# 0.9 Calculation of the discrete wavelet transform  (Page 4/5)

 Page 4 / 5

If we assume the length of the sequence of the signal is $L$ and the length of the sequence of scaling filter coefficients $h\left(n\right)$ is $N$ , then the number of multiplications necessary to calculate each scaling functionand wavelet expansion coefficient at the next scale, $c\left({J}_{1}-1,k\right)$ and $d\left({J}_{1}-1,k\right)$ , from the samples of the signal, $f\left(Tk\right)\approx c\left({J}_{1},k\right)$ , is $LN$ . Because of the downsampling, only half are needed to calculate thecoefficients at the next lower scale, $c\left({J}_{2}-1,k\right)$ and $d\left({J}_{2}-1,k\right)$ , and repeats until there is only one coefficient at a scale of $j={J}_{0}$ . The total number of multiplications is, therefore,

$\text{Mult}\phantom{\rule{4.pt}{0ex}}=LN+LN/2+LN/4+\cdots +N$
$=LN\left(1+1/2+1/4+\cdots +1/L\right)=2NL-N$

which is linear in $L$ and in $N$ . The number of required additions is essentially the same.

If the length of the signal is very long, essentially infinity, the coarsest scale ${J}_{0}$ must be determined from the goals of the particular signal processing problem being addressed. For this case, the number ofmultiplications required per DWT coefficient or per input signal sample is

$\text{Mult/sample}=N\left(2-{2}^{-{J}_{0}}\right)$

Because of the relationship of the scaling function filter $h\left(n\right)$ and the wavelet filter ${h}_{1}\left(n\right)$ at each scale (they are quadrature mirror filters), operations can be shared between them through the use of alattice filter structure, which will almost halve the computational complexity. That is developed in Chapter: Filter Banks and Transmultiplexers and [link] .

## The periodic case

In many practical applications, the signal is finite in length (finite support) and can be processed as single “block," much as the FourierSeries or discrete Fourier transform (DFT) does. If the signal to be analyzed is finite in length such that

$f\left(t\right)=\left\{\begin{array}{cc}0\hfill & t<0\hfill \\ 0\hfill & t>P\hfill \\ f\left(t\right)\hfill & 0

we can construct a periodic signal $\stackrel{˜}{f}\left(t\right)$ by

$\stackrel{˜}{f}\left(t\right)=\sum _{n}f\left(t+Pn\right)$

and then consider its wavelet expansion or DWT. This creation of a meaningful periodic function can still be done, even if $f\left(t\right)$ does not have finite support, if its energy is concentrated and some overlap isallowed in [link] .

Periodic Property 1: If $\stackrel{˜}{f}\left(t\right)$ is periodic with integer period $P$ such that $\stackrel{˜}{f}\left(t\right)=\stackrel{˜}{f}\left(t+Pn\right)$ , then the scaling function and wavelet expansion coefficients (DWT terms) at scale $J$ are periodic with period ${2}^{J}P$ .

$\text{If}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\stackrel{˜}{f}\left(t\right)=\stackrel{˜}{f}\left(t+P\right)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{then}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\stackrel{˜}{d}}_{j}\left(k\right)={\stackrel{˜}{d}}_{j}\left(k+{2}^{j}P\right)$

This is easily seen from

${\stackrel{˜}{d}}_{j}\left(k\right)={\int }_{-\infty }^{\infty }\stackrel{˜}{f}\left(t\right)\phantom{\rule{0.166667em}{0ex}}\psi \left({2}^{j}t-k\right)\phantom{\rule{0.166667em}{0ex}}dt=\int \stackrel{˜}{f}\left(t+Pn\right)\phantom{\rule{0.166667em}{0ex}}\psi \left({2}^{j}t-k\right)\phantom{\rule{0.166667em}{0ex}}dt$

which, with a change of variables, becomes

$=\int \stackrel{˜}{f}\left(x\right)\phantom{\rule{0.166667em}{0ex}}\psi \left({2}^{j}\left(x-Pn\right)-k\right)\phantom{\rule{0.166667em}{0ex}}dx=\int \stackrel{˜}{f}\left(x\right)\phantom{\rule{0.166667em}{0ex}}\psi \left({2}^{j}x-\left({2}^{j}Pn+k\right)\right)\phantom{\rule{0.166667em}{0ex}}dx={\stackrel{˜}{d}}_{j}\left(k+{2}^{j}Pn\right)$

and the same is true for ${\stackrel{˜}{c}}_{j}\left(k\right)$ .

