# 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

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
Do somebody tell me a best nano engineering book for beginners?
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
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