# 0.12 Convolution algorithms  (Page 5/7)

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## Block state formulation

It is possible to reduce the size of the matrix operators in the blockrecursive description [link] to give a form even more like a state variable equation [link] , [link] , [link] . If $K$ in [link] has several zero eigenvalues, it should be possible to reduce the size of $K$ until it has full rank. That was done in [link] and the result is

${\underline{z}}_{n}={K}_{1}\phantom{\rule{0.166667em}{0ex}}{\underline{z}}_{n-1}+{K}_{2}\phantom{\rule{0.166667em}{0ex}}{\underline{x}}_{n}$
${\underline{y}}_{n}={H}_{1}\phantom{\rule{0.166667em}{0ex}}{\underline{z}}_{n-1}+{H}_{0}\phantom{\rule{0.166667em}{0ex}}{\underline{x}}_{n}$

where ${H}_{0}$ is the same $N$ by $N$ convolution matrix, ${N}_{1}$ is a rectangular $L$ by $N$ partition of the convolution matrix $H$ , ${K}_{1}$ is a square $N$ by $N$ matrix of full rank, and ${K}_{2}$ is a rectangular $N$ by $L$ matrix.

This is now a minimal state equation whose input and output are blocks of the original input and output. Some of the matrix multiplications can becarried out using the FFT or other techniques.

## Block implementations of digital filters

The advantage of the block convolution and recursion implementations is a possible improvement in arithmetic efficiency by using the FFT or otherfast convolution methods for some of the multiplications in [link] or [link] [link] , [link] . There is the reduction of quantization effects due to an effective decrease in the magnitude of the eigenvalues and thepossibility of easier parallel implementation for IIR filters. The disadvantages are a delay of at least one block length and an increasedmemory requirement.

These methods could also be used in the various filtering methods for evaluating the DFT. This the chirp z-transform, Rader's method, andGoertzel's algorithm.

## Multidimensional formulation

This process of partitioning the data vectors and the operator matrices can be continued by partitioning [link] and [link] and creating blocks of blocks to give a higher dimensional structure. One should useindex mapping ideas rather than partitioned matrices for this approach [link] , [link] .

## Periodically time-varying discrete-time systems

Most time-varying systems are periodically time-varying and this allows special results to be obtained. If the block length is set equal to theperiod of the time variations, the resulting block equations are time invariant and all to the time varying characteristics are contained in thematrix multiplications. This allows some of the tools of time invariant systems to be used on periodically time-varying systems.

## Multirate filters, filter banks, and wavelets

Another area that is related to periodically time varying systems and to block processing is filter banks [link] , [link] . Recently the area of perfect reconstruction filter banks has been further developed and shownto be closely related to wavelet based signal analysis [link] , [link] , [link] , [link] . The filter bank structure has several forms with the polyphase and lattice being particularly interesting.

An idea that has some elements of multirate filters, perfect reconstruction, and distributed arithmetic is given in [link] , [link] , [link] . Parks has noted that design of multirate filters has some elements in common with complex approximation and of 2-D filterdesign [link] , [link] and is looking at using Tang's method for these designs.

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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