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7 votes
3 answers
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Real-world examples of unweighted directed graphs

Social Networks, Internet Traffic, Citation Networks, Transportation Networks, and Biological Networks exemplify real-world instances of unweighted directed graphs. Within these domains, the Graph ...
ABB's user avatar
  • 3,992
1 vote
0 answers
60 views

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$...
Bernard 's user avatar
1 vote
0 answers
62 views

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 ...
ABB's user avatar
  • 3,992
1 vote
0 answers
161 views

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. $(\operatorname{sh}_\tau f)(t) = f(t-\tau)$). ...
Rami's user avatar
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6 votes
1 answer
368 views

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')\...
Math_Y's user avatar
  • 311
2 votes
2 answers
178 views

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 ...
ABB's user avatar
  • 3,992
0 votes
2 answers
118 views

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 ...
AChem's user avatar
  • 761
0 votes
0 answers
40 views

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}}$...
BabaUtah's user avatar
2 votes
1 answer
101 views

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 ...
Will Rust's user avatar
0 votes
0 answers
103 views

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 ...
VojtaK's user avatar
  • 151
1 vote
1 answer
73 views

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 ...
r_l's user avatar
  • 190
3 votes
0 answers
104 views

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{...
ABB's user avatar
  • 3,992
2 votes
0 answers
41 views

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 ...
ABB's user avatar
  • 3,992
1 vote
0 answers
190 views

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, ...
AChem's user avatar
  • 761
7 votes
1 answer
248 views

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 ...
user avatar
6 votes
1 answer
414 views

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, ...
CfourPiO's user avatar
  • 159
0 votes
0 answers
62 views

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^\...
dohmatob's user avatar
  • 6,726
3 votes
2 answers
158 views

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/...
Trade Paul's user avatar
1 vote
1 answer
309 views

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 $F$ ...
ABB's user avatar
  • 3,992
2 votes
0 answers
117 views

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 $b \in \mathbb R$, ...
dohmatob's user avatar
  • 6,726
1 vote
0 answers
96 views

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, ...
Jiaji Huang's user avatar
4 votes
1 answer
474 views

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 ...
ABB's user avatar
  • 3,992
3 votes
1 answer
321 views

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}\...
NN2's user avatar
  • 250
3 votes
0 answers
143 views

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 ...
Fallen Apart's user avatar
  • 1,605
1 vote
1 answer
243 views

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 ...
Humberto Gimenes Macedo's user avatar
1 vote
0 answers
25 views

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 ...
e. sfe's user avatar
  • 39
3 votes
1 answer
578 views

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 ...
AChem's user avatar
  • 761
1 vote
1 answer
112 views

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=\...
Muzi's user avatar
  • 163
3 votes
1 answer
2k views

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 ...
AChem's user avatar
  • 761
6 votes
2 answers
329 views

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 ...
J. Swail's user avatar
  • 437
1 vote
0 answers
57 views

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_\...
ABB's user avatar
  • 3,992
2 votes
0 answers
98 views

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 ...
ABB's user avatar
  • 3,992
3 votes
1 answer
778 views

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 ...
LucioPhys's user avatar
3 votes
0 answers
202 views

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 ...
J1996's user avatar
  • 131
2 votes
0 answers
124 views

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.
ABB's user avatar
  • 3,992
4 votes
0 answers
228 views

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 (...
as1's user avatar
  • 91
0 votes
0 answers
112 views

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)} ...
Bazin's user avatar
  • 15.2k
1 vote
1 answer
221 views

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 ...
Yongj Tang's user avatar
2 votes
1 answer
2k views

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, ...
AChem's user avatar
  • 761
20 votes
2 answers
3k views

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 ...
AChem's user avatar
  • 761
1 vote
1 answer
88 views

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 ...
Magi's user avatar
  • 281
5 votes
3 answers
797 views

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 ...
AChem's user avatar
  • 761
52 votes
1 answer
5k views

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 ...
Piyush Grover's user avatar
1 vote
0 answers
38 views

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 ...
Paddy's user avatar
  • 111
1 vote
0 answers
56 views

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\...
Javani's user avatar
  • 11
1 vote
0 answers
137 views

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 ...
kreitz's user avatar
  • 53
1 vote
0 answers
91 views

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 ...
niloofar jamshidi's user avatar
2 votes
0 answers
21 views

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 ...
Xue Liu's user avatar
  • 21
3 votes
1 answer
201 views

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 ...
ABB's user avatar
  • 3,992
1 vote
1 answer
130 views

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}^...
Jiro's user avatar
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