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Building Block for the Discrete Wavelet Transform
Building Block for the Discrete Wavelet Transform

The computational complexity of the DWT algorithm can also be easily established. Let C D W T ( N ) 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 D W T ( N ) = O ( N ) + C D W T ( N / 2 ) ,

which gives rise to the solution

C D W T ( N ) = O ( N ) .

The operation in [link] can also be expressed in matrix form W N ; e.g., for Haar wavelet,

W 4 H a a r = 2 / 2 1 - 1 0 0 0 0 1 - 1 1 1 0 0 0 0 1 1 .

The orthogonality conditions on h and g ensure W N ' W N = I N . The matrix for multiscale DWT is formed by W N for different N ; e.g., for three scale DWT,

W N / 4 I N / 4 I N / 2 W N / 2 I N / 2 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 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:

F N = F N W N T W N = A N / 2 B N / 2 C N / 2 D N / 2 F N / 2 0 0 F N / 2 W N ,

where A N / 2 , B N / 2 , C N / 2 , and D N / 2 are all diagonal matrices. The values on the diagonal of A N / 2 and C N / 2 are the length-N DFT ( i.e., frequency response ) of h , and the values on the diagonal of B N / 2 and D N / 2 are the length-N DFT of 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.

Last stage of a length-8 DWT based FFT.
Last stage of a length-8 DWT based FFT.

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 { 0 , 1 , ... , N / 2 - 1 } . Now the twiddle factors are length-N DFT of h and 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 h = [ 1 , 0 ] and g = [ 0 , 1 ] . 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.

Butterfly Operations in a Radix-2 DIT FFT

Computational complexity

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

Questions & Answers

what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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