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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. | |||
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. | |||
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. | |||
(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.) | |||
==Algorithm== | |||
Recall that the DFT is defined by the formula: | |||
:<math>X_k = \sum_{n=0}^{N-1} x_n e^{-\frac{2\pi i}{N} nk } | |||
\qquad | |||
k = 0,\dots,N-1. </math> | |||
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). | |||
===Re-indexing=== | |||
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: | |||
:<math>n = n_1 N_2 + n_2 N_1 \mod N,</math> | |||
:<math>k = k_1 N_2^{-1} N_2 + k_2 N_1^{-1} N_1 \mod N,</math> | |||
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. | |||
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>. | |||
(One could instead use the Ruritanian mapping for the output ''k'' and the CRT mapping for the input ''n'', or various intermediate choices.) | |||
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). | |||
===DFT re-expression=== | |||
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: | |||
:<math>X_{k_1 N_2^{-1} N_2 + k_2 N_1^{-1} N_1} = | |||
\sum_{n_1=0}^{N_1-1} | |||
\left( \sum_{n_2=0}^{N_2-1} x_{n_1 N_2 + n_2 N_1} | |||
e^{-\frac{2\pi i}{N_2} n_2 k_2 } \right) | |||
e^{-\frac{2\pi i}{N_1} n_1 k_1 }. | |||
</math> | |||
The inner and outer sums are simply DFTs of size ''N''<sub>2</sub> and ''N''<sub>1</sub>, respectively. | |||
(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.) | |||
==References== | |||
*{{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}}. | |||
*{{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= }} | |||
*{{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 }} | |||
*{{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 }} | |||
[[Category:FFT algorithms]] | |||
Revision as of 16:39, 14 January 2014
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 = N1N2 as a two-dimensional N1×N2 DFT, but only for the case where N1 and N2 are relatively prime. These smaller transforms of size N1 and N2 can then be evaluated by applying PFA recursively or by using some other FFT algorithm.
PFA should not be confused with the mixed-radix generalization of the popular Cooley–Tukey algorithm, which also subdivides a DFT of size N = N1N2 into smaller transforms of size N1 and N2. 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 factors, 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 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.
PFA is also closely related to the nested Winograd FFT algorithm, where the latter performs the decomposed N1 by N2 transform via more sophisticated two-dimensional convolution techniques. Some older papers therefore also call Winograd's algorithm a PFA FFT.
(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.)
Algorithm
Recall that the DFT is defined by the formula:
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).
Re-indexing
Suppose that N = N1N2, where N1 and N2 are relatively prime. In this case, we can define a bijective re-indexing of the input n and output k by:
where N1−1 denotes the modular multiplicative inverse of N1 modulo N2 and vice-versa for N2−1; the indices ka and na run from 0,...,Na−1 (for a = 1, 2). These inverses only exist for relatively prime N1 and N2, and that condition is also required for the first mapping to be bijective.
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 = k1 mod N1 and k = k2 mod N2.
(One could instead use the Ruritanian mapping for the output k and the CRT mapping for the input n, or various intermediate choices.)
A great deal of research has been devoted to schemes for evaluating this re-indexing efficiently, ideally in-place, while minimizing the number of costly modulo (remainder) operations (Chan, 1991, and references).
DFT re-expression
The above re-indexing is then substituted into the formula for the DFT, and in particular into the product nk in the exponent. Because e2πi = 1, this exponent is evaluated modulo N: any N1N2 = N cross term in the nk product can be set to zero. (Similarly, Xk and xn are implicitly periodic in N, so their subscripts are evaluated modulo N.) The remaining terms give:
The inner and outer sums are simply DFTs of size N2 and N1, respectively.
(Here, we have used the fact that N1−1N1 is unity when evaluated modulo N2 in the inner sum's exponent, and vice-versa for the outer sum's exponent.)
References
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Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
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In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang