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This collection of modules is from a Rice University, ECE Department Technical Report written around September 1994. It grew out of the doctoral and post doctoral research of Ivan Selesnick working with Prof. C. Sidney Burrus at Rice. Earlier reports on this work were published in the ICASSP and ISCAS conference proceedings in 1992-94 and a fairly complete report was published in the IEEE Transaction on Signal Processing in January 1996.

Preliminaries

Because we compute prime point DFTs by converting them in to circular convolutions, most of this and the next section is devotedto an explanation of the split nesting convolution algorithm. In this section we introduce the various operations neededto carry out the split nesting algorithm. In particular, we describe the prime factor permutationthat is used to convert a one-dimensional circular convolution into a multi-dimensional one.We also discuss the reduction operations needed when the Chinese Remainder Theorem for polynomials is usedin the computation of convolution. The reduction operations needed for the split nesting algorithmare particularly well organized. We give an explicit matrix description of the reduction operationsand give a program that implements the action of these reduction operations.

The presentation relies upon the notions of similarity transformations, companion matrices and Kronecker products.With them, we describe the split nesting algorithm in a manner that brings out its structure.We find that when companion matrices are used to describe convolution, the reduction operations block diagonalizesthe circular shift matrix.

The companion matrix of a monic polynomial, M ( s ) = m 0 + m 1 s + + m n - 1 s n - 1 + s n is given by

C M = - m 0 1 1 - m 1 1 - m n - 1 .

Its usefulness in the following discussion comes from the following relation which permits a matrix formulationof convolution. Let

X ( s ) = x 0 + x 1 s + x n - 1 s n - 1 H ( s ) = h 0 + h 1 s + h n - 1 s n - 1 Y ( s ) = y 0 + y 1 s + y n - 1 s n - 1 M ( s ) = m 0 + m 1 s + m n - 1 s n - 1 + s n

Then

Y ( s ) = H ( s ) X ( s ) M ( s ) y = k = 0 n - 1 h k C M k x

where y = ( y 0 , , y n - 1 ) t , x = ( x 0 , , x n - 1 ) t , and C M is the companion matrix of M ( s ) . In [link] , we say y is the convolution of x and h with respect to M ( s ) . In the case of circular convolution, M ( s ) = s n - 1 and C s n - 1 is the circular shift matrixdenoted by S n ,

S n = 1 1 1

Notice that any circulant matrix can be written as k h k S n k .

Similarity transformations can be used to interpret the action of some convolution algorithms. If C M = T - 1 A T for some matrix T ( C M and A are similar, denoted C M A ), then [link] becomes

y = T - 1 k = 0 n - 1 h k A k T x .

That is, by employing the similarity transformation given by T in this way, the action of S n k is replaced by that of A k . Many circular convolution algorithms can be understood,in part, by understanding the manipulations made to S n and the resulting new matrix A . If the transformation T is to be useful, it must satisfy two requirements:(1) T x must be simple to compute, and (2) A must have some advantageous structure. For example, by the convolution property of the DFT,the DFT matrix F diagonalizes S n ,

S n = F - 1 w 0 w 1 w n - 1 F

so that it diagonalizes every circulant matrix. In this case, T x can be computed by using an FFT and the structure of A is the simplest possible. So the two above mentioned conditions are met.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
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
Google
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
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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.
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
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
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Source:  OpenStax, Automatic generation of prime length fft programs. OpenStax CNX. Sep 09, 2009 Download for free at http://cnx.org/content/col10596/1.4
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