# 0.10 Wavelet-based signal processing and applications  (Page 4/13)

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The computational complexity of the DWT algorithm can also be easily established. Let ${C}_{DWT}\left(N\right)$ be the complexity for a length-N DWT. Since after each scale, we only further operate on half of the output data, wecan show

${C}_{DWT}\left(N\right)=O\left(N\right)+{C}_{DWT}\left(N/2\right),$

which gives rise to the solution

${C}_{DWT}\left(N\right)=O\left(N\right).$

The operation in [link] can also be expressed in matrix form ${\mathbf{W}}_{N};$ e.g., for Haar wavelet,

${\mathbf{W}}_{4}^{Haar}=\sqrt{2}/2\left[\begin{array}{cccc}\hfill 1& \hfill -1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1& \hfill -1\\ \hfill 1& \hfill 1& \hfill 0& \hfill 0\\ \hfill 0& \hfill 0& \hfill 1& \hfill 1\end{array}\right].$

The orthogonality conditions on $\mathbf{h}$ and $\mathbf{g}$ ensure ${\mathbf{W}}_{N}^{\text{'}}{\mathbf{W}}_{N}={\mathbf{I}}_{N}$ . The matrix for multiscale DWT is formed by ${\mathbf{W}}_{N}$ for different $N$ ; e.g., for three scale DWT,

$\left[\begin{array}{cc}\left[\begin{array}{cc}{\mathbf{W}}_{N/4}& \\ & {\mathbf{I}}_{N/4}\end{array}\right]& \\ & {\mathbf{I}}_{N/2}\end{array}\right]\left[\begin{array}{cc}{\mathbf{W}}_{N/2}& \\ & {\mathbf{I}}_{N/2}\end{array}\right]{\mathbf{W}}_{N}.$

We could further iterate the building block on some of the highpass outputs. This generalization is called the wavelet packets [link] .

## The algorithm development

The key to the fast Fourier transform is the factorization of ${\mathbf{F}}_{N}$ into several sparse matrices, and one of the sparse matrices represents two DFTs of half the length. In a manner similar to the DIT FFT, the following matrixfactorization can be made:

${\mathbf{F}}_{N}={\mathbf{F}}_{N}{\mathbf{W}}_{N}^{T}{\mathbf{W}}_{N}=\left[\begin{array}{cc}{\mathbf{A}}_{N/2}& {\mathbf{B}}_{N/2}\\ {\mathbf{C}}_{N/2}& {\mathbf{D}}_{N/2}\end{array}\right]\left[\begin{array}{cc}{\mathbf{F}}_{N/2}& 0\\ 0& {\mathbf{F}}_{N/2}\end{array}\right]{\mathbf{W}}_{N},$

where ${\mathbf{A}}_{N/2},{\mathbf{B}}_{N/2},{\mathbf{C}}_{N/2}$ , and ${\mathbf{D}}_{N/2}$ are all diagonal matrices. The values on the diagonal of ${\mathbf{A}}_{N/2}$ and ${\mathbf{C}}_{N/2}$ are the length-N DFT ( i.e., frequency response ) of $\mathbf{h}$ , and the values on the diagonal of ${\mathbf{B}}_{N/2}$ and ${\mathbf{D}}_{N/2}$ are the length-N DFT of $\mathbf{g}$ . We can visualize the above factorization as

where we image the real part of DFT matrices, and the magnitude of the matrices for butterfly operations and the one-scale DWT using length-16Daubechies' wavelets [link] , [link] . Clearly we can see that the new twiddle factors have non-unit magnitudes.

The above factorization suggests a DWT-based FFT algorithm. The block diagram of the last stage of a length-8 algorithm is shown in [link] . This scheme is iteratively applied to shorter length DFTs to get the full DWT based FFTalgorithm. The final system is equivalent to a full binary tree wavelet packet transform [link] followed by classical FFT butterfly operations, where the new twiddle factors are the frequency response of thewavelet filters.

The detail of the butterfly operation is shown in [link] , where $i\in \left\{0,1,...,$ $N/2\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}1\right\}$ . Now the twiddle factors are length-N DFT of $\mathbf{h}$ and $\mathbf{g}$ . For well defined wavelet filters, they have well known properties; e.g., forDaubechies' family of wavelets, their frequency responses are monotone, and nearly half of which have magnitude close to zero. This fact can beexploited to achieve speed vs. accuracy tradeoff. The classical radix-2 DIT FFT is a special case of the above algorithm when $\mathbf{h}=\left[1,\phantom{\rule{0.277778em}{0ex}}0\right]$ and $\mathbf{g}=\left[0,\phantom{\rule{0.277778em}{0ex}}1\right]$ . Although they do not satisfy some of the conditions required for wavelets, they do constitute a legitimate(and trivial) orthogonal filter bank and are often called the lazy wavelets in the context of lifting.

## Computational complexity

For the DWT-based FFT algorithm, the computational complexity is on the same order of the FFT — $O\left(N{log}_{2}N\right)$ , since the recursive relation in [link] is again satisfied. However, the constant appearing before $N{log}_{2}N$ depends on the wavelet filters used.

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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
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da
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Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
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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
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Professor
I think
Professor
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Alexandre
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Rafiq
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Damian
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LITNING
scanning tunneling microscope
Sahil
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Santosh
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Rafiq
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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 .
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Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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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?
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Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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