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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 Constant Q transform ： ウィキペディア英語版 Constant Q transform In mathematics and signal processing, the Constant Q Transform transforms a data series to the frequency domain. It is related to the Fourier Transform,〔Judith C. Brown, (Calculation of a constant Q spectral transform ), ''J. Acoust. Soc. Am.'', 89(1):425–434, 1991.〕 and very closely related to the complex Morlet wavelet transform.〔(Continuous Wavelet Transform ) "When the mother wavelet can be interpreted as a windowed sinusoid (such as the Morlet wavelet), the wavelet transform can be interpreted as a constant-Q Fourier transform. Before the theory of wavelets, constant-Q Fourier transforms (such as obtained from a classic third-octave filter bank) were not easy to invert, because the basis signals were not orthogonal."〕 The transform can be thought of as a series of logarithmically spaced filters, with the ''k''-th filter having a spectral width some multiple of the previous filter's width, i.e. $\backslash begin$ \delta f_k &= 2^ } * \delta f_ \\ &= \left ( }} \right )^ * \delta f_ where δ''fk'' is the bandwidth of the kth filter, ''f''min is the centre frequency of the lowest filter, and ''n'' is the number of filters per octave. == Calculation of the Transform == The short-time Fourier Transform of ''x''() for a frame shifted to sample ''m'' is calculated as follows: :$X()\; =\; \backslash sum\_^\; W()\; x()\; e^\; \}$ Given a data series, sampled at ''fs'' = 1/''T'', ''T'' being the sampling period of our data, for each frequency bin we can define the following: * Filter width, δ''fk'' * ''Q'', the "quality factor". This is shown below to be the integer number of cycles processed at a center frequency ''fk''. As such, this somewhat defines the time complexity of the transform. ::$Q\; =\; \backslash frac$ * Window length for the ''k''-th bin ::$N()\; =\; \backslash left(\; \backslash frac\; \backslash right)\; =\; \backslash left(\; \backslash frac\; \backslash right)\; Q$ :As ''S''/''fk'' is the number of samples processed per cycle at frequency ''fk'', ''Q'' is the number of integer cycles processed at this center frequency. The equivalent transform kernel can be found by using the following substitutions: * The window length of each bin is now a function of the bin number: ::$N\; =\; N()\; =\; Q\; \backslash frac$ * The relative power of each bin will decrease with higher frequencies, as these sum over fewer terms. To compensate for this, we normalize by ''N''(). * Any windowing function will be a function of window length, and likewise a function of window number. For example, the equivalent Hamming window would be, ::$W()\; =\; \backslash alpha\; -\; \backslash left(1\; -\; \backslash alpha\; \backslash right)\; \backslash cos\; \backslash left(\; \backslash frac\; \backslash right),\; \backslash alpha\; =\; 25/46\; ,\; 0\; \backslash leqslant\; n\; \backslash leqslant\; N()\; -\; 1$ * Our digital frequency, $\backslash frac$, becomes $\backslash frac$ After these modifications, we are left with: :$X()\; =\; \backslash frac\; \backslash sum\_^\; W()\; x()\; e^\}$ 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「Constant Q 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.