Questions tagged [signal-analysis]

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2
votes
0answers
32 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
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0answers
96 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 (...
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0answers
81 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
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1answer
145 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
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1answer
105 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, ...
17
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2answers
1k 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
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1answer
71 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
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3answers
363 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
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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 ...
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0answers
29 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
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0answers
42 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\...
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0answers
129 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
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0answers
81 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
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0answers
17 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
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1answer
176 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
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1answer
82 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}^...
7
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1answer
1k 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
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1answer
82 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
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0answers
85 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: $$ ...
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0answers
51 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
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1answer
102 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
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0answers
202 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 $\mathrm{Im}(\psi(\cdot))$ ...
3
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0answers
78 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
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2answers
308 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}...
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0answers
53 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
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0answers
576 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
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2answers
270 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 ...
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0answers
53 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
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1answer
105 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
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0answers
768 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
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0answers
102 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
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1answer
74 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
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1answer
193 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
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3answers
380 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
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2answers
670 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
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0answers
427 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 ...
5
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1answer
175 views

Discrete Wavelets

I am looking for research that has been done in Discrete wavelets. Let me be specific as Google doesn't give me what I want when I say "discrete wavelets". I don't want countable basis for $ L^2(\...
4
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0answers
2k views

Weighted Median Filtering

Let's begin with a little review of unweighted median filtering. Suppose I have a list of $N$ real-valued numbers, $x=x_1,...,x_N$. Let $m_i$ be the median of $K$ consecutive values: $m_i=$ median$(...
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0answers
103 views

In which sense Daubechies wavelets converge to the Shannon wavelet?

My question is about wavelets theory. Consider $\psi_n$ the Daubechies wavelet of order $n \geq 1$; that is, the Daubechies wavelet with $n$ vanishing moments. We also define the Shannon wavelet in ...
3
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1answer
110 views

Why is it important to know if a frame is a Parseval frame?

I understand that a Parseval frame is one in which both upper and lower frame bounds equal 1. What's the main advantage to having this be the case? Or, more specifically, if I'm constructing a frame ...
2
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1answer
407 views

What kind of role has Functional Analysis played in Signal Processing? [closed]

Does it serve mainly as a narration or is there any substantive consequence which might not be derived without tools of functional analysis?
1
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1answer
947 views

Correlation between two continuous-time stochastic processes

Consider two continous-time stochastic processes $\{A(t)\}_{t \ge 0}$ and $\{B(t)\}_{t \ge 0}$ with $A(t)=t$ and $B(t)=t$. Each process starts at $t=0$ and emits "ticks" at increasing time slots. For ...
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0answers
34 views

Linearizing a multifrequency signal

I have a component of a signal $$\sin (k\omega_1t + \ell\omega_2t)$$ with wavenumbers $k, \ell \in \mathbb{Z}$, frequencies $\omega_1, \omega_2 \in \mathbb{R^+}$ and time $t \in \mathbb{R^+}$. (This ...
1
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0answers
39 views

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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0answers
31 views

Approximate rank of the set formed by all delayed replicas of a bandlimited signals between 0 and T

Given a complex-valued signal with a certain delay $s(t-\tau)$ for which we sample $N$ instants $$ \mathbf{s(\tau)}=\left[s(0-\tau),\ldots,s\left(\frac{N-1}{f_s}-\tau\right)\right]^T $$ at Nyquist ...
0
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0answers
85 views

Wavelet transform stability to deformations

I've come across the following claim in a paper of Mallat: "High frequency instabilities [of a signal representation] to deformations can be avoided by grouping frequencies into dyadic packets in $\...
2
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0answers
44 views

Analogous filter to Kalman filter that maximized mode instead (as opposed to minimizing variance)

I may have a potential application where maximizing the mode (as opposed to typically minimizing the variance) would be useful for state estimates. The situation may arise from skewed distributions ...
1
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1answer
1k views

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 ...
5
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1answer
726 views

A palindromic polynomial and its derivative have the same number of zeros outside the unit circle. Reference?

I am trying to find the original reference for a lemma attributed to Cohn (as in Schur-Cohn method): Let $A(z)$ be a palindromic or skew-palindromic polynomial, and denote its derivative by $A'(z)$....
4
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1answer
4k views

Spatial and temporal covariance matrices

Suppose $(x_i(t))$ is a $n$-dimensional time-series, where $t$ is an integer between $1$ and $T$ (time is discrete) and $i$ an integer between $1$ and $n$, and I assume $n<T$. From this time-series,...