# 0.5 The scaling function and scaling coefficients, wavelet and  (Page 12/13)

 Page 12 / 13

Because of the iterative form of this algorithm, applying the same process over and over, it is sometimes called the cascade algorithm [link] , [link] .

## Iterating the filter bank

An interesting method for calculating the scaling function also uses an iterative procedure which consists of the stages of the filterstructure of Chapter: Filter Banks and the Discrete Wavelet Transform which calculates wavelet expansions coefficients (DWT values) at one scale from those at another. A scalingfunction, wavelet expansion of a scaling function itself would be a single nonzero coefficient at the scale of $j=1$ . Passing this single coefficient through the synthesis filter structure of Figure: Two-Stage Two-Band Synthesis Tree and [link] would result in a fine scale output that for large $j$ would essentially be samples of the scaling function.

## Successive approximation in the frequency domain

The Fourier transform of the scaling function defined in [link] is an important tool for studying and developing wavelet theory. It could beapproximately calculated by taking the DFT of the samples of $\phi \left(t\right)$ but a more direct approach is available using the infinite product in [link] . From this formulation we can see how the zeros of $H\left(\omega \right)$ determine the zeros of $\text{Φ}\left(\omega \right)$ . The existence conditions in Theorem 5 require $H\left(\pi \right)=0$ or, more generally, $H\left(\omega \right)=0$ for $\omega =\left(2k+1\right)\pi$ . Equation [link] gives the relation of these zeros of $H\left(\omega \right)$ to the zeros of $\text{Φ}\left(\omega \right)$ . For the index $k=1$ , $H\left(\omega /2\right)=0$ at $\omega =2\left(2k+1\right)\pi$ . For $k=2$ , $H\left(\omega /4\right)=0$ at $\omega =4\left(2k+1\right)\pi$ , $H\left(\omega /8\right)=0$

at $\omega =8\left(2k+1\right)\pi$ , etc. Because [link] is a product of stretched versions of $H\left(\omega \right)$ , these zeros of $H\left(\omega /{2}^{j}\right)$ are the zeros of the Fourier transform of $\phi \left(t\right)$ . Recall from  Theorem 15 that $H\left(\omega \right)$ has no zeros in $-\pi /3<\omega <\pi /3$ . All of this gives a picture of the shape of $\text{Φ}\left(\omega \right)$ and the location of its zeros. From an asymptotic analysis of $\text{Φ}\left(\omega \right)$ as $\omega \to \infty$ , one can study the smoothness of  $\phi \left(t\right)$ .

A Matlab program that calculates $\text{Φ}\left(\omega \right)$ using this frequency domain successive approximations approach suggested by [link] is given in Appendix C . Studying this program gives further insight into the structure of $\text{Φ}\left(\omega \right)$ . Rather than starting the calculations given in [link] for the index $j=1$ , they are started for the largest $j=J$ and worked backwards. If we calculate a length-N DFT consistent with $j=J$ using the FFT, then the samples of $H\left(\omega /{2}^{j}\right)$ for $j=J-1$ are simply every other sample of the case for $j=J$ . The next stage for $j=J-2$ is done likewise and if the original $N$ is chosen a power of two, the process in continued down to $j=1$ without calculating any more FFTs. This results in a very efficient algorithm. The details are in theprogram itself.

This algorithm is so efficient, using it plus an inverse FFT might be a good way to calculate $\phi \left(t\right)$ itself. Examples of the algorithm are illustrated in [link] where the transform is plotted for each step of the iteration.

## The dyadic expansion of the scaling function

The next method for evaluating the scaling function uses a completely different approach. It starts by calculating the values of the scalingfunction at integer values of $t$ , which can be done exactly (within our ability to solve simultaneous linear equations). Consider the basicrecursion equation [link] for integer values of $t=k$

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
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
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
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?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Got questions? Join the online conversation and get instant answers!