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| The '''prime-factor algorithm (PFA)''', also called the '''Good–Thomas algorithm''' (1958/1963), is a [[fast Fourier transform]] (FFT) algorithm that re-expresses the [[discrete Fourier transform]] (DFT) of a size ''N'' = ''N''<sub>1</sub>''N''<sub>2</sub> as a two-dimensional ''N''<sub>1</sub>×''N''<sub>2</sub> DFT, but ''only'' for the case where ''N''<sub>1</sub> and ''N''<sub>2</sub> are [[relatively prime]]. These smaller transforms of size ''N''<sub>1</sub> and ''N''<sub>2</sub> can then be evaluated by applying PFA [[recursion|recursively]] or by using some other FFT algorithm.
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| PFA should not be confused with the '''mixed-radix''' generalization of the popular [[Cooley–Tukey FFT algorithm|Cooley–Tukey algorithm]], which also subdivides a DFT of size ''N'' = ''N''<sub>1</sub>''N''<sub>2</sub> into smaller transforms of size ''N''<sub>1</sub> and ''N''<sub>2</sub>. The latter algorithm can use ''any'' factors (not necessarily relatively prime), but it has the disadvantage that it also requires extra multiplications by roots of unity called [[twiddle factor]]s, in addition to the smaller transforms. On the other hand, PFA has the disadvantages that it only works for relatively prime factors (e.g. it is useless for [[power of two|power-of-two]] sizes) and that it requires a more complicated re-indexing of the data based on the [[Chinese remainder theorem]] (CRT). Note, however, that PFA can be combined with mixed-radix Cooley–Tukey, with the former factorizing ''N'' into relatively prime components and the latter handling repeated factors.
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| PFA is also closely related to the nested [[Winograd FFT algorithm]], where the latter performs the decomposed ''N''<sub>1</sub> by ''N''<sub>2</sub> transform via more sophisticated two-dimensional convolution techniques. Some older papers therefore also call Winograd's algorithm a PFA FFT.
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| (Although the PFA is distinct from the Cooley–Tukey algorithm, Good's 1958 work on the PFA was cited as inspiration by Cooley and Tukey in their famous 1965 paper, and there was initially some confusion about whether the two algorithms were different. In fact, it was the only prior FFT work cited by them, as they were not then aware of the earlier research by Gauss and others.)
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| ==Algorithm==
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| Recall that the DFT is defined by the formula:
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| :<math>X_k = \sum_{n=0}^{N-1} x_n e^{-\frac{2\pi i}{N} nk }
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| \qquad
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| k = 0,\dots,N-1. </math>
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| The PFA involves a re-indexing of the input and output arrays, which when substituted into the DFT formula transforms it into two nested DFTs (a two-dimensional DFT).
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| ===Re-indexing===
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| Suppose that ''N'' = ''N''<sub>1</sub>''N''<sub>2</sub>, where ''N''<sub>1</sub> and ''N''<sub>2</sub> are relatively prime. In this case, we can define a [[bijection|bijective]] re-indexing of the input ''n'' and output ''k'' by:
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| :<math>n = n_1 N_2 + n_2 N_1 \mod N,</math>
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| :<math>k = k_1 N_2^{-1} N_2 + k_2 N_1^{-1} N_1 \mod N,</math>
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| where ''N''<sub>1</sub><sup>−1</sup> denotes the [[modular multiplicative inverse]] of ''N''<sub>1</sub> [[modular arithmetic|modulo]] ''N''<sub>2</sub> and vice-versa for ''N''<sub>2</sub><sup>−1</sup>; the indices ''k''<sub>''a''</sub> and ''n''<sub>''a''</sub> run from 0,...,''N''<sub>''a''</sub>−1 (for ''a'' = 1, 2). These inverses only exist for relatively prime ''N''<sub>1</sub> and ''N''<sub>2</sub>, and that condition is also required for the first mapping to be bijective.
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| This re-indexing of ''n'' is called the ''Ruritanian'' mapping (also ''Good's'' mapping), while this re-indexing of ''k'' is called the ''CRT'' mapping. The latter refers to the fact that ''k'' is the solution to the Chinese remainder problem ''k'' = ''k''<sub>1</sub> mod ''N''<sub>1</sub> and ''k'' = ''k''<sub>2</sub> mod ''N''<sub>2</sub>.
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| (One could instead use the Ruritanian mapping for the output ''k'' and the CRT mapping for the input ''n'', or various intermediate choices.)
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| A great deal of research has been devoted to schemes for evaluating this re-indexing efficiently, ideally [[in-place algorithm|in-place]], while minimizing the number of costly modulo (remainder) operations (Chan, 1991, and references).
