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This module introduces continuous wavelet transform.

The STFT provided a means of (joint) time-frequency analysis with the property that spectral/temporal widths (or resolutions)were the same for all basis elements. Let's now take a closer look at the implications of uniform resolution.

Consider two signals composed of sinusoids with frequency 1 Hz and 1.001 Hz, respectively. It may be difficult to distinguishbetween these two signals in the presence of background noise unless many cycles are observed, implying the need for amany-second observation. Now consider two signals with pure frequencies of 1000 Hz and 1001 Hz-again, a 0.1%difference. Here it should be possible to distinguish the two signals in an interval of much less than one second. In otherwords, good frequency resolution requires longer observation times as frequency decreases. Thus, it might be more convenientto construct a basis whose elements have larger temporal width at low frequencies.

The previous example motivates a multi-resolution time-frequency tiling of the form ( [link] ):

The Continuous Wavelet Transform (CWT) accomplishes the above multi-resolution tiling by time-scaling and time-shifting aprototype function ψ t , often called the mother wavelet . The a -scaled and τ -shifted basis elements is given by ψ a , τ t 1 a ψ t τ a where a τ t ψ t 0 C ψ Ω ψ Ω 2 Ω The conditions above imply that ψ t is bandpass and sufficiently smooth. Assuming that ψ t 1 , the definition above ensures that ψ a , τ t 1 for all a and τ . The CWT is then defined by the transform pair X CWT a τ t x t ψ a , τ t x t 1 C ψ a τ X CWT a τ ψ a , τ t a 2 In basis terms, the CWT says that a waveform can be decomposed into a collection of shifted and stretched versions of themother wavelet ψ t . As such, it is usually said that wavelets perform a "time-scale" analysis rather than a time-frequency analysis.

The Morlet wavelet is a classic example of the CWT. It employs a windowed complex exponential as the motherwavelet: ψ t 1 2 Ω 0 t t 2 2 Ψ Ω Ω Ω 0 2 2 where it is typical to select Ω 0 2 2 . (See illustration .) While this wavelet does not exactly satisfy the conditions established earlier, since Ψ 0 7 -7 0 , it can be corrected, though in practice the correction is negligible and usually ignored.

While the CWT discussed above is an interesting theoretical and pedagogical tool, the discrete wavelet transform (DWT) is muchmore practical. Before shifting our focus to the DWT, we take a step back and review some of the basic concepts from the branchof mathematics known as Hilbert Space theory ( Vector Space , Normed Vector Space , Inner Product Space , Hilbert Space , Projection Theorem ). These concepts will be essential in our development of the DWT.

Questions & Answers

how can chip be made from sand
Eke Reply
are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
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
Google
da
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Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
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
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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
Smarajit Reply
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Source:  OpenStax, Digital signal processing (ohio state ee700). OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10144/1.8
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