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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

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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
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.
Bharti
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Damian Reply
absolutely yes
Daniel
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Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
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
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
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
Abhijith Reply
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?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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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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