Questions tagged [signal-analysis]
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116 questions
1 vote
0 answers
73 views
Space of interpolating functions with constraints on interpolation
Disclaimer: I am a first year mathematics student who is trying to write an applied math paper, so my question might seem trivial. Definitions: Let $N \in 2 \mathbb{N}$ and $u \in \mathbb{R}^N $ be a ...
0 votes
2 answers
125 views
Calculating the Coefficients of a Sinusoid
Question: what is known about calculating the coefficients $a,\phi,\theta,d$ of $f(x)= a\sin(\phi x+\theta)+d$, resp. of $g(x)= a\sinh(\phi x+\theta)+d$ $\phantom{}$ that interpolate $\lbrace(x_0,...
- 13.9k
2 votes
0 answers
156 views
On the expected quality of a rank-$1$ approximation of a complex Gaussian noise covariance matrix
I am developing quality coefficients for a specific type of approximation of covariance matrices. I want to to specify meaningful lower bounds (or rather thresholds), for which I would like to use the ...
- 21
0 votes
0 answers
40 views
When can the action of a large convolution kernel be replicated by a sequence of convolutions with smaller kernels?
Suppose I have some target filter $f(x)$ supported on the $r$-ball. I am curious what the necessary/sufficient conditions are for when the convolution operator $f \star h$ can be computed via a ...
- 204
2 votes
0 answers
184 views
Error in discrete FFT
I am interested in taking an FFT of an image which is periodic in space (does not decay) across a finite window of size $L\times L$. The image has triangular symmetry; for simplicity one could imagine ...
- 21
5 votes
0 answers
126 views
Rational maps from the circle to the unitary group (energy-preserving convolutive mixtures)
Consider a rational map $A : S^1 \to U(n)$, i.e. a matrix of rational functions such that evaluation at any $z \in S^1 \subset \mathbf C$ is unitary. These objects show up in digital signal processing ...
- 151
1 vote
0 answers
117 views
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 ...
0 votes
1 answer
149 views
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 ...
- 4,140
7 votes
3 answers
908 views
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 ...
- 4,140
1 vote
0 answers
110 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$...
- 437
1 vote
0 answers
125 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 ...
- 4,140
1 vote
0 answers
193 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)$). ...
- 2,669
6 votes
1 answer
527 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')\...
- 189
2 votes
2 answers
296 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 ...
- 4,140
0 votes
2 answers
211 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 ...
- 873
2 votes
1 answer
217 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 ...
- 29
0 votes
0 answers
151 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},$$ where $\lambda > 0$ is real constant and $k = \frac{2\pi}{\lambda}$. This ...
- 151
1 vote
1 answer
115 views
Discrete uniqueness sets for the two-sided Laplace transform?
Let $f : \mathbb R_+ \to \mathbb C$ be a measurable and integrable function where $\mathbb R_+ = [0,\infty)$. The Laplace transform of $f$ is given by $$ Lf(s) = \int_0^\infty f(x)e^{-sx} \, dx. $$ A ...
- 250
3 votes
0 answers
122 views
A new arranging of discrete sine transform
Let $n$ be even and consider the discrete sine transform of type 5 which is the matrix $$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{...
- 4,140
2 votes
0 answers
57 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 ...
- 4,140
1 vote
0 answers
261 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, ...
- 873
7 votes
1 answer
334 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 ...
user491917
6 votes
1 answer
556 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, ...
- 159
0 votes
0 answers
79 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^\...
- 7,033
3 votes
2 answers
211 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/...
- 65
1 vote
1 answer
371 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$ ...
- 4,140
2 votes
0 answers
146 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$, ...
- 7,033
1 vote
0 answers
129 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, ...
- 215
4 votes
1 answer
580 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 ...
- 4,140
3 votes
1 answer
658 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 $$ where $\varphi(x)= \frac{1}{\sqrt{2\pi}}\exp{\left(-\frac{x^2}{2}\...
