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The problem to be solved

In early versions of FFTW, the only choice made by the planner was the sequence of radices [link] , and so each step of the plan took a DFT of a given size n , possibly with discontiguous input/output, and reduced it (via a radix r ) to DFTs of size n / r , which were solved recursively. That is, each step solved the following problem: given a size n , an input pointer I , an input stride ι , an output pointer O , and an output stride o , it computed the DFT of I [ ι ] for 0 < n and stored the result in O [ k o ] for 0 k < n . However, we soon found that we could not easily express many interesting algorithms within thisframework; for example, in-place ( I = O ) FFTs that do not require a separate bit-reversalstage [link] , [link] , [link] , [link] . It became clear that the key issue was not the choice of algorithms, as we hadfirst supposed, but the definition of the problem to be solved. Because only problems that can be expressed can be solved, therepresentation of a problem determines an outer bound to the space of plans that the planner can explore, and therefore it ultimatelyconstrains FFTW's performance.

The difficulty with our initial ( n , I , ι , O , o ) problem definition was that it forced each algorithmic step to address only asingle DFT. In fact, FFTs break down DFTs into multiple smaller DFTs, and it is the combination of these smaller transforms that is best addressed by many algorithmicchoices, especially to rearrange the order of memory accesses between the subtransforms. Therefore, we redefined our notion of a problemin FFTW to be not a single DFT, but rather a loop of DFTs, and in fact multiple nested loops of DFTs. The following sections describe some of the new algorithmic steps thatsuch a problem definition enables, but first we will define the problem more precisely.

DFT problems in FFTW are expressed in terms of structures called I/O tensors, I/O tensors are unrelated to the tensor-product notation used by some other authors to describe FFTalgorithms [link] , [link] . which in turn are described in terms of ancillary structures called I/O dimensions. An I/O dimension d is a triple d = ( n , ι , o ) , where n is a non-negative integer called the length , ι is an integer called the input stride , and o is an integer called the output stride . An I/O tensor t = d 1 , d 2 , ... , d ρ is a set of I/O dimensions. The non-negative integer ρ = t is called the rank of the I/O tensor. A DFT problem , denoted by dft ( N , V , I , O ) , consists of two I/O tensors N and V , and of two pointers I and O . Informally, this describes V nested loops of N -dimensional DFTs with input data starting at memory location I and output data starting at O .

For simplicity, let us consider only one-dimensional DFTs, so that N = ( n , ι , o ) implies a D F T of length n on input data with stride ι and output data with stride o , much like in the original FFTW as described above. The main newfeature is then the addition of zero or more “loops” V . More formally, dft ( N , ( n , ι , o ) V , I , O ) is recursively defined as a “loop” of n problems: for all 0 k < n , do all computations in dft ( N , V , I + k · ι , O + k · o ) . The case of multi-dimensional DFTs is defined more preciselyelsewhere [link] , but essentially each I/O dimension in N gives one dimension of the transform.

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
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John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
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David Reply
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David
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emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
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Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
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Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
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Muhammad Reply
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Mohammed
hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
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Source:  OpenStax, Fast fourier transforms. OpenStax CNX. Nov 18, 2012 Download for free at http://cnx.org/content/col10550/1.22
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