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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

z ̲ n = K 1 z ̲ n - 1 + K 2 x ̲ n
y ̲ n = H 1 z ̲ n - 1 + H 0 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.

The PTV system is analyzed in [link] , [link] , [link] , [link] , the filter analysis and design problem, which includes the decimation–interpolationstructure, is addressed in [link] , [link] , [link] , and the bandwidth compression problem in [link] . These structures can take the form of filter banks [link] .

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.

Questions & Answers

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.
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
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
How can I make nanorobot?
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Fast fourier transforms. OpenStax CNX. Nov 18, 2012 Download for free at http://cnx.org/content/col10550/1.22
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