- 250
3 votes
0 answers
179 views
Is there any injective mapping from smooth functions on closed interval to smooth functions on circle? Motivated by signal processing
One advantage of Discrete Cosine Transform (DCT) over Discrete Fourier Transform (DFT) is that DCT maps any "continuous" signal defined on interval to a continuous one defined on circle. I ...
- 1,675
1 vote
1 answer
311 views
Continuous wavelet transform of a periodic function
I have a question regarding the Continuous Wavelet Transform (CWT) of non finite energy functions, such as $g(t) = a\exp(i\omega_0t)$. We know that the CWT is defined for functions in the Hilbert ...
1 vote
0 answers
38 views
Message Passing algorithm: misadjustement, study of convergence, for inexact MPA
I am looking for resources (articles or other information) on the derivation of mis-adjustments and on the study of convergence for the message passing algorithm (MPA) and/or the inexact message ...
- 39
3 votes
1 answer
839 views
van Cittert deconvolution method
In the early 1930s, van Cittert published a deconvolution method. Although his method was not perfect but it is the forefather of many improved spectral deconvolution methods. The basic idea is that ...
- 873
1 vote
1 answer
150 views
How many Fourier coefficients of a sparse signal $f=\sum_{n=1}^Nc_n\delta_{t_n}$ are needed to determine $f$ uniquely?
Let $N \in \mathbb N$ and $c_n \in \mathbb C$, $t_n \in \mathbb R$ for $n=1, \dots, N$. Suppose that $f$ is a linear combination of dirac-deltas with locations $t_n$ and coefficients $c_n$, i.e. $$ f=\...
- 123
3 votes
1 answer
3k 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 ...
- 873
6 votes
2 answers
369 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 ...
- 66
1 vote
0 answers
69 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. $$ \hat{x}(\omega)=\int_\mathbb{R} x(t)e^{-it\omega} dt~~~,~~~x(t)=\frac{1}{2\pi} \int_\...
- 4,140
2 votes
0 answers
133 views
Fourier Transform diagonalizes time-invariant convolution operators [closed]
I got the following paragraph from the book "A wavelet tour of signal processing" chapter one, page 2. The Fourier transform is everywhere in physics and mathematics because it diagonalizes ...
- 4,140
3 votes
1 answer
1k views
Relation between signal derivative and frequency spectrum
I want to sample a signal whose derivative I know to be bounded by physical constraints. The sampling is disturbed by gaussian noise, hence I need to filter the sample with a lowpass filter. Since I ...
- 33
3 votes
0 answers
220 views
Compressed sensing for partitioning instead of recovery
Let $x_0 \in \mathbb{R}^{m}$ be a signal whose support $T_0 = \{ t \mid x_{0}(t) \neq 0\}$ is assumed to be of small cardinality. The recovery of $x_0$ from a small number of $n \ll m$ linear ...
- 131
2 votes
0 answers
149 views
eigenvectors of a graph Laplacian VS Fourier basis
Could you please illustrate the following statement: the eigenvectors of a graph Laplacian behave similarly to a Fourier basis, motivating the development of graph-based Fourier analysis theory.
- 4,140
4 votes
0 answers
255 views
Convergence of the expectation of a random variable when conditioned on its sum with another, independent but not identically distributed
Suppose that for all $n \in \mathbf{N}$, $X_n$ and $Y_n$ are independent random variables with $$X_n \sim \mathtt{Binomial}(n,1-q),$$ and $$Y_n \sim \mathtt{Poisson}(n(q+\epsilon_n)),$$ where $q \in (...
- 91
0 votes
0 answers
147 views
Wigner distribution
The Wigner distribution of $u\in L^2(\mathbb R)$ is defined as a function $W(u)$ on $\mathbb R^2$ given by $$ W(u)(x,\xi)=\int_\mathbb R u\left(x+\tfrac z2\right) \overline{u\left(x-\tfrac z2\right)} ...
- 16.6k
1 vote
1 answer
282 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 ...
- 13
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, ...
- 873
24 votes
2 answers
4k 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 ...
- 873
1 vote
1 answer
116 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 ...
- 281
5 votes
3 answers
1k 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 ...
- 873
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 ...
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