<< Chapter < Page Chapter >> Page >
Introduction to the filterbanks interpretation of the DWT.

Assume that we start with a signal x t 2 . Denote the best approximation at the 0 th level of coarseness by x 0 t . (Recall that x 0 t is the orthogonal projection of x t onto V 0 .) Our goal, for the moment, is to decompose x 0 t into scaling coefficients and wavelet coefficients at higher levels. Since x 0 t V 0 and V 0 V 1 W 1 , there exist coefficients c 0 n , c 1 n , and d 1 n such that

x 0 t n n c 0 n 0 n t n n c 1 n 1 n t n n d 1 n 1 n t
Using the fact that 1 n t n is an orthonormal basis for V 1 , in conjunction with the scaling equation,
c 1 n x 0 t 1 n t m m c 0 m 0 m t 1 n t m m c 0 m 0 m t 1 n t m m c 0 m t m h t 2 n m m c 0 m h t m t 2 n m m c 0 m h m 2 n
where t 2 n t m t 2 n . The previous expression ( ) indicates that c 1 n results from convolving c 0 m with a time-reversed version of h m then downsampling by factor two ( ).

Using the fact that 1 n t n is an orthonormal basis for W 1 , in conjunction with the wavelet scaling equation,

d 1 n x 0 t 1 n t m m c 0 m 0 m t 1 n t m m c 0 m 0 m t 1 n t m m c 0 m t m g t 2 n m m c 0 m g t m t 2 n m m c 0 m g m 2 n
where t 2 n t m t 2 n .

The previous expression ( ) indicates that d 1 n results from convolving c 0 m with a time-reversed version of g m then downsampling by factor two ( ).

Putting these two operations together, we arrive at what looks like the analysis portion of an FIR filterbank ( ):

We can repeat this process at the next higher level. Since V 1 W 2 V 2 , there exist coefficients c 2 n and d 2 n such that

x 1 t n n c 1 n 1 n t n n d 2 n 2 n t n n c 2 n 2 n t
Using the same steps as before we find that
c 2 n m m c 1 m h m 2 n
d 2 n m m c 1 m g m 2 n
which gives a cascaded analysis filterbank ( ):

If we use V 0 W 1 W 2 W 3 W k V k to repeat this process up to the k th level, we get the iterated analysis filterbank ( ).

As we might expect, signal reconstruction can be accomplished using cascaded two-channel synthesis filterbanks. Using thesame assumptions as before, we have:

c 0 m x 0 t 0 m t n n c 1 n 1 n t n n d 1 n 1 n t 0 m t n n c 1 n 1 n t 0 m t n n d 1 n 1 n t 0 m t n n c 1 n h m 2 n n n d 1 n g m 2 n
where h m 2 n 1 n t 0 m t and g m 2 n 1 n t 0 m t which can be implemented using the block diagram in .

The same procedure can be used to derive

c 1 m n n c 2 n h m 2 n n n d 2 n g m 2 n
from which we get the diagram in .

To reconstruct from the k th level, we can use the iterated synthesis filterbank ( ).

The table makes a direct comparison between wavelets and the two-channelorthogonal PR-FIR filterbanks.

Discrete Wavelet Transform 2-Channel Orthogonal PR-FIR Filterbank
Analysis-LPF H z -1 H 0 z
Power Symmetry H z H z -1 H z H z -1 2 H 0 z H 0 z -1 H 0 z H 0 z -1 1
Analysis HPF G z -1 H 1 z
Spectral Reverse P P is odd G z z P H z -1 N N is even H 1 z z N 1 H 0 z -1
Synthesis LPF H z G 0 z 2 z N 1 H 0 z -1
Synthesis HPF G z G 1 z 2 z N 1 H 1 z -1

From the table, we see that the discrete wavelet transform that we have been developing is identical to two-channel orthogonalPR-FIR filterbanks in all but a couple details.

  • Orthogonal PR-FIR filterbanks employ synthesis filters with twice the gain of the analysis filters, whereas in the DWTthe gains are equal.
  • Orthogonal PR-FIR filterbanks employ causal filters of length N , whereas the DWT filters are not constrained to be causal.
For convenience, however, the wavelet filters H z and G z are usually chosen to be causal. For both to have even impulse response length N , we require that P N 1 .

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

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Fundamentals of signal processing. OpenStax CNX. Nov 26, 2012 Download for free at http://cnx.org/content/col10360/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Fundamentals of signal processing' conversation and receive update notifications?

Ask