}$ both of the following Fourier identities hold.
$$ \hat{x}(\omega)=\int_\mathbb{R} x(t)e^{-it\omega} dt~~~,~~~x(t)=\frac{1}{2\pi} \int_\...
- 4,058
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105
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Fourier Transform diagonalizes time-invariant convolution operators [closed]
I got the following paragraph from the book "A wavelet tour of signal processing" chapter one, page 2.
The Fourier transform is everywhere in physics and mathematics because it diagonalizes ...
- 4,058
2
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127
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eigenvectors of a graph Laplacian VS Fourier basis
Could you please illustrate the following statement:
the eigenvectors of a
graph Laplacian behave similarly to a Fourier basis, motivating
the development of graph-based Fourier analysis theory.
- 4,058
1
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0
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40
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Fitting an inverse DFT within predefined bounds
My problems starts out with a variable length of samples. Usually, it is 1024 or higher powers of 2. The DFT of this "signal" is taken and only the amplitude spectrum is retained and the phase ...
- 111
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138
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Proof that two vectors can not have the same power spectrum when one is a permutation (excluding rotations) of the other?
The power spectrum being the absolute value of the DFT of the vector.
Has it been proven that two vectors can not have the same power spectrum if one is a permutation of the other? Where, in this ...
1
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1
answer
134
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Maximum Magnitude Deviation between DFT and DTFT
This is a cross-post from signal processing forum as it was not conclusive.
Let $x[n]$ be a finite-length sequence with length $N$. The continuous DTFT $X(\omega)$ is then
$$
X(\omega) = \sum_{n = 0}^...
- 909
1
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1
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1k
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Relationship between Fourier series & DFT
Sources like http://www.dsprelated.com/dspbooks/mdft/Relation_DFT_Fourier_Series.html explain the equivalence between FS and DFT.
However, isn't there a flaw? When I integrate over the continuous ...
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171
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Convergence of integral formula for Fourier inversion (and Hilbert transform) for integrable piecewise-smooth functions
I asked the question below on Math Stack Exchange, https://math.stackexchange.com/questions/2592555/convergence-of-integral-formula-for-fourier-inversion-and-hilbert-transform-fo, but [despite it ...
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1
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138
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How can obtain energy of a signal using stockwell´s transform?
The stockwell´s transform is defined as: $$S(t,f) = \int_{-\infty}^\infty x(\tau)w(t-τ,f)e^{-2\pi if\tau}d\tau$$ Where $$w(t-τ,f)$$ is the gaussian window.
I need obtain the energy of a signal using ...
- 188k
3
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2
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354
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Bandwidth approximation for a nonlinear problem
Can anyone please help me with this problem.
I must let you know from the beginning that it's not an easy one.
"Two functions are given: $u, y \in L^{2}(-\infty,\infty), y(t)=\frac{u(t)}{u(t)+b}$ ,
...
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1
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Multidimensional Filters
Say you want to design a LP FIR filter with low pass cutoff $fc$, transition band $fc$ to $fs$ and ripple factor $dp$ at passband and $ds$ at stop band. If one divides the frequencies by $\pi$, then $...
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