Introduction to concise signal models

 Page 1 / 1
This collection reviews fundamental concepts underlying the use of concise models for signal processing. Topics are presented from a geometric perspective and include low-dimensional linear, sparse, and manifold-based signal models, approximation, compression, dimensionality reduction, and Compressed Sensing.

Overview

In characterizing a given problem in signal processing, one is often able to specify a model for the signals to be processed. This model may distinguish (either statistically or deterministically)classes of interesting signals from uninteresting ones, typical signals from anomalies, information from noise, etc.

Very commonly, models in signal processing deal with some notion of structure, constraint, or conciseness. Roughly speaking, one often believes that a signal has “few degrees of freedom”relative to the size of the signal. This notion of conciseness is a very powerful assumption, and it suggests the potential for dramatic gains via algorithms that capture and exploit the true underlyingstructure of the signal.

In these modules, we survey three common examples of concise models: linear models, sparse nonlinear models, and manifold-based models. In each case, we discuss an important phenomenon:the conciseness of the model corresponds to a low-dimensional geometric structure along which the signals of interest tend to cluster. This low-dimensional geometry again has important implicationsin the understanding and the development of efficient algorithms for signal processing.

We discuss this low-dimensional geometry in several contexts, including projecting a signal onto the model class (i.e., forming a concise approximation to a signal), encoding such an approximation(i.e., data compression), and reducing the dimensionality of signals and data sets. We conclude with an important and emerging application area known as Compressed Sensing (CS), which is a novel methodfor data acquisition that relies on concise models and builds upon strong geometric principles. We discuss CS in its traditional, sparsity-based context and also discuss extensions of CS to otherconcise models such as manifolds.

Signal notation

We will treat signals as real- or complex-valued functions having domains that are either discrete (and finite) or continuous (andeither compact or infinite). Each of these assumptions will be made clear as needed. As a generalrule, however, we will use $x$ to denote a discrete signal in ${\mathbb{R}}^{N}$ and $f$ to denote a function over a continuousdomain $\mathcal{D}$ . We also commonly refer to these as discrete- or continuous- time signals, though the domain need not actually be temporal in nature.

Lp and lp norms

As measures for signal energy, fidelity, or sparsity, we will employ the ${L}_{p}$ and ${\ell }_{p}$ norms. For continuous-time functions, the ${L}_{p}$ norm is defined as

$\begin{array}{ccc}\hfill {\parallel f\parallel }_{{L}_{p}\left(\mathcal{D}\right)}& =& {\left({\int }_{\mathcal{D}},{|f|}^{p}\right)}^{1/p},\phantom{\rule{1.em}{0ex}}p\in \left(0,\infty \right),\hfill \end{array}$
and for discrete-time functions, the ${\ell }_{p}$ norm is defined as
${\parallel x\parallel }_{{\ell }_{p}}=\left\{\begin{array}{cc}\left(\sum _{i=1}^{N}{|x\left(i\right)|}^{p}{\right)}^{1/p},\hfill & p\in \left(0,\infty \right),\hfill \\ \underset{i=1,\cdots ,N}{max}|x\left(i\right)|,\hfill & p=\infty ,\hfill \\ \sum _{i=1}^{N}{\mathbf{1}}_{x\left(i\right)\ne 0},\hfill & p=0,\hfill \end{array}\right)$
where $\mathbf{1}$ denotes the indicator function. (While we often refer to these measures as “norms,” they actually do not meetthe technical criteria for norms when $p<1$ .)

Linear algebra

Let $A$ be a real-valued $M×N$ matrix. We denote the nullspace of $A$ as $\mathcal{N}\left(A\right)$ (note that $\mathcal{N}\left(A\right)$ is a linear subspace of ${\mathbb{R}}^{N}$ ), and we denote the transpose of $A$ as $A{}^{T}$ .

We call $A$ an orthoprojector from ${\mathbb{R}}^{N}$ to ${\mathbb{R}}^{M}$ if it has orthonormal rows. From such a matrix we call ${A}^{{}^{T}}A$ the corresponding orthogonal projection operator onto the $M$ -dimensional subspace of ${\mathbb{R}}^{N}$ spanned by the rows of $A$ .

how can chip be made from sand
is this allso about nanoscale material
Almas
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!