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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 Wigner distribution function ： ウィキペディア英語版 Wigner distribution function The Wigner distribution function (WDF) is used in signal processing as a transform in time-frequency analysis. The WDF was first proposed in physics to account for quantum corrections to classical statistical mechanics in 1932 by Eugene Wigner, and it is of importance in quantum mechanics in phase space (see, by way of comparison: ''Wigner quasi-probability distribution'', also called the ''Wigner function'' or the ''Wigner–Ville distribution''). Given the shared algebraic structure between position-momentum and time-frequency conjugate pairs, it also usefully serves in signal processing, as a transform in time-frequency analysis, the subject of this article. Compared to a short-time Fourier transform, such as the Gabor transform, the Wigner distribution function can furnish higher clarity in some cases. == Mathematical definition == There are several different definitions for the Wigner distribution function. The definition given here is specific to time-frequency analysis. Given the time series $x()$, its non-stationary autocorrelation function is given by : $C\_x(t\_1,\; t\_2)\; =\; \backslash left\backslash langle\; \backslash left(x()\; -\; \backslash mu()\backslash right)\; \backslash left(x()\; -\; \backslash mu()\backslash right)^$ * \right\rangle , where $\backslash langle\; \backslash cdots\; \backslash rangle$ denotes the average over all possible realizations of the process and $\backslash mu(t)$ is the mean, which may or may not be a function of time. The Wigner function $W\_x(t,f)$ is then given by first expressing the autocorrelation function in terms of the average time $t\; =\; (t\_1+t\_2)/2$ and time lag $\backslash tau\; =\; t\_1\; -\; t\_2$, and then Fourier transforming the lag. : $W\_x(t,f)=\backslash int\_^\; C\_x\backslash left(t\; +\; \backslash frac,\; t\; -\; \backslash frac\backslash right)\; \backslash ,\; e^\; \backslash ,\; d\backslash tau\; .$ So for a single (mean-zero) time series, the Wigner function is simply given by : $W\_x(t,f)=\backslash int\_^x\backslash left(t\; +\; \backslash frac\backslash right)\; \backslash ,\; x\backslash left(t\; -\; \backslash frac\backslash right)^$ * \, e^\,d\tau . The motivation for the Wigner function is that it reduces to the spectral density function at all times $t$ for stationary processes, yet it is fully equivalent to the non-stationary autocorrelation function. Therefore, the Wigner function tells us (roughly) how the spectral density changes in time 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「Wigner distribution function」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.