Questions tagged [signal-analysis]
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108
questions
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Gain of a steady state Kálmán filter
It is well known that the state covariance of a steady-state Kálmán filter is the solution of a discrete Riccati equation.
$$P_\infty = F(P_\infty - P_\infty H^T(HP_\infty H^T+R)^{-1}HP_\infty)F^T + Q$...
1
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0
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58
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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 ...
1
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0
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157
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Fast algorithm for computing certain signal transformations
Let $f,g,h:\mathbb Z\to\mathbb C$ supported on $[-n,n]$. For $\tau\in \mathbb Z$, let $\operatorname{sh}_\tau f$ be the shift of $f$ by $\tau$ (i.e. $(\operatorname{sh}_\tau f)(t) = f(t-\tau)$). ...
6
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1
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365
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When are the chirp signals orthogonal?
Assume that we have two bounded-time chirp signals,
\begin{align}
x(t)&=\exp\Big(j\pi(\alpha t^2+\beta t+\gamma)\Big),\quad 0\leq t\leq T,\\
y(t)&=\exp\Big(j\pi(\alpha' t^2+\beta' t+\gamma')\...
2
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2
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159
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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 ...
0
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2
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115
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Reshaping data vector into a matrix for deconvolution using a circulant matrix
Suppose we have a circulant matrix S made from pseudorandom binary sequence of length $N$ consisting of $0$'s or/and $1$'s. $1$ means that we can inject something for chemical analysis and $0$ means ...
0
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0
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40
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When wavelet estimates fail?
I am interested in some models studied in non-parametric estimation, more precisely the Gaussian white noise model,
$$dX_{t_{1},...,t_{d}}=f(t_{1},...,t_{d})dt_{1}...dt_{d}+\theta dW_{t_{1},...,t_{d}}$...
2
votes
1
answer
94
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PCA-like method for filtering known variances
Principal Component Analysis is used to reduced the dimensions of atmospheric pressure grids (lat X long X time) into their most important modes of behaviour (e.g, the North Atlantic Oscillation is ...
0
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0
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102
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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 ...
1
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1
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72
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Discrete uniqueness sets for the two-sided Laplace transform?
Let $f : \mathbb R_+ \to \mathbb C$ be a measurable and integrable function where $\mathbb R_+ = [0,\infty)$. The Laplace transform of $f$ is given by
$$
Lf(s) = \int_0^\infty f(x)e^{-sx} \, dx.
$$
A ...
3
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0
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103
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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{...
2
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40
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Selecting some linearly independent columns of a particular matrix
Let us consider the matrix $C=A_1+A_2$ where :
$A_1=(a_{k,l})_{k,l=0}^{n-1}$ is the $n$ by $n$ matrix given by $a_{k,l}=\frac{2}{\sqrt{n}}(\cos\frac{2kl\pi}{n})$
$A_2$ is the the $n$ by $n$ block ...
1
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184
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Special function: Pulse peak modified with a power term
PeakFit (Systat, v. 4.12) is a software for fitting experimental peaks obtained in physics or chemical experiments. Under the miscellenous peak functions, it shows the following equations with a name, ...
7
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246
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Square-root lattices: where do they appear?
As an experimental physicist working on crystallography I'm often dealing with the reconstruction of an object from intensity data that emerge from an imaging device. In mathematics the problem is ...
6
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1
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405
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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, ...
0
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0
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62
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Direct (first-order ?) algorithm to minimize $u(x) := \|x-a\|_C + r\|x\|_p$
Fix $a \in \mathbb R^n$, $r \ge 0$, $p \in \{1,2\}$, and a positive-definite matrix $C$ of order $n$. Define $u:\mathbb R^n \to \mathbb R$ by $u(x) := \|x-a\|_C + r\|x\|_p$, where $\|z\|_C := \sqrt{z^\...
3
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2
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157
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On finding an upper bound on the error of a sparse approximation
I posted this question on math.stackexchange earlier, but didn't see any response. So, I am posting it here, in case someone else has an answer.
Original question: https://math.stackexchange.com/...
