# 5.4 Continuous wavelet transform

 Page 1 / 1
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 $\psi (t)$ , often called the mother wavelet . The $a$ -scaled and $\tau$ -shifted basis elements is given by ${\psi }_{a,\tau }(t)=\frac{1}{\sqrt{\left|a\right|}}\psi (\frac{t-\tau }{a})$ where $(a\land \tau )\in \mathbb{R}$ $\int_{()} \,d t$ ψ t 0 ${C}_{\psi }=\int_{()} \,d \Omega$ ψ Ω 2 Ω The conditions above imply that $\psi (t)$ is bandpass and sufficiently smooth. Assuming that $(\psi (t))=1$ , the definition above ensures that $({\psi }_{a,\tau }(t))=1$ for all $a$ and $\tau$ . The CWT is then defined by the transform pair ${X}_{\mathrm{CWT}}(a, \tau )=\int_{()} \,d t$ x t ψ a , τ t $x(t)=\frac{1}{{C}_{\psi }}\int_{()} \,d a$ 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 $\psi (t)$ . 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: $\psi (t)=\frac{1}{\sqrt{2\pi }}e^{-(i{\Omega }_{0}t)}e^{-\left(\frac{t^{2}}{2}\right)}$ $\Psi (\Omega )=e^{-\left(\frac{(\Omega -{\Omega }_{0})^{2}}{2}\right)}$ where it is typical to select ${\Omega }_{0}=\pi \sqrt{\frac{2}{\lg 2}}$ . (See illustration .) While this wavelet does not exactly satisfy the conditions established earlier, since $\Psi (0)\approx 7E-7\neq 0$ , 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.

Preparation and Applications of Nanomaterial for Drug Delivery
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
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
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
hi
Loga
Got questions? Join the online conversation and get instant answers!