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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 Bilinear transform ： ウィキペディア英語版 Bilinear transform The bilinear transform (also known as Tustin's method) is used in digital signal processing and discrete-time control theory to transform continuous-time system representations to discrete-time and vice versa. The bilinear transform is a special case of a conformal mapping (namely, the Möbius transformation), often used to convert a transfer function $H\_a(s)\; \backslash$ of a linear, time-invariant (LTI) filter in the continuous-time domain (often called an analog filter) to a transfer function $H\_d(z)\; \backslash$ of a linear, shift-invariant filter in the discrete-time domain (often called a digital filter although there are analog filters constructed with switched capacitors that are discrete-time filters). It maps positions on the $j\; \backslash omega\; \backslash$ axis, $Re()=0\; \backslash$, in the s-plane to the unit circle, $|z|\; =\; 1\; \backslash$, in the z-plane. Other bilinear transforms can be used to warp the frequency response of any discrete-time linear system (for example to approximate the non-linear frequency resolution of the human auditory system) and are implementable in the discrete domain by replacing a system's unit delays $\backslash left(\; z^\; \backslash right)\; \backslash$ with first order all-pass filters. The transform preserves stability and maps every point of the frequency response of the continuous-time filter, $H\_a(j\; \backslash omega\_a)\; \backslash$ to a corresponding point in the frequency response of the discrete-time filter, $H\_d(e^)\; \backslash$ although to a somewhat different frequency, as shown in the Frequency warping section below. This means that for every feature that one sees in the frequency response of the analog filter, there is a corresponding feature, with identical gain and phase shift, in the frequency response of the digital filter but, perhaps, at a somewhat different frequency. This is barely noticeable at low frequencies but is quite evident at frequencies close to the Nyquist frequency. == Discrete-time approximation == The bilinear transform is a first-order approximation of the natural logarithm function that is an exact mapping of the ''z''-plane to the ''s''-plane. When the Laplace transform is performed on a discrete-time signal (with each element of the discrete-time sequence attached to a correspondingly delayed unit impulse), the result is precisely the Z transform of the discrete-time sequence with the substitution of :$$ \begin z &= e^ \\ &= \frac} \\ &\approx \frac \end where $T\; \backslash$ is the numerical integration step size of the trapezoidal rule used in the bilinear transform derivation; or, in other words, the sampling period. The above bilinear approximation can be solved for $s\; \backslash$ or a similar approximation for $s\; =\; (1/T)\; \backslash ln(z)\; \backslash \; \backslash$ can be performed. The inverse of this mapping (and its first-order bilinear approximation) is :$$ \begin s &= \frac \ln(z) \\ &= \frac \left(+ \frac \left( \frac \right)^3 + \frac \left( \frac \right)^5 + \frac \left( \frac \right)^7 + \cdots \right ) \\ &\approx \frac \frac \\ &= \frac \frac} \end The bilinear transform essentially uses this first order approximation and substitutes into the continuous-time transfer function, $H\_a(s)\; \backslash$ :$s\; \backslash leftarrow\; \backslash frac\; \backslash frac.$ That is :$H\_d(z)\; =\; H\_a(s)\; \backslash bigg|\_\; \backslash frac\}=\; H\_a\; \backslash left(\; \backslash frac\; \backslash frac\; \backslash right).\; \backslash$ 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「Bilinear transform」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.