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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 we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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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