<< Chapter < Page Chapter >> Page >

The prime factor algorithm (PFA) and the Winograd Fourier transform algorithm (WFTA) are methods for efficiently calculatingthe DFT which use, and in fact, depend on the Type-1 index map from Multidimensional Index Mapping: Equation 10 and Multidimensional Index Mapping: Equation 6 . The use of this index map preceded Cooley and Tukey's paper [link] , [link] but its full potential was not realized until it was combined with Winograd's short DFT algorithms.The modern PFA was first presented in [link] and a program given in [link] . The WFTA was first presented in [link] and programs given in [link] , [link] .

The number theoretic basis for the indexing in these algorithms may, at first, seem more complicated than in theCooley-Tukey FFT; however, if approached from the general index mapping point of view of Multidimensional Index Mapping , it is straightforward, and part of a common approach to breaking large problems intosmaller ones. The development in this section will parallel that in The Cooley-Tukey Fast Fourier Transform Algorithm .

The general index maps of Multidimensional Index Mapping: Equation 6 and Multidimensional Index Mapping: Equation 12 must satisfy the Type-1 conditions of Multidimensional Index Mapping: Equation 7 and Multidimensional Index Mapping: Equation 10 which are

K 1 = a N 2 and K 2 = b N 1 with ( K 1 , N 1 ) = ( K 2 , N 2 ) = 1
K 3 = c N 2 and K 4 = d N 1 with ( K 3 , N 1 ) = ( K 4 , N 2 ) = 1

The row and column calculations in Multidimensional Index Mapping: Equation 15 are uncoupled by Multidimensional Index Mapping: Equation 16 which for this case are

( ( K 1 K 4 ) ) N = ( ( K 2 K 3 ) ) N = 0

In addition, to make each short sum a DFT, the K i must also satisfy

( ( K 1 K 3 ) ) N = N 2 a n d ( ( K 2 K 4 ) ) N = N 1

In order to have the smallest values for K i , the constants in [link] are chosen to be

a = b = 1 , c = ( ( N 2 - 1 ) ) N , d = ( ( N 1 - 1 ) ) N

which gives for the index maps in [link]

n = ( ( N 2 n 1 + N 1 n 2 ) ) N
k = ( ( K 3 k 1 + K 4 k 2 ) ) N

The frequency index map is a form of the Chinese remainder theorem. Using these index maps, the DFT in Multidimensional Index Mapping: Equation 15 becomes

X = n 2 = 0 N 2 - 1 n 1 = 0 N 1 - 1 x W N 1 n 1 k 1 W N 2 n 2 k 2

which is a pure two-dimensional DFT with no twiddle factors and the summations can be done in either order. Choices other than [link] could be used. For example, a = b = c = d = 1 will cause the input and output index map to be the same and, therefore,there will be no scrambling of the output order. The short summations in (96), however, will no longer be short DFT's [link] .

An important feature of the short Winograd DFT's described in Winograd’s Short DFT Algorithms that is useful for both the PFA and WFTA is the fact that the multiplier constants in Winograd’s Short DFT Algorithms: Equation 6 or Winograd’s Short DFT Algorithms: Equation 8 are either real or imaginary, never a general complex number. For thatreason, multiplication by complex data requires only two real multiplications, not four. That is a very significant feature. It isalso true that the j multiplier can be commuted from the D operator to the last part of the A T operator. This means the D operator has only real multipliers and the calculations on real data remains real until the last stage. This can be seen by examining theshort DFT modules in [link] , [link] and in the appendices.

Questions & Answers

physiological Psychology
Nwosu Reply
How can I develope my cognitive domain
Amanyire Reply
why is communication effective
Dakolo Reply
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
Wekolamo Reply
please i need answer
Wekolamo
because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).
Manix Reply
Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!
ARC
A child is a member of community not society elucidate ?
JESSY Reply
Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?
Simon Reply
compare and contrast skinner's perspective on personality development on freud
namakula Reply
Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning
war
explain how nature and nurture affect the development and later the productivity of an individual.
Amesalu Reply
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
Zyryn Reply
good👍
Jonathan
and having a good philosophy of the world is like a sandwich and a peanut butter 👍
Jonathan
generally amnesi how long yrs memory loss
Kelu Reply
interpersonal relationships
Abdulfatai Reply
What would be the best educational aid(s) for gifted kids/savants?
Heidi Reply
treat them normal, if they want help then give them. that will make everyone happy
Saurabh
What are the treatment for autism?
Magret Reply
hello. autism is a umbrella term. autistic kids have different disorder overlapping. for example. a kid may show symptoms of ADHD and also learning disabilities. before treatment please make sure the kid doesn't have physical disabilities like hearing..vision..speech problem. sometimes these
Jharna
continue.. sometimes due to these physical problems..the diagnosis may be misdiagnosed. treatment for autism. well it depends on the severity. since autistic kids have problems in communicating and adopting to the environment.. it's best to expose the child in situations where the child
Jharna
child interact with other kids under doc supervision. play therapy. speech therapy. Engaging in different activities that activate most parts of the brain.. like drawing..painting. matching color board game. string and beads game. the more you interact with the child the more effective
Jharna
results you'll get.. please consult a therapist to know what suits best on your child. and last as a parent. I know sometimes it's overwhelming to guide a special kid. but trust the process and be strong and patient as a parent.
Jharna
what is learning? discuss how any three scholars propounded on how learning is
Grace Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




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