Questions tagged [signal-analysis]
The signal-analysis tag has no usage guidance.
108
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How Does My Radio Work?
Bear with me for a moment while I invoke the real world; the main question at the end is purely mathematical.
I live in an area with $n$ AM radio stations and $m$ FM radio stations.
AM station ...
52
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1
answer
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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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2
answers
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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 ...
20
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Signal processing reference for pure mathematician
Before giving a more detailed question below, the basic one is: can anyone recommend a good signal-processing reference which would be maximally readable by a pure mathematician (who nevertheless ...
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What is a rigorous statement for "linear time-invariant systems can be represented as convolutions"?
In Signal Processing books, a fundamental theorem is that linear time invariant systems can be represented as a convolution with a distribution. Could you give a mathematically rigorous statement of ...
9
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2
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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 : ...
9
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3
answers
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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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3
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Spectra of 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,...
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How would you compute that "average" ?
I created a DJ-ing application that allows you to mix your MP3s with a real turntable.
So I generated an audio timecode to burn on a CD, left channel is the absolute position, right channel is a ...
8
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1
answer
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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 ...
7
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1
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246
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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 ...
6
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2
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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 ...
6
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1
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367
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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')\...
6
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1
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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(\...
6
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1
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408
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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, ...
6
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3
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Signal Analysis/Processing Textbook
Can anybody recommend me a decent Signal Analysis/Processing textbook. If possible one that deals a little with MATLAB. I have an little knowledge of Real Analysis and fourier transforms. Wavelets i ...
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3
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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 ...
5
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1
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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 $A'(z)$....
5
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1
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Are there interesting problems involving arbitrarily long time series of small matrices?
Are there well-known or interesting applied problems (especially of the real-time signal processing sort) where arbitrarily long time series of small (say $d \equiv \dim \le 30$ for a nominal bound, ...
5
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1
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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 ...
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Decomposing a 1-d signal into arbitary basis functions
Hi all,
The short-time fourier transform decomposes a signal window into a sin/cosine series.
How would one approximate a signal in the same way, but using a set of arbitrary basis functions instead ...
4
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1
answer
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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
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2
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345
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Convolutive noise removal
I have the time domain signal
$$
u_o(t) = u(t)e^{-t/\tau}\eta(t) + \sigma(t)
$$
where $\tau$ is known, $\eta$ is non-Gaussian noise, and $\sigma$ is Gaussian noise. The distribution of $\eta(t)$ is ...
4
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1
answer
363
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Most orthogonal lattice basis
Let $n \in \mathbf{N}$ be a natural number and $v_1,\cdots,v_n$ a set of basis vectors in $\mathbb{R}^n$. How does one find the matrix $g \in \mathbf{GL}_n(\mathbb{Z})$ orthogonalizing these best ...
4
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Rotationally-Invariant 2D Discrete Transforms
I'm interested in 2D discrete transforms (such as discrete wavelet transforms, Curvelets, Ridgelets, Beamlets etc.) that operate on a discrete unit disk and:
Are invariant to rotations only
Output a ...
4
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0
answers
227
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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 (...
4
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0
answers
213
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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
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0
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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 ...
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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$(...
3
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2
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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/...
3
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1
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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
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1
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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
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1
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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 ...
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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 ...
3
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1
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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 ...
3
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2
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Bandwidth approximation for a nonlinear problem
Can anyone please help me with this problem.
I must let you know from the beginning that it's not an easy one.
"Two functions are given: $u, y \in L^{2}(-\infty,\infty), y(t)=\frac{u(t)}{u(t)+b}$ ,
...
3
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2
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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}...
3
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1
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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 ...
3
votes
1
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Levelset of band limited function
In a practical application problem I encountered such a question: Given a subset of a N*N Cartesian grid, how to determine if it is a sublevel-set of a band-limited (discrete) function? Here band-...
3
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0
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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{...
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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 ...
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202
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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 ...
3
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Interpolating Wavelet Coefficients
Hi! I was instructed via reddit that this place would be the best place to post this question. Fingers cross you can help...
Ive been writing some code to get rid of noise "spikes" in a signal. I'm ...
2
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2
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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 ...
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1
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Complete formulas book for Communication System engineer
I'm looking for a formulas book.
I'm currently student in Communication Systems and we have several courses involving mainly complex analysis, fourier analysis, signal processing, information theory ...
2
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1
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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?
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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 ...
2
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2
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Periodicity of data
I have some real data (data packets in a router). When I plot it I can see there is a clear periodicity on the dataset (24hours+-).
But how can I discover the periodicity of the data without being by ...
2
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1
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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, ...
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1
answer
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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.
...