1
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1
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309
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A particular commutator of the discrete Fourier matrix
For $N$ be a fixed natural number, define $w=e^{\frac{2\pi i}{N}}$ and $z=e^{\frac{\pi i}{N}}$, so that $z^2=w$. Let $D$ be the diagonal matrix $D=\operatorname{diag}(1,z,z^2,\ldots,z^{N-1})$ and $F$ ...
2
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0
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116
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Consistent approximation of weighted Radon transform of smooth probability density, using kernel density estimation
Let $X$ be a random vector in $\mathbb R^d$, with "sufficiently smooth" probability density function on $\rho$. For unit-vectors $w$ and $u$ in $\mathbb R^d$, and a scalar $b \in \mathbb R$, ...
1
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0
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96
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Optimal bandwidth for a Gaussian filter
I have an $n \times n$ image $A$, and an $m\times m$ image $B$, where $n>m$. As the smaller image looks like a lower-resolution version of the larger one, I'm interested in the relative loss, ...
4
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1
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470
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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 ...
3
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1
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305
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Fast computation of convolution integral of a gaussian function
Given a convolution integral
$$
g(y) =\int_a^b\varphi(y-x)f(x)dx=\int_{-\infty}^{+\infty}\varphi(y-x)f(x)\mathbb{I}_{[a,b]}(x)dx
$$
where
$\varphi(x)= \frac{1}{\sqrt{2\pi}}\exp{\left(-\frac{x^2}{2}\...
3
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0
answers
140
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Is there any injective mapping from smooth functions on closed interval to smooth functions on circle? Motivated by signal processing
One advantage of Discrete Cosine Transform (DCT) over Discrete Fourier Transform (DFT) is that DCT maps any "continuous" signal defined on interval to a continuous one defined on circle.
I ...
1
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1
answer
229
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Continuous wavelet transform of a periodic function
I have a question regarding the Continuous Wavelet Transform (CWT) of non finite energy functions, such as $g(t) = a\exp(i\omega_0t)$. We know that the CWT is defined for functions in the Hilbert ...
1
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0
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25
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Message Passing algorithm: misadjustement, study of convergence, for inexact MPA
I am looking for resources (articles or other information) on the derivation of mis-adjustments and on the study of convergence for the message passing algorithm (MPA) and/or the inexact message ...
3
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532
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van Cittert deconvolution method
In the early 1930s, van Cittert published a deconvolution method. Although his method was not perfect but it is the forefather of many improved spectral deconvolution methods. The basic idea is that ...
1
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1
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112
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How many Fourier coefficients of a sparse signal $f=\sum_{n=1}^Nc_n\delta_{t_n}$ are needed to determine $f$ uniquely?
Let $N \in \mathbb N$ and $c_n \in \mathbb C$, $t_n \in \mathbb R$ for $n=1, \dots, N$. Suppose that $f$ is a linear combination of dirac-deltas with locations $t_n$ and coefficients $c_n$, i.e.
$$
f=\...
3
votes
1
answer
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Deconvolution using the discrete Fourier transform
Summary: From discrete convolution theorem, it is understandable that we need 2N-1 point DFT of both sequences in order to avoid circular convolution. If we need to do deconvolution of a given ...
6
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2
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327
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Is there a way to reconstruct the convolution $(f * g)(x)$ of $f$ with a Gaussian $g$ from sampled values, $(f*g)(a), a \in A$?
Suppose that $f: \mathbb{R} \to \mathbb{C}$ is a function which has support in $[-1,1]$. Let $g = g_\sigma$ be a centered Gaussian with variance $\sigma^2$. Is there a way to reconstruct the ...
1
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0
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56
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The meaning of the frequency in continuous signals
Suppose that for a given signal $x:\mathbb{R}\to \mathbb{C}$ both of the following Fourier identities hold.
$$ \hat{x}(\omega)=\int_\mathbb{R} x(t)e^{-it\omega} dt~~~,~~~x(t)=\frac{1}{2\pi} \int_\...
2
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96
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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 ...
3
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1
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755
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Relation between signal derivative and frequency spectrum
I want to sample a signal whose derivative I know to be bounded by physical constraints. The sampling is disturbed by gaussian noise, hence I need to filter the sample with a lowpass filter.
Since I ...
