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The '''cyclotomic fast Fourier transform''' is a type of [[fast Fourier transform]] algorithm over [[finite field]]s.<ref>S.V. Fedorenko and P.V. Trifonov, {{cite journal |last=Fedorenko |first=S. V. |first2=P. V.. |last2=Trifonov |url=http://dcn.ftk.spbstu.ru/~petert/papers/pushkin2.pdf |title=On Computing the Fast Fourier Transform over Finite Fields |journal=Proceedings of International Workshop on Algebraic and Combinatorial Coding Theory |pages=108–111|year=2003}}</ref> This algorithm first decomposes a DFT into several circular convolutions, and then derives the DFT results from the circular convolution results. When applied to a DFT over GF(2<sup''>''m''</sup>), this algorithm has a very low multiplicative complexity. In practice, since there usually exist efficient algorithms for circular convolutions with specific lengths, this algorithm is very efficient.<ref name="complexityAnalysis">{{cite journal | last=Wu | first=Xuebin | last2=Wang | first2=Ying |last3=Yan | first3=Zhiyuan | title=On Algorithms and Complexities of Cyclotomic Fast Fourier Transforms Over Arbitrary Finite Fields| journal=IEEE Transactions on Signal Processing|volume=60|issue=3|year=2012|pages=1149–1158}}</ref>
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== Background ==
The [[discrete Fourier transform]] over [[finite field]]s finds widespread application in the decoding of [[error-correcting code]]s such as [[BCH codes]] and [[Reed–Solomon error correction|Reed–Solomon codes]]. Generalized from the [[complex number|complex field]], a discrete Fourier transform of a sequence <math>\{f_i\}_{0}^{N-1}</math> over a finite field GF(''p''<sup>''m''</sup>) is defined as
 
:<math>F_j=\sum_{i=0}^{N-1}f_i\alpha^{ij}, 0\le j\le N-1, </math>
 
where <math>\alpha</math> is the ''N''-th [[primitive root]] of 1 in GF(''p''<sup>''m''</sup>). If we define the polynomial representation of <math>\{f_i\}_{0}^{N-1}</math> as
 
:<math>f(x) = f_0+f_1x+f_2x^2+\cdots+f_{N-1}x^{N-1}=\sum_{0}^{N-1}f_ix^i,</math>
 
it is easy to see that <math>F_j</math> is simply <math>f(\alpha^j)</math>. That is, the discrete Fourier transform of a sequence converts it to a polynomial evaluation problem.  
 
Written in matrix format,
:<math>\mathbf{F}=\left[\begin{matrix}F_0\\F_1\\ \vdots \\ F_{N-1}\end{matrix}\right]=
\left[\begin{matrix}
\alpha^0&\alpha^0 &\cdots & \alpha^0\\
\alpha^0 & \alpha^1 &\cdots &\alpha^{N-1}\\
\vdots &\vdots & \ddots & \vdots \\
\alpha^{0} & \alpha^{N-1} &\cdots & \alpha^{(N-1)(N-1)}
\end{matrix}\right]
\left[\begin{matrix}f_0\\f_1\\\vdots\\f_{N-1}\end{matrix}\right]=\mathcal{F}\mathbf{f}.
</math>
 
Direct evaluation of DFT has an <math>O(N^2)</math> complexity. Fast Fourier transforms are just efficient algorithms evaluating the above matrix-vector product.
 