Periodic Property 2: The scaling function and wavelet expansion coefficients (DWT terms) can be calculated from the inner product of $\stackrel{˜}{f}\left(t\right)$ with $\phi \left(t\right)$ and $\psi \left(t\right)$ or, equivalently, from the inner product of $f\left(t\right)$ with the periodized $\stackrel{˜}{\phi }\left(t\right)$ and $\stackrel{˜}{\psi }\left(t\right)$ .

${\stackrel{˜}{c}}_{j}\left(k\right)=〈\stackrel{˜}{f},\left(t\right),,,\phi ,\left(t\right)〉=〈f,\left(t\right),,,\stackrel{˜}{\phi },\left(t\right)〉$

and

${\stackrel{˜}{d}}_{j}\left(k\right)=〈\stackrel{˜}{f},\left(t\right),,,\psi ,\left(t\right)〉=〈f,\left(t\right),,,\stackrel{˜}{\psi },\left(t\right)〉$

where $\stackrel{˜}{\phi }\left(t\right)={\sum }_{n}\phi \left(t+Pn\right)$ and $\stackrel{˜}{\psi }\left(t\right)={\sum }_{n}\psi \left(t+Pn\right)$ .

This is seen from

${\stackrel{˜}{d}}_{j}\left(k\right)={\int }_{-\infty }^{\infty }\stackrel{˜}{f}\left(t\right)\phantom{\rule{0.166667em}{0ex}}\psi \left({2}^{j}t-k\right)\phantom{\rule{0.166667em}{0ex}}dt=\sum _{n}{\int }_{0}^{P}f\left(t\right)\phantom{\rule{0.166667em}{0ex}}\psi \left({2}^{j}\left(t+Pn\right)-k\right)\phantom{\rule{0.166667em}{0ex}}dt={\int }_{0}^{P}f\left(t\right)\phantom{\rule{0.166667em}{0ex}}\sum _{n}\psi \left({2}^{j}\left(t+Pn\right)-k\right)\phantom{\rule{0.166667em}{0ex}}dt$
${\stackrel{˜}{d}}_{j}\left(k\right)={\int }_{0}^{P}f\left(t\right)\phantom{\rule{0.166667em}{0ex}}\stackrel{˜}{\psi }\left({2}^{j}t-k\right)\phantom{\rule{0.166667em}{0ex}}dt$

where $\stackrel{˜}{\psi }\left({2}^{j}t-k\right)={\sum }_{n}\psi \left({2}^{j}\left(t+Pn\right)-k\right)$ is the periodized scaled wavelet.

Periodic Property 3: If $\stackrel{˜}{f}\left(t\right)$ is periodic with period $P$ , then Mallat's algorithm for calculating the DWT coefficients in [link] becomes

${\stackrel{˜}{c}}_{j}\left(k\right)=\sum _{m}h\left(m-2k\right)\phantom{\rule{0.166667em}{0ex}}{\stackrel{˜}{c}}_{j+1}\left(m\right)$

#### Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
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
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
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
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

### Read also:

#### Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Wavelets and wavelet transforms. OpenStax CNX. Aug 06, 2015 Download for free at https://legacy.cnx.org/content/col11454/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Wavelets and wavelet transforms' conversation and receive update notifications?

 By Saylor Foundation By JavaChamp Team By Stephen Voron By Lakeima Roberts By OpenStax By Vanessa Soledad By Anonymous User By OpenStax By Jessica Collett By Mackenzie Wilcox