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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 Continuous wavelet transform ： ウィキペディア英語版 Continuous wavelet transform In mathematics, a continuous wavelet transform (CWT) is used to divide a continuous-time function into wavelets. Unlike Fourier transform, the continuous wavelet transform possesses the ability to construct a time-frequency representation of a signal that offers very good time and frequency localization. The continuous wavelet transform of a function $x(t)$ at a scale (a>0) $a\backslash in\backslash mathbb$ is expressed by the following integral :$X\_w(a,b)=\backslash frac^\; x(t)\backslash overline\backslash psi\backslash left(\backslash frac\backslash right)\backslash ,\; dt$ where $\backslash psi(t)$ is a continuous function in both the time domain and the frequency domain called the mother wavelet and the overline represents operation of complex conjugate. The main purpose of the mother wavelet is to provide a source function to generate the daughter wavelets which are simply the translated and scaled versions of the mother wavelet. To recover the original signal $x(t)$, the first inverse continuous wavelet transform can be exploited. :$x(t)=C\_\backslash psi^\backslash int\_^\backslash int\_^\; X\_w(a,b)\backslash frac\backslash right)\backslash ,\; db\backslash \; \backslash frac$ $\backslash tilde\backslash psi(t)$ is the dual function of $\backslash psi(t)$ and :$C\_\backslash psi=\backslash int\_^\backslash frac(\backslash omega)\}\backslash ,\; d\backslash omega$ is admissible constant, where hat means Fourier transform operator. Sometimes, $\backslash tilde\backslash psi(t)=\backslash psi(t)$, then the admissible constant becomes :$C\_\backslash psi\; =\; \backslash int\_^$ \frac d\omega Traditionally, this constant is called wavelet admissible constant. A wavelet whose admissible constant satisfies : is called an admissible wavelet. An admissible wavelet implies that $\backslash hat(0)\; =\; 0$, so that an admissible wavelet must integrate to zero. To recover the original signal $x(t)$, the second inverse continuous wavelet transform can be exploited. :$x(t)=\backslash frac\backslash int\_^\backslash int\_^\; \backslash fracX\_w(a,b)\backslash exp\backslash left(i\backslash frac\backslash right)\backslash ,\; db\backslash \; da$ This inverse transform suggests that a wavelet should be defined as :$\backslash psi(t)=w(t)\backslash exp(it)$ where $w(t)$ is a window. Such defined wavelet can be called as an analyzing wavelet, because it admits to time-frequency analysis. An analyzing wavelet is unnecessary to be admissible. ==Scale factor== The scale factor $a$ either dilates or compresses a signal. When the scale factor is relatively low, the signal is more contracted which in turn results in a more detailed resulting graph. However, the drawback is that low scale factor does not last for the entire duration of the signal. On the other hand, when the scale factor is high, the signal is stretched out which means that the resulting graph will be presented in less detail. Nevertheless, it usually lasts the entire duration of the signal. 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「Continuous wavelet 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.