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In applied mathematics, the starred transform, or star transform, is a discrete-time variation of the Laplace transform, so-named because of the asterisk or "star" in the customary notation of the sampled signals. The transform is an operator of a continuous-time function , which is transformed to a function in the following manner:〔Jury, Eliahu I. ''Analysis and Synthesis of Sampled-Data Control Systems''., Transactions of the American Institute of Electrical Engineers- Part I: Communication and Electronics, 73.4, 1954, p. 332-346.〕 : where is a Dirac comb function, with period of time T. The starred transform is a convenient mathematical abstraction that represents the Laplace transform of an ''impulse sampled'' function , which is the output of an ''ideal sampler'', whose input is a continuous function, . The starred transform is similar to the Z transform, with a simple change of variables, where the starred transform is explicitly declared in terms of the sampling period (T), while the Z transform is performed on a discrete signal and is independent of the sampling period. This makes the starred transform a de-normalized version of the one-sided Z-transform, as it restores the dependence on sampling parameter ''T''. == Relation to Laplace transform == Since , where: : Then per the convolution theorem, the starred transform is equivalent to the complex convolution of and }}, hence:〔 : This line integration is equivalent to integration in the positive sense along a closed contour formed by such a line and an infinite semicircle that encloses the poles of X(s) in the left half-plane of ''p''. The result of such an integration (per the residue theorem) would be: : ■ウィキペディアで「Starred transform」の詳細全文を読む スポンサード リンク
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