# Discrete wavelet transform: main concepts

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## Main concepts

The discrete wavelet transform (DWT) is a representation of a signal $x(t)\in {ℒ}_{2}$ using an orthonormal basis consisting of a countably-infinite set of wavelets . Denoting the wavelet basis as $\{{\psi }_{k,n}(t)\colon k\in \mathbb{Z}\land n\in \mathbb{Z}\}$ , the DWT transform pair is

$x(t)=\sum_{k=()}$ n d k , n ψ k , n t
${d}_{k,n}={\psi }_{k,n}(t)\dot x(t)=\int_{()} \,d t$ ψ k , n t x t
where $\{{d}_{k,n}\}$ are the wavelet coefficients. Note the relationship to Fourier series and to the sampling theorem: in both caseswe can perfectly describe a continuous-time signal $x(t)$ using a countably-infinite ( i.e. , discrete) set of coefficients. Specifically, Fourier seriesenabled us to describe periodic signals using Fourier coefficients $\{X(k)\colon k\in \mathbb{Z}\}$ , while the sampling theorem enabled us to describe bandlimited signals using signal samples $\{x(n)\colon n\in \mathbb{Z}\}$ . In both cases, signals within a limited class are represented using a coefficient set with a single countableindex. The DWT can describe any signal in ${ℒ}_{2}$ using a coefficient set parameterized by two countable indices: $\{{d}_{k,n}\colon k\in \mathbb{Z}\land n\in \mathbb{Z}\}$ .

Wavelets are orthonormal functions in ${ℒ}_{2}$ obtained by shifting and stretching a mother wavelet , $\psi (t)\in {ℒ}_{2}$ . For example,

$\forall k, n, (k\land n)\in \mathbb{Z}\colon {\psi }_{k,n}(t)=2^{-\left(\frac{k}{2}\right)}\psi (2^{-k}t-n)$
defines a family of wavelets $\{{\psi }_{k,n}(t)\colon k\in \mathbb{Z}\land n\in \mathbb{Z}\}$ related by power-of-two stretches. As $k$ increases, the wavelet stretches by a factor of two; as $n$ increases, the wavelet shifts right.
When $(\psi (t))=1$ , the normalization ensures that $({\psi }_{k,n}(t))=1$ for all $k\in \mathbb{Z}$ , $n\in \mathbb{Z}$ .
Power-of-two stretching is a convenient, though somewhat arbitrary, choice. In our treatment of the discrete wavelettransform, however, we will focus on this choice. Even with power-of two stretches, there are various possibilities for $\psi (t)$ , each giving a different flavor of DWT.

Wavelets are constructed so that $\{{\psi }_{k,n}(t)\colon n\in \mathbb{Z}\}$ ( i.e. , the set of all shifted wavelets at fixed scale $k$ ), describes a particular level of 'detail' in the signal. As $k$ becomes smaller ( i.e. , closer to $()$ ), the wavelets become more "fine grained" and the level of detail increases. In this way, the DWT can give a multi-resolution description of a signal, very useful in analyzing "real-world" signals. Essentially, theDWT gives us a discrete multi-resolution description of a continuous-time signal in ${ℒ}_{2}$ .

In the modules that follow, these DWT concepts will be developed "from scratch" using Hilbert space principles. Toaid the development, we make use of the so-called scaling function $\phi (t)\in {ℒ}_{2}$ , which will be used to approximate the signal up to a particular level of detail . Like with wavelets, a family of scaling functions can beconstructed via shifts and power-of-two stretches

$\forall k, n, (k\land n)\in \mathbb{Z}\colon {\phi }_{k,n}(t)=2^{-\left(\frac{k}{2}\right)}\phi (2^{-k}t-n)$
given mother scaling function $\phi (t)$ . The relationships between wavelets and scaling functions will be elaborated upon later via theory and example .
The inner-product expression for ${d}_{k,n}$ , is written for the general complex-valued case. In our treatment of the discrete wavelet transform,however, we will assume real-valued signals and wavelets. For this reason, we omit the complex conjugations in theremainder of our DWT discussions

where we get a research paper on Nano chemistry....?
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
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
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
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
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