== Algorithm ==
 
First, we define a [[linearized polynomial]] over GF(p<sup>m</sup>) as
 
:<math>L(x) = \sum_{i} l_i x^{p^i}, l_i \in \mathrm{GF}(p^m).</math>
 
<math>L(x)</math> is called linearized because <math>L(x_1+x_2) = L(x_1) + L(x_2)</math>, which comes from the fact that for elements <math>x_1,x_2 \in \mathrm{GF}(p^m),</math><math>(x_1+x_2)^p=x_1^p+x_2^p.</math>
 
The elements <math>\{0, 1, 2, \ldots, N-1\}</math> can be partitioned into <math>l+1</math> cyclotomic cosets modulo <math>N</math>:
:<math>\{0\},</math>
:<math>\{k_1, pk_1, p^2k_1, \ldots, p^{m_1-1}k_1\},</math>
:<math>\ldots,</math>
:<math>\{k_l, pk_l, p^2k_l, \ldots, p^{m_l-1}k_l\},</math>
and <math>k_i=p^{m_i}k_i \pmod{N}</math>. Therefore, the Fourier transform can be rewritten as  
 
:<math>f(x)=\sum_{i=0}^l L_i(x^{k_i}), L_i(y) = \sum_{j=0}^{m_j-1}f_{p^jk_i\bmod{N}}y^{p^j},</math>
 
In this way, the polynomial representation is decomposed into a sum of linear polynomials, and hence <math>F_j</math> is given by
:<math>F_j=f(\alpha^j)=\sum_{i=0}^lL_i(\alpha^{jk_i})</math>.
Expanding <math>\alpha^{jk_i} \in \mathrm{GF}(p^{m_i})</math> with the proper basis <math>\{\beta_{i,0}, \beta_{i,1}, \ldots, \beta_{i,m_i-1}\}</math>, we have <math>\alpha^{jk_i} = \sum_{s=0}^{m_i-1}a_{ijs}\beta_{i,s}</math> where <math>a_{ijs} \in \mathrm{GF}(p)</math>, and by the property of the linearized polynomial <math>L_i(x)</math>, we have
 
:<math>F_j=\sum_{i=0}^l\sum_{s=0}^{m_i-1}a_{ijs}\left(\sum_{t=0}^{m_i-1}\beta_{i,t}^{p^t}f_{p^{t}k_i\bmod{N}}\right)</math>
 
This equation can be rewritten in matrix form as <math>\mathbf{F}=\mathbf{AL\Pi f}</math>, where <math>\mathbf{A}</math> is an <math>N\times N</math> matrix over GF(''p'') that contains the elements <math>a_{ijs}</math>, <math>\mathbf{L}</math> is a block diagonal matrix, and <math>\mathbf{\Pi}</math> is a permutation matrix regrouping the elements in <math>\mathbf{f}</math> according to the cyclotomic coset index.  
 
Note that if the [[normal basis]] <math>\{\gamma_i^{p^0}, \gamma_i^{p^1}, \cdots, \gamma_i^{p^{m_i-1}}\} </math> is used to expand the field elements of <math>\mathrm{GF}(p^{m_i})</math>, the i-th block of <math>\mathbf{L}</math> is given by:
:<math>\mathbf{L}_i=
\begin{bmatrix}
  \gamma_i^{p^0}  &\gamma_i^{p^1}  &\cdots  &\gamma_i^{p^{m_i-1}}\\
  \gamma_i^{p^1}  &\gamma_i^{p^2}  &\cdots  &\gamma_i^{p^{0}}\\
  \vdots & \vdots & \ddots & \vdots\\
  \gamma_i^{p^{m_i-1}}  &\gamma_i^{p^0}  &\cdots  &\gamma_i^{p^{m_i-2}}\\
\end{bmatrix}
</math>
which is a [[circulant matrix]]. It is well known that a circulant matrix-vector product can be efficiently computed by [[convolution]]s. Hence we successfully reduce the discrete Fourier transform into short convolutions.  
 
