<< Chapter < Page Chapter >> Page >
u ( D F T ( N ) ) = 2 N - m 2 - m - 2

This result is not practically useful because the number of additions necessary to realize this minimum of multiplicationsbecomes very large for lengths greater than 16. Nevertheless, it proves the minimum number of multiplications required of an optimalalgorithm is a linear function of N rather than of N log N which is that required of practical algorithms. The best practical power-of-two algorithm seems to the Split-Radix [link] FFT discussed in The Cooley-Tukey Fast Fourier Transform Algorithm: The Split-Radix FFT Algorithm .

All of these theorems use ideas based on residue reduction, multiplication of the residues, and then combination by the CRT. Itis remarkable that this approach finds the minimum number of required multiplications by a constructive proof which generates analgorithm that achieves this minimum; and the structure of the optimal algorithm is, within certain variations, unique. For shorterlengths, the optimal algorithms give practical programs. For longer lengths the uncounted operations involved with the multiplication ofthe higher degree residue polynomials become very large and impractical. In those cases, efficient suboptimal algorithms can begenerated by using the same residue reduction as for the optimal case, but by using methods other than the Toom-Cook algorithm of Theorem 1 to multiply the residue polynomials.

Practical long DFT algorithms are produced by combining short prime length optimal DFT's with the Type 1 index map from Multidimensional Index Mapping to give the Prime Factor Algorithm (PFA) and the Winograd Fourier Transform Algorithm (WFTA) discussed in The Prime Factor and Winograd Fourier Transform Algorithms . It is interesting to note that the index mapping technique is useful inside the short DFT algorithms to replace the Toom-Cookalgorithm and outside to combine the short DFT's to calculate long DFT's.

The automatic generation of winograd's short dfts

by Ivan Selesnick, Polytechnic Institute of New York University

Introduction

Efficient prime length DFTs are important for two reasons. A particular application may require a prime length DFT and secondly, the maximum lengthand the variety of lengths of a PFA or WFTA algorithm depend upon the availability of prime length modules.

This [link] , [link] , [link] , [link] discusses automation of the process Winograd used for constructing prime length FFTs [link] , [link] for N < 7 and that Johnson and Burrus [link] extended to N < 19 . It also describes a program that will design any prime length FFT in principle,and will also automatically generate the algorithm as a C program and draw the corresponding flow graph.

Winograd's approach uses Rader's method to convert a prime length DFT into a P - 1 length cyclic convolution, polynomial residue reduction to decompose the problem into smaller convolutions [link] , [link] , and the Toom-Cook algorithm [link] , [link] . The Chinese Remainder Theorem (CRT) for polynomials is then used to recombine theshorter convolutions. Unfortunately, the design procedure derived directly from Winograd's theory becomes cumbersome for longer length DFTs, and this has oftenprevented the design of DFT programs for lengths greater than 19.

Questions & Answers

the art of managing the production, distribution and consumption.
Satangthem Reply
what is economics
Khawar Reply
okk
damfash
marginal utility is the additional satisfaction one derives from consuming additional unit of a good or service.
Fred
It's the allocation of scarce resources.
Fred
marginal utility please?
Dishan
marginal utility is the additional satisfaction one derives from consuming additional unit of a good or service.
Fred
I know the definition, but I don't understand its meaning.
Dishan
what is the must definition of economic please?
Nurudeen
demand lfs
Alpha
Economics is derived from the word Oikonomia which means management of household things. Thus, Economics is a study of household things with the constrains of allocating scare resources.
Dishan
what is Open Market Operation
Adu Reply
dominating middlemen men activities circumstances
Christy Reply
what Equilibrium price
Adji Reply
what is gap
mirwais
who is good with the indifference curve
Dexter
What is diseconomic
Alixe Reply
what are the types of goods
WARIDI
how can price determination be the central problem of micro economics
simon Reply
marginal cost formula
Nandu Reply
you should differentiate the total cost function in order to get marginal cost function then you can get marginal cost from it
boniphace
What about total cost
Foday
ok
Foday
how can price determination be the central problem if micro economics
simon
formula of cross elasticity of demand
Theresia Reply
what is ceteris paribus
Priyanka Reply
what is ceteris parabus
Priyanka
Ceteris paribus - Literally, "other things being equal"; usually used in economics to indicate that all variables except the ones specified are assumed not to change.
Abdullah
What is broker
scor
land is natural resources that is made by nature
scor
What is broker
scor
what is land
kafui
What is broker
scor
land is natural resources that is made by nature
scor
whats poppina nigga turn it up for a minute get it
amarsyaheed Reply
what is this?
Philo
am from nigeria@ pilo
Frank
am from nigeria@ pilo
Frank
so
owusu
what is production possibility frontier
owusu
it's a summary of opportunity cost depicted on a curve.
okhiria
please help me solve this question with the aid of appropriate diagrams explain how each of the following changes will affect the market price and quantity of bread 1. A
Manuela Reply
please l need past question about economics
Prosper Reply
ok let me know some of the questions please.
Effah
ok am not wit some if den nw buh by tommorow I shall get Dem
adepojurafiu
Hi guys can I get Adam Smith's WEALTH OF NATIONS fo sale?
Ukpen
hello I'm Babaisa alhaji Mustapha. I'm studying Economics in the university of Maiduguri
Babaisa
okay
Humaira
my name is faisal Yahaya. i studied economics at Kaduna state university before proceeding to West African union university benin republic for masters
Faisal
Hi guys..I am from Bangladesh..
Mannan
Wat d meaning of management
igwe Reply
disaster management cycle
Gogul Reply
cooperate social responsibility
igwe
Fedric Wilson Taylor also define management as the act of knowing what to do and seeing that it is done in the best and cheapest way
OLANIYI
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
Difference between extinct and extici spicies
Amanpreet Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Fast fourier transforms. OpenStax CNX. Nov 18, 2012 Download for free at http://cnx.org/content/col10550/1.22
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Fast fourier transforms' conversation and receive update notifications?

Ask