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| ===DFT re-expression===
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| The above re-indexing is then substituted into the formula for the DFT, and in particular into the product ''nk'' in the exponent. Because ''e''<sup>2''πi''</sup> = 1, this exponent is evaluated modulo ''N'': any ''N''<sub>1</sub>''N''<sub>2</sub> = ''N'' cross term in the ''nk'' product can be set to zero. (Similarly, ''X''<sub>''k''</sub> and ''x''<sub>''n''</sub> are implicitly periodic in ''N'', so their subscripts are evaluated modulo ''N''.) The remaining terms give:
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| :<math>X_{k_1 N_2^{-1} N_2 + k_2 N_1^{-1} N_1} =
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| \sum_{n_1=0}^{N_1-1}
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| \left( \sum_{n_2=0}^{N_2-1} x_{n_1 N_2 + n_2 N_1}
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| e^{-\frac{2\pi i}{N_2} n_2 k_2 } \right)
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| e^{-\frac{2\pi i}{N_1} n_1 k_1 }.
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| </math>
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| The inner and outer sums are simply DFTs of size ''N''<sub>2</sub> and ''N''<sub>1</sub>, respectively.
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| (Here, we have used the fact that ''N''<sub>1</sub><sup>−1</sup>''N''<sub>1</sub> is unity when evaluated modulo ''N''<sub>2</sub> in the inner sum's exponent, and vice-versa for the outer sum's exponent.)
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| ==References==
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| *{{cite journal |first=I. J. |last=Good |title=The interaction algorithm and practical Fourier analysis |journal=Journal of the Royal Statistical Society, Series B |volume=20 |issue=2 |pages=361–372 |year=1958 |jstor=2983896 }} Addendum, ''ibid.'' '''22''' (2), 373-375 (1960) {{JSTOR|2984108}}.
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| *{{cite book |first=L. H. |last=Thomas |chapter=Using a computer to solve problems in physics |title=Applications of Digital Computers |publisher=Ginn |location=Boston |year=1963 |isbn= }}
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| *{{cite journal |first=P. |last=Duhamel |first2=M. |last2=Vetterli |title=Fast Fourier transforms: a tutorial review and a state of the art |journal=Signal Processing |volume=19 |issue=4 |pages=259–299 |year=1990 |doi=10.1016/0165-1684(90)90158-U }}
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| *{{cite journal |first=S. C. |last=Chan |first2=K. L. |last2=Ho |title=On indexing the prime-factor fast Fourier transform algorithm |journal=IEEE Trans. Circuits and Systems |volume=38 |issue=8 |pages=951–953 |year=1991 |doi=10.1109/31.85638 }}
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| [[Category:FFT algorithms]]
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Over time, the data on your difficult drive gets scattered. Defragmenting your hard drive puts your data back to sequential order, creating it simpler for Windows to access it. As a outcome, the performance of your computer may boost. An excellent registry cleaner allows perform this task. But in the event you would like to defrag the PC with Windows software. Here a link to show we how.
Firstly, we should utilize a Antivirus or security tool and run a scan on the computer. It can be done that a computer is afflicted with virus or malware which slows down a computer. If there is nothing found inside the scanning report, it might be your RAM that cause the problem.
Your PC might moreover have a fragmented hard drive or the windows registry might have been corrupted. It may equally be because of the dust plus dirt that should be cleaned. Whatever the problem, you can constantly find a solution. Here are certain tricks on how to create your PC run quicker.
There are tricks to create your slow computer work efficient and swiftly. In this article, I usually tell we just 3 best tricks or techniques to avoid a computer of being slow and rather of which create it quicker and work even greater than before.
In a word, to speed up windows XP, Vista startup, it's very significant to disable certain business items plus clean plus optimize the registry. You can follow the steps above to disable unwanted programs. To optimize the registry, I recommend we employ a registry reviver software. Because it really is quite risky for you to edit the registry by oneself.
The many probable cause of the trouble is the program issue - Registry Errors! That is the reason why folks who already have over 2 G RAM on their computers are nevertheless consistently bothered by the issue.
It is critical which we remove obsolete registry entries from a program regularly, if you need the system to run faster, that is. If you don't keep a registry clean, a time comes whenever your system usually stop working completely. Then, the only choice would be to reformat a difficult drive and begin over!
What I would recommend is to look on your for registry products. You are able to do this with a Google look. Whenever you find treatments, look for critiques plus testimonials regarding the product. Next we can see how others like the product, plus how effectively it works.