<< Chapter < Page Chapter >> Page >

We call N the size of the problem. The rank of a problem is defined to be the rank of its size (i.e., the dimensionality of the DFT). Similarly, we call V the vector size of the problem, and the vector rank of a problem is correspondingly defined to be the rank of its vector size.Intuitively, the vector size can be interpreted as a set of “loops” wrapped around a single DFT, and we therefore refer to a singleI/O dimension of V as a vector loop . (Alternatively, one can view the problem as describing a DFT over a V -dimensional vector space.) The problem does not specify the order of execution of these loops, however, and thereforeFFTW is free to choose the fastest or most convenient order.

Dft problem examples

A more detailed discussion of the space of problems in FFTW can be found in [link] , but a simple understanding can be gained by examining a few examples demonstrating that the I/O tensorrepresentation is sufficiently general to cover many situations that arise in practice, including some that are not usually considered tobe instances of the DFT.

A single one-dimensional DFT of length n , with stride-1 input X and output Y , as in [link] , is denoted by the problem dft ( ( n , 1 , 1 ) , , X , Y ) (no loops: vector-rank zero).

As a more complicated example, suppose we have an n 1 × n 2 matrix X stored as n 1 consecutive blocks of contiguous length- n 2 rows (this is called row-major format). The in-place DFT of all the rows of this matrix would be denoted by the problem dft ( ( n 2 , 1 , 1 ) , ( n 1 , n 2 , n 2 ) , X , X ) : a length- n 1 loop of size- n 2 contiguous DFTs, where each iteration of the loop offsets its input/output data by a stride n 2 . Conversely, the in-place DFT of all the columns of this matrix would be denoted by dft ( ( n 1 , n 2 , n 2 ) , ( n 2 , 1 , 1 ) , X , X ) : compared to the previous example, N and V are swapped. In the latter case, each DFT operates on discontiguous data, andFFTW might well choose to interchange the loops: instead of performing a loop of DFTs computed individually, the subtransformsthemselves could act on n 2 -component vectors, as described in "The space of plans in FFTW" .

A size-1 DFT is simply a copy Y [ 0 ] = X [ 0 ] , and here this can also be denoted by N = (rank zero, a “zero-dimensional” DFT). This allows FFTW's problems to represent many kinds of copies andpermutations of the data within the same problem framework, which is convenient because these sorts of operations arise frequently inFFT algorithms. For example, to copy n consecutive numbers from I to O , one would use the rank-zero problem dft ( , ( n , 1 , 1 ) , I , O ) . More interestingly, the in-place transpose of an n 1 × n 2 matrix X stored in row-major format, as described above, is denoted by dft ( , ( n 1 , n 2 , 1 ) , ( n 2 , 1 , n 1 ) , X , X ) (rank zero, vector-rank two).

The space of plans in fftw

Here, we describe a subset of the possible plans considered by FFTW; while not exhaustive [link] , this subset is enough to illustrate the basic structure of FFTW and the necessity ofincluding the vector loop(s) in the problem definition to enable several interesting algorithms. The plans that we now describeusually perform some simple “atomic” operation, and it may not be apparent how these operations fit together to actually computeDFTs, or why certain operations are useful at all. We shall discuss those matters in "Discussion" .

Questions & Answers

I'm interested in biological psychology and cognitive psychology
Tanya Reply
what does preconceived mean
sammie Reply
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
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