3
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0
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202
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Compressed sensing for partitioning instead of recovery
Let $x_0 \in \mathbb{R}^{m}$ be a signal whose support $T_0 = \{ t \mid x_{0}(t) \neq 0\}$ is assumed to be of small cardinality. The recovery of $x_0$ from a small number of $n \ll m$ linear ...
2
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0
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122
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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
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225
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Convergence of the expectation of a random variable when conditioned on its sum with another, independent but not identically distributed
Suppose that for all $n \in \mathbf{N}$, $X_n$ and $Y_n$ are independent random variables with
$$X_n \sim \mathtt{Binomial}(n,1-q),$$
and
$$Y_n \sim \mathtt{Poisson}(n(q+\epsilon_n)),$$
where $q \in (...
0
votes
0
answers
108
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Wigner distribution
The Wigner distribution of $u\in L^2(\mathbb R)$ is defined as a function $W(u)$ on $\mathbb R^2$ given by
$$
W(u)(x,\xi)=\int_\mathbb R u\left(x+\tfrac z2\right) \overline{u\left(x-\tfrac z2\right)} ...
1
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1
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219
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The derivative of a filter with respect to a output signal [closed]
I have two signals, $d(t)$ and $p(t)$, respectively the input and the output of the matching filter $w(t)$, i.e.
$$
d(t)*w(t)=p(t)
$$
where $*$ denotes convolution.The impulse response $w(t)$ may be ...
2
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1
answer
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History- calculating convolution by tabular method
I often see a trick for calculating convolution of discrete data by a so-called Tabular method. There are a lot of Youtube videos and many Indian textbooks on Signal Processing [Books].1
Basically, ...
20
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2
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Origin of the term "sinc" function
Is the sinc function defined here, really a short form of "sinus cardinalis" as proposed by Wikipedia? This information is deleted now but it existed some time ago. Even if we search Google Books for ...
1
vote
1
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87
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Additional structures for sparse recovery
The problem of sparse recovery using $l_1$ minimization is well known. Using random Gaussian matrices, we are able to achieve recovery with high probability in $O(k\log(d/k)$ measurements. It is ...
5
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3
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756
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Mathematical Techniques to Reduce the Width of a Gaussian Peak
In the chemical analysis by instruments, the signals of several molecules are overlapped which makes it difficult to determine the true area of each peak, such as those shown in red. I simulated this ...
52
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1
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5k
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Mathematics of imaging the black hole
The first ever black hole was "pictured" recently, per an announcement made on 10th April, 2019. See for example: https://www.bbc.com/news/science-environment-47873592 .
It has been claimed that ...
1
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0
answers
38
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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 ...
1
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0
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56
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The significant role of dual frames in the progress of Frame theory
For a given frame $\{\zeta_i\}_{i=1}^\infty$, any Bessel sequence $\{\eta_i\}_{i=1}^\infty$ satisfying in the following identity for every $\xi\in H$
$$\xi=\sum_{i=1}^\infty \langle \xi, \eta_i\...
1
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0
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137
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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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0
answers
91
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What is intuitive perception of $T_{\alpha_1} \circ T_{\alpha_2} \circ ... \circ T_{\alpha_M} $ in graph domain?
if $G(V,E,W)$ be a weighted graph and $\vert V \vert =N $. for any vertex $i \in \lbrace 1,2,...,,N \rbrace $ define a generalized translation operator $T_i:\mathbb{R}^N \to \mathbb{R}^N $via ...
2
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0
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21
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Optimum frequency estimation over a Rayleigh fading channel
This question comes from the book "Autonomous Software-Defined Radio Receivers for Deep Space Applications"1.
The chapter 4.1.3 is "Optimum Frequency Estimation over a Rayleigh Fading Channel". The ...
3
votes
1
answer
200
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Why do we consider some weakening frames like K-frames, frame sequences, and upper semi-frames?
I have found some applications of the Frame Theory in engineering sciences like signal processing, image processing, data compression, sampling theory, optics, filter-banks, signal detection.
As we ...
1
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1
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130
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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}^...
8
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1
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2k
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graph signal processing
I have read this article
https://arxiv.org/abs/1307.5708
about vertix-frequency analysis on graph.
David IShuman
in this article claims that,"we generalize one of the most important signal ...