# Questions tagged [signal-analysis]

The signal-analysis tag has no usage guidance.

89
questions

**3**

votes

**1**answer

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

**0**answers

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**

votes

**0**answers

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

**1**answer

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**

vote

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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**

votes

**0**answers

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**

vote

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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**

votes

**0**answers

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

**0**answers

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**

votes

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**3**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**0**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**3**answers

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

**2**answers

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

**0**answers

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 ...