# 0.1 Signal dictionaries and representations

 Page 1 / 2
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.

For a wide variety of signal processing applications (including analysis, compression, noise removal, and so on) it is useful toconsider the representation of a signal in terms of some dictionary  [link] . In general, a dictionary $\phantom{\rule{2pt}{0ex}}\Psi \phantom{\rule{2pt}{0ex}}$ issimply a collection of elements drawn from the signal space whose linear combinations can be used to represent or approximatesignals.

Considering, for example, signals in ${\mathbb{R}}^{N}$ , we may collect and represent the elements of the dictionary $\Psi$ as an $N×Z$ matrix, which we also denote as $\Psi$ . From this dictionary, a signal $x\in {\mathbb{R}}^{N}$ can be constructed as a linear combination of the elements (columns) of $\Psi$ . We write

$x=\Psi \alpha$
for some $\alpha \in {\mathbb{R}}^{Z}$ . (For much of our notation in this section, we concentrate on signals in ${\mathbb{R}}^{N}$ , though the basic concepts translate to other vector spaces.)

Dictionaries appear in a variety of settings. The most common may be the basis, in which case $\Psi$ has exactly $N$ linearly independent columns, and each signal $x$ has a unique set of expansion coefficients $\alpha ={\Psi }^{-1}x$ . The orthonormal basis (where the columns are normalized and orthogonal) is also ofparticular interest, as the unique set of expansion coefficients $\alpha ={\Psi }^{-1}x={\Psi }^{{}^{T}}x$ can be obtained as the inner products of $x$ against the columns of $\Psi$ . That is, $\alpha \left(i\right)=〈x,,,{\psi }_{i}〉,i=1,2,\cdots ,N$ , which gives us the expansion

$x=\sum _{i=1}^{N}〈x,,,{\psi }_{i}〉{\psi }_{i}.$

We also have that ${∥x∥}_{2}^{2}={\sum }_{i=1}^{N}{〈x,,,{\psi }_{i}〉}^{2}$ .

Frames are another special type of dictionary  [link] . A dictionary $\Psi$ is a frame if there exist numbers $A$ and $B$ , $0 such that, for any signal $x$

$A{∥x∥}_{2}^{2}\le \sum _{z}{〈x,,,{\psi }_{z}〉}^{2}\le B{∥x∥}_{2}^{2}.$
The elements of a frame may be linearly dependent in general (see [link] ), and so there may exist many ways to express a particular signal among the dictionary elements.However, frames do have a useful analysis/synthesis duality: for any frame $\Psi$ there exists a dual frame $\stackrel{˜}{\Psi }$ such that
$x=\sum _{z}〈x,,,{\psi }_{z}〉{\stackrel{˜}{\psi }}_{z}=\sum _{z}〈x,,,{\stackrel{˜}{\psi }}_{z}〉{\psi }_{z}.$
In the case where the frame vectors are represented as columns of the $N$ x $Z$ matrix $\Psi$ , the matrix $\stackrel{˜}{\Psi }$ containing the dual frame elements is simply the transpose of the pseudoinverse of $\Psi$ . A frame is called tight if the frame bounds $A$ and $B$ are equal. Tight frames have the special properties of (i) being theirown dual frames (after a rescaling by $1/A$ ) and (ii) preserving norms, i.e., ${\sum }_{i=1}^{N}{〈x,,,{\psi }_{i}〉}^{2}=A{∥x∥}_{2}^{2}$ . The remainder of this section discusses several importantdictionaries.

## The canonical basis

The standard basis for representing a signal is the canonical (or “spike”) basis. In ${\mathbb{R}}^{N}$ , this corresponds to a dictionary $\Psi ={I}_{N}$ (the $N×N$ identity matrix). When expressed in the canonical basis, signals are often said tobe in the “time domain.”

## Fourier dictionaries

The frequency domain provides one alternative representation to the time domain. The Fourier series and discrete Fourier transformare obtained by letting $\Psi$ contain complex exponentials and allowing the expansion coefficients $\alpha$ to be complex as well. (Such a dictionary can be used to represent real or complexsignals.) A related “harmonic” transform to express signals in ${\mathbb{R}}^{N}$ is the discrete cosine transform (DCT), in which $\Psi$ contains real-valued, approximately sinusoidal functions and the coefficients $\alpha$ are real-valued as well.

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
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
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
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!