# 4.14 Filterbanks interpretation of the discrete wavelet transform

 Page 1 / 1
Introduction to the filterbanks interpretation of the DWT.

Assume that we start with a signal $x(t)\in {}_{2}$ . Denote the best approximation at the ${0}^{\mathrm{th}}$ 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 ${x}_{0}(t)\in {V}_{0}$ and ${V}_{0}={V}_{1}\mathop{\mathrm{xor}}{W}_{1}$ , there exist coefficients $\{{c}_{0}(n)\}$ , $\{{c}_{1}(n)\}$ , and $\{{d}_{1}(n)\}$ such that

${x}_{0}(t)=\sum {c}_{0}(n){}_{0n}(t)=\sum {c}_{1}(n){}_{1n}(t)+\sum {d}_{1}(n){}_{1n}(t)$
Using the fact that $\{{}_{1n}(t)\colon n\in \mathbb{Z}\}$ is an orthonormal basis for ${V}_{1}$ , in conjunction with the scaling equation,
${c}_{1}(n)={x}_{0}(t)\dot {}_{1n}(t)=\sum {c}_{0}(m){}_{0m}(t)\dot {}_{1n}(t)=\sum {c}_{0}(m)({}_{0m}(t)\dot {}_{1n}(t))=\sum {c}_{0}(m)((t-m)\dot \sum h()(t--2n))=\sum {c}_{0}(m)\sum h()((t-m)\dot (t--2n))=\sum {c}_{0}(m)h(m-2n)$
where $(t--2n)=(t-m)\dot (t--2n)$ . 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 $\{{}_{1n}(t)\colon n\in \mathbb{Z}\}$ is an orthonormal basis for ${W}_{1}$ , in conjunction with the wavelet scaling equation,

${d}_{1}(n)={x}_{0}(t)\dot {}_{1n}(t)=\sum {c}_{0}(m){}_{0m}(t)\dot {}_{1n}(t)=\sum {c}_{0}(m)({}_{0m}(t)\dot {}_{1n}(t))=\sum {c}_{0}(m)((t-m)\dot \sum g()(t--2n))=\sum {c}_{0}(m)\sum g()((t-m)\dot (t--2n))=\sum {c}_{0}(m)g(m-2n)$
where $(t--2n)=(t-m)\dot (t--2n)$ .

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 ${V}_{1}={W}_{2}\mathop{\mathrm{xor}}{V}_{2}$ , there exist coefficients $\{{c}_{2}(n)\}$ and $\{{d}_{2}(n)\}$ such that

${x}_{1}(t)=\sum {c}_{1}(n){}_{1n}(t)=\sum {d}_{2}(n){}_{2n}(t)+\sum {c}_{2}(n){}_{2n}(t)$
Using the same steps as before we find that
${c}_{2}(n)=\sum {c}_{1}(m)h(m-2n)$
${d}_{2}(n)=\sum {c}_{1}(m)g(m-2n)$
which gives a cascaded analysis filterbank ( ):

If we use ${V}_{0}={W}_{1}\mathop{\mathrm{xor}}{W}_{2}\mathop{\mathrm{xor}}{W}_{3}\mathop{\mathrm{xor}}\mathop{\mathrm{xor}}{W}_{k}\mathop{\mathrm{xor}}{V}_{k}$ to repeat this process up to the ${k}^{\mathrm{th}}$ 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:

${c}_{0}(m)={x}_{0}(t)\dot {}_{0m}(t)=\sum {c}_{1}(n){}_{1n}(t)+\sum {d}_{1}(n){}_{1n}(t)\dot {}_{0m}(t)=\sum {c}_{1}(n)({}_{1n}(t)\dot {}_{0m}(t))+\sum {d}_{1}(n)({}_{1n}(t)\dot {}_{0m}(t))=\sum {c}_{1}(n)h(m-2n)+\sum {d}_{1}(n)g(m-2n)$
$\text{where}h(m-2n)={}_{1n}(t)\dot {}_{0m}(t)$ $\text{and}g(m-2n)={}_{1n}(t)\dot {}_{0m}(t)$ which can be implemented using the block diagram in .

The same procedure can be used to derive

${c}_{1}(m)=\sum {c}_{2}(n)h(m-2n)+\sum {d}_{2}(n)g(m-2n)$
from which we get the diagram in .

To reconstruct from the ${k}^{\mathrm{th}}$ 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 $\forall P, \text{P is odd}\colon G(z)=(z^{-P}H(-z^{-1}))$ $\forall N, \text{N is even}\colon {H}_{1}(z)=(z^{-(N-1)}{H}_{0}(-z^{-1}))$
Synthesis LPF $H(z)$ ${G}_{0}(z)=2z^{-(N-1)}{H}_{0}(z^{-1})$
Synthesis HPF $G(z)$ ${G}_{1}(z)=2z^{-(N-1)}{H}_{1}(z^{-1})$

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

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
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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
Daniel
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
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
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
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
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.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
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, 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?   By By Eric Crawford By By By Rhodes   By