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A cepstrum () is the result of taking the Inverse Fourier transform (IFT) of the logarithm of the estimated spectrum of a signal. It may be pronounced in the two ways given, the second having the advantage of avoiding confusion with 'kepstrum' which also exists (see below). There is a ''complex cepstrum'', a ''real cepstrum'', a ''power cepstrum'', and a ''phase cepstrum''. The power cepstrum in particular finds applications in the analysis of human speech. The name "cepstrum" was derived by reversing the first four letters of "spectrum". Operations on cepstra are labelled ''quefrency analysis'' (aka ''quefrency alanysis''〔), ''liftering'', or ''cepstral analysis''. ==Origin and definition== The ''power cepstrum'' was defined in a 1963 paper by Bogert et al.〔B. P. Bogert, M. J. R. Healy, and J. W. Tukey: "The Quefrency of Time Series for Echoes: Cepstrum, Pseudo Autocovariance, Cross-Cepstrum and Saphe Cracking". ''Proceedings of the Symposium on Time Series Analysis'' (M. Rosenblatt, Ed) Chapter 15, 209-243. New York: Wiley, 1963.〕 The power cepstrum of a signal is defined as the squared magnitude of the inverse Fourier transform of the logarithm of the squared magnitude of the Fourier transform of a signal. :power cepstrum of signal A short-time cepstrum analysis was proposed by Schroeder and Noll for application to pitch determination of human speech.〔 A. Michael Noll and Manfred R. Schroeder, "Short-Time 'Cepstrum' Pitch Detection," (abstract) Journal of the Acoustical Society of America, Vol. 36, No. 5, p. 1030〕〔A. Michael Noll (1964), “Short-Time Spectrum and Cepstrum Techniques for Vocal-Pitch Detection,” Journal of the Acoustical Society of America, Vol. 36, No. 2, pp. 296-302.〕〔A. Michael Noll (1967), “Cepstrum Pitch Determination,” Journal of the Acoustical Society of America, Vol. 41, No. 2, pp. 293-309.〕 The ''complex cepstrum'' was defined by Oppenheim in his development of homomorphic system theory〔A. V. Oppenheim, "Superposition in a class of nonlinear systems" (Ph.D. dissertation), Res. Lab. Electronics, Massachusetts Institute of Technology, Cambridge, MA, 1965.〕 and is defined as the Inverse Fourier transform of the logarithm (with unwrapped phase) of the Fourier transform of the signal. This is sometimes called the spectrum of a spectrum. : complex cepstrum of signal ) (where is the integer required to properly unwrap the angle or imaginary part of the complex log function) The ''real cepstrum'' uses the logarithm function defined for real values. The real cepstrum is related to the power via the relationship (4 * real cepstrum)^2 = power cepstrum, and is related to the complex cepstrum as real cepstrum = 0.5 *(complex cepstrum + time reversal of complex cepstrum). The complex cepstrum uses the complex logarithm function defined for complex values. The ''phase cepstrum'' is related to the complex cepstrum as phase spectrum = (complex cepstrum - time reversal of complex cepstrum)^2. The complex cepstrum holds information about magnitude and phase of the initial spectrum, allowing the reconstruction of the signal. The real cepstrum uses only the information of the magnitude of the spectrum. Many texts define the process as FT → abs() → log → IFT, i.e., that the cepstrum is the "inverse Fourier transform of the log-magnitude Fourier spectrum". The ''kepstrum'', which stands for "Kolmogorov equation power series time response", is similar to the cepstrum and has the same relation to it as expected value has to statistical average, i.e. cepstrum is the empirically measured quantity while kepstrum is the theoretical quantity.〔 "Use of the kepstrum in signal analysis", M.T.Silvia and W.A.Robinson, Geoexploration, Volume 16, Issues 1-2, April 1978, Pages 55-73. 〕〔 "A kepstrum approach to filtering, smoothing and prediction with application to speech enhancement",T.J.Moir and J.F.Barrett,Proc Royal Society A,Vol.459,2003, pp.2957-2976 〕 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「Cepstrum」の詳細全文を読む スポンサード リンク
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