Questions tagged [signal-analysis]

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3
votes
1answer
73 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}\...
3
votes
0answers
98 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 ...
0
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0answers
27 views

signal large wiggles filter

I am trying to filter out large wiggles showing in a signal. Those wiggles are happening due to many signal processing layers and a final sine Fourier transform (no phase) Here's the python code i am ...
0
votes
1answer
69 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
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0answers
13 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 ...
3
votes
1answer
97 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 ...
1
vote
1answer
95 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=\...
2
votes
1answer
486 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 ...
6
votes
2answers
284 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 ...
0
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0answers
17 views

Estimate of the value of the function parameter at which its minimum/maximum is reached

Suppose we have a Gaussian function of the following form: $f(x) = e^{-(x-x_*)^2}$ where $x$ - parameter of the function, $x_*$ - the value at which its extremum is reached (in this case, the maximum) ...
1
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0answers
55 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_\...
2
votes
0answers
55 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 ...
3
votes
1answer
171 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 ...
3
votes
0answers
191 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 ...
2
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0answers
58 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
votes
0answers
108 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 (...
0
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0answers
86 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)} ...
1
vote
1answer
170 views

The derivative of a filter with respect to a output singal [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
votes
1answer
763 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, ...
20
votes
2answers
2k 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 ...
1
vote
1answer
73 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 ...
5
votes
3answers
439 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 ...
52
votes
1answer
4k 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 ...
1
vote
0answers
32 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 ...
1
vote
0answers
45 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\...
1
vote
0answers
134 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 ...
1
vote
0answers
83 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 ...
2
votes
0answers
18 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 ...
3
votes
1answer
184 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 ...
1
vote
1answer
96 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}^...
8
votes
1answer
2k views

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 ...
2
votes
1answer
89 views

Are the Prolate Spheroidal Wave Functions absolutely integrable?

I would like to know if the Prolate Spheroidal Wavefunctions (PSWFs, defined below) are in $L^1(\mathbb{R})$. I know that they are square integrable, but cannot decide about absolute integrability. ...
2
votes
0answers
89 views

Is it possible to find an atlas for the set: $\{F:FE = I, E \text{ is a frame for } \mathbb{R}^n\}$

Let $E$ be the matrix whos rows are $ \{e_i^{\top}\}_{i=1}^m$. Let $E$ also be a frame of $m$ elements for $\mathbb{R}^n$, $m \geq n$. This means there exist two constants $A, B > 0$ such that: $$ ...
1
vote
0answers
66 views

Discrete Wavelet Transform and Gaussian decay

I have a question regarding the possibility of constructing a Discrete Wavelet Transform based on a scaling function having Gaussian decay (and no more decay than that). More specifically, I am ...
2
votes
1answer
107 views

Determinant of a matrix involving the Prolate Spheroidal Wave Functions

The Prolate Spheroidal Wave Functions are eigenfunctions of the following integral equation: $$\int_{-T}^T\varphi_n(x) \text{sinc}(t-x) dx = \lambda_n \varphi_n(t)$$ where $\text{sinc}(t) = \sin(\pi ...
4
votes
0answers
203 views

Can the wavelet bispectrum be normalised so that its integral "gives the right answer"?

Fix a rapidly decreasing function $\psi \in \mathcal{S}(\mathbb{R})$ with the properties that $\int_\mathbb{R} \psi = 0$, $\mathrm{Re}(\psi(\cdot))$ is an even function, and $\mathrm{Im}(\psi(\cdot))$ ...
4
votes
0answers
100 views

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 ...
3
votes
2answers
313 views

How far can the domain of definition of multiplier operators be extended?

Given any $g \in L^\infty(\mathbb{R})$, we define the associated multiplier operator $T_g \colon L^2(\mathbb{R}) \to L^2(\mathbb{R})$ by $$ \mathcal{F}(T_g f) \ = \ g.\mathcal{F}f $$ where $\mathcal{F}...
1
vote
0answers
58 views

Problem with state dimensions in IMM algorithm

I'm working on tracking algorithm for radar system. I have 3 motion models with state vector: $x_1=[x,y,v_x,v_y]$, $x_2=[x,y,v_x,v_y,a_x,a_y]$, $x_3=[x,y,v_{abs},\phi,\omega]$. The first two models ...
1
vote
0answers
624 views

L0 norm compressed sensing vs L1 norm compressed sensing

Suppose we have an very efficient way to perform L0 norm compressed vs L1 norm compressed sensing. Specifically: L0 norm compressed sensing is: $$\eqalign{ & \min \quad {x^T}Qx + {b^T}x + \mu {\...
2
votes
2answers
273 views

On a number theoretic problem coming from multiuser coding?

Can Chinese remainder theorem be used to solve this problem in multiuser coding? We have two transmitters sending integers $q,q'>0$ to a common receiver. The duty of the receiver is to recover ...
1
vote
0answers
54 views

Cramer Rao bound for relative estimation

I have an observed vector ${\bf y}$ from which I would like to estimate a parameter vector ${\bf c}$ (denote the estimate $\hat{{\bf c}}$). A feature of our estimation problem is that the involved ...
0
votes
1answer
121 views

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 ...
2
votes
0answers
886 views

Distinguishing signals with same frequency but different phase

Fourier decomposition of a mixed signal can straightforwardly give me the frequencies of the different components and their relative amplitudes, but how can I extract the components of a mixed signal ...
2
votes
0answers
104 views

Wiener-Ikehara Theorem and Signal Processing

I am trying to understand the Wiener-Ikehara Tauberian theorem which can be a step to understanding the prime number theorem. Let $$ \hat{a}(s) = \int_0^\infty e^{-us}\, da(u) $$ with $a(u)$ some ...
1
vote
1answer
79 views

Finite Parseval Frame

Assume that $G$ is a finite vector space over a finite field with order $|G|$. (For example, $G=Z_p^k$). Assume that $\{f_n\}_n$ is a Parseval frame for $l^2(G)$. Can we say that the sequence $\{f_n\}...
5
votes
1answer
223 views

Boundary behavior of harmonic function on the square

Is there a constant $C$ such that if $u:[0,1]^2\to \mathbb{R}$ is harmonic with $u\in L^\infty(\partial [0,1]^2)$ (if you prefer you can also assume $\|u\|_\infty = 1$ on the boundary and $u$ smooth ...
9
votes
3answers
451 views

Finite realization of irrational transfer functions

In the field of digital signal processing, linear time-invariant systems play a distinguished role. These are the systems for which there exists an impulse response, a function $h:\mathbb{Z}\to\...
9
votes
2answers
744 views

When is a mapping the proximity operator of some convex function?

Is there a characterization of mappings $p : \mathbb R^n \rightarrow \mathbb R^n$ which are proximity operators (in the sense of Moreau) of l.s.c (extended) real-valued functions ? That is, given $p : ...
2
votes
0answers
447 views

Is there a Bayesian theory of deterministic signal? Prequel and motivation for my previous question

This is a prequel to my question: What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory) Clearly my ...