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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 Similarities between Wiener and LMS ： ウィキペディア英語版 Similarities between Wiener and LMS The Least mean squares filter solution converges to the Wiener filter solution, assuming that the unknown system is LTI and the noise is stationary. Both filters can be used to identify the impulse response of an unknown system, knowing only the original input signal and the output of the unknown system. By relaxing the error criterion to reduce current sample error instead of minimizing the total error over all of n, the LMS algorithm can be derived from the Wiener filter. == Derivation of the Wiener filter for system identification == Given a known input signal $s()$, the output of an unknown LTI system $x()$ can be expressed as: $x()\; =\; \backslash sum\_^\; h\_ks()\; +\; w()$ where $h\_k$ is an unknown filter tap coefficients and $w()$ is noise. The model system $\backslash hat()$, using a Wiener filter solution with an order N, can be expressed as: $\backslash hat()\; =\; \backslash sum\_^\backslash hat\_ks()$ where $\backslash hat\_k$ are the filter tap coefficients to be determined. The error between the model and the unknown system can be expressed as: $e()\; =\; x()\; -\; \backslash hat()$ The total squared error $E$ can be expressed as: $E\; =\; \backslash sum\_^e()^2$ $E\; =\; \backslash sum\_^(x()\; -\; \backslash hat())^2$ $E\; =\; \backslash sum\_^(x()^2\; -\; 2x()\backslash hat()\; +\; \backslash hat()^2)$ Use the Minimum mean-square error criterion over all of $n$ by setting its gradient to zero: $\backslash nabla\; E\; =\; 0$ which is $\backslash frac\; =\; 0$ for all $i\; =\; 0,\; 1,\; 2,\; ...,\; N-1$ $\backslash frac\; =\; \backslash frac\; \backslash sum\_^\backslash hat()\; +\; \backslash hat()^2\; ]$ Substitute the definition of $\backslash hat()$: $\backslash frac\; =\; \backslash frac\; \backslash sum\_^\backslash sum\_^\backslash hat\_ks()\; +\; (\backslash sum\_^\backslash hat\_ks())^2\; ]$ Distribute the partial derivative: $\backslash frac\; =\; \backslash sum\_^\; +\; 2(\backslash sum\_^\backslash hat\_ks())s()\; ]$ Using the definition of discrete cross-correlation: $R\_(i)\; =\; \backslash sum\_^\; x()y()$ $\backslash frac\; =\; -2R\_()\; +\; 2\backslash sum\_^\backslash hat\_kR\_(-\; k)\; =\; 0$ Rearrange the terms: $R\_()\; =\; \backslash sum\_^\backslash hat\_kR\_(-\; k)$ for all $i\; =\; 0,\; 1,\; 2,\; ...,\; N-1$ This system of N equations with N unknowns can be determined. 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「Similarities between Wiener and LMS」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.