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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Circular convolution ： ウィキペディア英語版 Circular convolution The circular convolution, also known as cyclic convolution, of two aperiodic functions (i.e. Schwartz functions) occurs when one of them is convolved in the normal way with a periodic summation of the other function.  That situation arises in the context of the Circular convolution theorem.  The identical operation can also be expressed in terms of the periodic summations of both functions, if the infinite integration interval is reduced to just one period.  That situation arises in the context of the discrete-time Fourier transform (DTFT) and is also called periodic convolution.  In particular, the DTFT of the product of two discrete sequences is the periodic convolution of the DTFTs of the individual sequences.〔If a sequence, ''x''(), represents samples of a continuous function, ''x''(''t''), with Fourier transform ''X''(ƒ), its DTFT is a periodic summation of ''X''(ƒ).  (see Discrete-time_Fourier_transform#Relationship_to_sampling)〕 Let ''x'' be a function with a well-defined periodic summation, ''x''''T'', where: :$x\_T(t)\; \backslash \; \backslash stackrel\; \backslash \; \backslash sum\_^\backslash infty\; x(t\; -\; kT)\; =\; \backslash sum\_^\backslash infty\; x(t\; +\; kT).$ If ''h'' is any other function for which the convolution ''x''''T'' ∗ ''h'' exists, then the convolution ''x''''T'' ∗ ''h'' is periodic and identical to: :$$ \begin (x_T * h)(t)\quad &\stackrel \ \int_^\infty h(\tau)\cdot x_T(t - \tau)\,d\tau \\ &\equiv \int_^ h_T(\tau)\cdot x_T(t - \tau)\,d\tau, \end 〔Proof: :$\backslash int\_^\backslash infty\; h(\backslash tau)\backslash cdot\; x\_T(t\; -\; \backslash tau)\backslash ,d\backslash tau$ :::$$ \begin &= \sum_^\infty \left(h(\tau)\cdot x_T(t - \tau)\ d\tau\right ) \\ &\stackrel\ \sum_^\infty \left(h(\tau+kT)\cdot x_T(t - \tau -kT)\ d\tau\right ) \\ &= \int_^ \left(h(\tau+kT)\cdot \underbrace_^ \underbrace^\infty"> h(\tau+kT)\right )}_ \ h_T(\tau)}\cdot x_T(t - \tau)\ d\tau \quad \quad \scriptstyle \end 〕 where ''t''o is an arbitrary parameter and ''h''''T'' is a periodic summation of ''h''. The second integral is called the periodic convolution〔Jeruchim 2000, pp 73-74.〕〔Udayashankara 2010, p 189.〕 of functions ''x''''T'' and ''h''''T'' and is sometimes normalized by 1/''T''.〔Oppenheim, pp 388-389〕 When ''x''''T'' is expressed as the periodic summation of another function, ''x'', the same operation may also be referred to as a circular convolution〔〔Priemer 1991, pp 286-289.〕 of functions ''h'' and ''x''. == Discrete sequences == Similarly, for discrete sequences and period N, we can write the circular convolution of functions ''h'' and ''x'' as: :$$ \begin (x_N * h)() \ &\stackrel \ \sum_^\infty h() \cdot x_N() \\ &= \sum_^\infty \left( h() \cdot \sum_^\infty x(-m -kN ) \right). \end For the special case that the non-zero extent of both ''x'' and ''h'' are ''≤ N'', this is reducible to matrix multiplication where the kernel of the integral transform is a circulant matrix. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Circular convolution」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.