== Complexity ==
 
When applied to a [[Characteristic (algebra)|characteristc]]-2 field GF(2<sup>''m''</sup>), the matrix <math>\mathbf{A}</math> is just a binary matrix. Only addition is used when calculating the matrix-vector product of <math>\mathrm{A}</math> and <math>\mathrm{L\Pi f}</math>. It has been shown that the multiplicative complexity of the cyclotomic algorithm is given by <math>O(n(\log_2n)^{\log_2\frac{3}{2}})</math>, and the additive complexity is given by <math>O(n^2/(\log_2 n)^{\log_2\frac{8}{3}})</math>.<ref name="complexityAnalysis"/>
 
== References ==
{{reflist}}
 
[[Category:Discrete transforms]]
[[Category:FFT algorithms]]

Latest revision as of 23:30, 28 September 2014

I am a property agent from DTZ, a world leader in property services. My providers embody leasing, rental, sale and purchase of commercial and residential properties. I value relationships and hope to serve you with Sincerity, Honesty and Integrity. My objective is to supply the best level of talent, service and care to each consumer during every transaction.

Since coming into this line, I've had the opportunity to work with many clients of numerous backgrounds. If you are in search of somebody to help you together with your actual estate wants, I am just a telephone call away. Contact me for a friendly, no obligation discussion. Many have finished so and are glad that they did – see their testimonials right here Try to have the tenancy agreement cut out in your company's identify, not your personal identify. This way, you are not bound should you determine to vary jobs (and the brand new employer would love to give you an even bigger house - however you possibly can't since you're personally happily sure to stay put until the lease runs out). until a minimum of one companion is a registered patent agent who has in drive a practising certificates. Your Most well-liked Agent

This document has been ready by All Property Options Singapore Pte Ltd. The knowledge including all supplies, estimates, calculations, opinions or suggestions contained in this doc have been supplied in good faith and have been based mostly on data received from sources All Property Options Singapore Pte Ltd has accepted in good faith. No guarantee is made as to the accuracy or reliability of any information contained in this document and neither All Property Options Singapore Pte Ltd nor any persons involved within the preparation of this doc settle for any type of liability for its content material.

I've met many agents. Those who stay the industries for lengthy are mostly laborious working and ethical. I personally welcome this Financial Advisory Trade Evaluate, those who have been doing good jobs for their clients will keep. Why ought to insurance brokers provide the shoppers "free recommendation" and be perceived as wealthy rogue salesmen where they themselves are struggling to feed their very own household? However wait! Is this the BEST strategy to be a Property Agent? Effectively, there's all the time one other means and I want to invite you to for a friendly dialogue and let you know extra about joining our Workforce in OrangeTee. Be part of OrangeTee – Be a part of sgDistrict Team As a way to be so registered, candidates must meet the necessities found within the PatentsAgents) Guidelines 2001

With the steam of an overheated property market dying down, the excitement has now shifted to builders enjoying low cost games to push new tasks or clear previous stock. With so many ‘great offers', consumers are spoiled for selections. In case you overprice your own home, consumers will be delay by the value. Because of this your home may take longer to promote, and find yourself being offered at a lower price. Consumers might imagine that you are determined in selling (so that you interact a number of agents to sell it off fast). Since they think you're desperate, they could not offer you an excellent price. Moreover, estate brokers are obligation-sure to keep away from another potential conflicts of interest (comparable to if a party to the transaction is related to the agent) unless the shopper's waiver is obtained.

I personally do not mind a busy property agent. Because the saying goes, "If you'd like one thing accomplished, ask a busy individual to do it". Irrespective of how busy although, a good property agent will call you again even when he misses your name. Or not less than SMS you. They are going to name you up once they see a superb alternative at hand, be it buying,promoting or renting a space. They will update you on the scenario even without you asking. 2. Have Consumer's Best Pursuits in Thoughts

Making the leap to turning into a property investor We 'assumed' we might just contact an estate agent and they would have a listing of properties for rent, if we didn't like what they'd we'd then move on to another agent and keep doing this till we located an condo that was suitable for us. Stamp responsibility is to be paid within 14 days from the date of acceptance of the OTP or Sale and Buy (S&P) Settlement. For more data, please visit www.iras.gov.sg Estaland property market singapore Associates Pte Ltd Totally different brokers could deliver the same buyer to view your own home a second time, leading to confusion. Agents who would not have unique rights to an inventory could also be less willing to work for that listing. together with Estate Brokers Act and regulation of real estate trade)

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